SI Base Unit of Length
1 m (metre)
The SI base unit of length is metre
Dimensional Formula of Velocity
[v] = [L T⁻¹]
Velocity has dimensions of length per time
Dimensional Formula of Force
[F] = [M L T⁻²]
Force has dimensions of mass times acceleration
Dimensional Formula of Energy
[E] = [M L² T⁻²]
Energy has dimensions of mass times velocity squared
Percentage Error
% error = (Δx/x) × 100
Percentage error in measurement
Variables: Δx = absolute error, x = measured value
Error in Sum
Δ(A + B) = ΔA + ΔB
Absolute error in sum is sum of absolute errors
Relative Error in Product
Δ(AB)/(AB) = ΔA/A + ΔB/B
Relative error in product is sum of relative errors
Light Year
1 ly = 9.46 × 10¹⁵ m
Distance light travels in one year
Average Velocity
v̄ = Δx/Δt
Average velocity is displacement per unit time
Variables: Δx = displacement, Δt = time interval
Instantaneous Velocity
v = dx/dt
Velocity at a specific instant
Average Acceleration
ā = Δv/Δt
Average acceleration is change in velocity per unit time
Variables: Δv = change in velocity, Δt = time interval
First Equation of Motion
v = u + at
Velocity after time t with constant acceleration
Variables: v = final velocity, u = initial velocity, a = acceleration, t = time
Second Equation of Motion
s = ut + ½at²
Displacement with constant acceleration
Variables: s = displacement, u = initial velocity, a = acceleration, t = time
Third Equation of Motion
v² = u² + 2as
Velocity-displacement relation
Variables: v = final velocity, u = initial velocity, a = acceleration, s = displacement
Average Velocity for Uniform Acceleration
v̄ = (u + v)/2
Average of initial and final velocities
Displacement in nth Second
sₙ = u + a(n - ½)
Distance traveled in the nth second
Free Fall Velocity
v = gt
Velocity of freely falling body after time t
Variables: g = acceleration due to gravity ≈ 9.8 m/s²
Height in Free Fall
h = ½gt²
Height fallen under gravity
Horizontal Range
R = (u² sin 2θ)/g
Horizontal range of projectile
Variables: u = initial velocity, θ = angle of projection, g = acceleration due to gravity
Maximum Height of Projectile
H = (u² sin² θ)/(2g)
Maximum height reached by projectile
Time of Flight
T = (2u sin θ)/g
Total time projectile remains in air
Trajectory Equation
y = x tan θ - (gx²)/(2u² cos² θ)
Path followed by projectile
Angular Velocity
ω = dθ/dt = 2π/T
Rate of change of angular displacement
Variables: θ = angular displacement, T = time period
Linear and Angular Velocity Relation
v = rω
Linear velocity in circular motion
Variables: r = radius, ω = angular velocity
Centripetal Acceleration
aᶜ = v²/r = rω²
Acceleration directed towards center
Centripetal Force
Fᶜ = mv²/r = mrω²
Force required for circular motion
Relative Velocity
v⃗ₐᵦ = v⃗ₐ - v⃗ᵦ
Velocity of A relative to B
River Crossing Time
t = d/vᵣ
Minimum time to cross river of width d
Variables: vᵣ = velocity of boat in still water perpendicular to flow
Newton's Second Law
F⃗ = ma⃗ = dp⃗/dt
Force equals mass times acceleration or rate of change of momentum
Variables: F = force, m = mass, a = acceleration, p = momentum
Momentum
p⃗ = mv⃗
Linear momentum is mass times velocity
Impulse
J⃗ = F⃗Δt = Δp⃗
Impulse equals change in momentum
Newton's Third Law
F⃗₁₂ = -F⃗₂₁
Action and reaction forces are equal and opposite
Static Friction
fₛ ≤ μₛN
Maximum static friction
Variables: μₛ = coefficient of static friction, N = normal force
Kinetic Friction
fₖ = μₖN
Kinetic friction force
Variables: μₖ = coefficient of kinetic friction, N = normal force
Acceleration on Smooth Incline
a = g sin θ
Acceleration of object on frictionless incline
Acceleration on Rough Incline
a = g(sin θ - μₖ cos θ)
Acceleration with friction on incline
Banking Angle
tan θ = v²/(rg)
Angle of banking for no friction at speed v
Tension in Pulley System
T = 2m₁m₂g/(m₁ + m₂)
Tension in Atwood machine
Acceleration in Atwood Machine
a = (m₁ - m₂)g/(m₁ + m₂)
Acceleration of masses in Atwood machine
Pseudo Force
F̄ₚ = -mā₀
Pseudo force in non-inertial frame
Variables: a₀ = acceleration of frame
Rocket Equation
F = v dm/dt
Thrust force on rocket
Variables: v = exhaust velocity, dm/dt = rate of mass ejection
Conservation of Momentum
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Total momentum before equals total momentum after collision
Coefficient of Restitution
e = (v₂ - v₁)/(u₁ - u₂)
Ratio of relative velocity after to before collision
Variables: e = 1 for elastic, e = 0 for inelastic
Work Done
W = F⃗·s⃗ = Fs cos θ
Work is dot product of force and displacement
Work-Energy Theorem
W = ΔKE = KEf - KEi
Net work equals change in kinetic energy
Kinetic Energy
KE = ½mv²
Energy due to motion
Gravitational Potential Energy
PE = mgh
Potential energy near Earth's surface
Variables: h = height above reference level
Elastic Potential Energy
PE = ½kx²
Potential energy in spring
Variables: k = spring constant, x = compression/extension
Conservation of Mechanical Energy
KE + PE = constant
Total mechanical energy is conserved in absence of friction
Average Power
P̄ = W/t
Work done per unit time
Instantaneous Power
P = F⃗·v⃗ = Fv cos θ
Power at any instant
Elastic Collision Velocity (1D)
v₁ = ((m₁-m₂)u₁ + 2m₂u₂)/(m₁+m₂)
Final velocity of first body in elastic collision
Loss in KE in Inelastic Collision
ΔKE = ½(m₁m₂/(m₁+m₂))(u₁-u₂)²
Kinetic energy lost in perfectly inelastic collision
Work by Variable Force
W = ∫F·ds
Work done by variable force
Kinetic Energy in Terms of Momentum
KE = p²/(2m)
Alternative form of kinetic energy
Efficiency
η = (Output/Input) × 100%
Percentage of input energy converted to useful output
Work Done in Stretching Spring
W = ½k(x₂² - x₁²)
Work to stretch spring from x₁ to x₂
Potential Energy Function
F = -dU/dx
Force is negative gradient of potential energy
Angular Displacement
θ = s/r
Angular displacement in radians
Variables: s = arc length, r = radius
Angular Velocity
ω = dθ/dt
Rate of change of angular displacement
Angular Acceleration
α = dω/dt = d²θ/dt²
Rate of change of angular velocity
First Equation of Rotational Motion
ω = ω₀ + αt
Angular velocity with constant angular acceleration
Second Equation of Rotational Motion
θ = ω₀t + ½αt²
Angular displacement with constant angular acceleration
Third Equation of Rotational Motion
ω² = ω₀² + 2αθ
Angular velocity-displacement relation
Torque
τ⃗ = r⃗ × F⃗ = rF sin θ
Torque is moment of force
Moment of Inertia Definition
I = Σmᵢrᵢ² = ∫r²dm
Rotational inertia
MI of Ring about Axis
I = MR²
Moment of inertia of ring about perpendicular axis through center
MI of Disc about Axis
I = ½MR²
Moment of inertia of disc about perpendicular axis through center
MI of Solid Sphere
I = (2/5)MR²
Moment of inertia of solid sphere about diameter
MI of Hollow Sphere
I = (2/3)MR²
Moment of inertia of hollow sphere about diameter
MI of Rod about Center
I = ML²/12
Moment of inertia of rod about perpendicular axis through center
MI of Rod about End
I = ML²/3
Moment of inertia of rod about perpendicular axis through end
Parallel Axis Theorem
I = Iᴄᴍ + Md²
MI about parallel axis at distance d from center of mass
Perpendicular Axis Theorem
Iᴢ = Iₓ + Iʏ
For planar bodies, MI about perpendicular axis equals sum of MIs about two perpendicular axes in plane
Torque-Angular Acceleration Relation
τ = Iα
Rotational analogue of Newton's second law
Angular Momentum
L⃗ = r⃗ × p⃗ = Iω⃗
Angular momentum for rotating body
Relation between Torque and Angular Momentum
τ⃗ = dL⃗/dt
Torque is rate of change of angular momentum
Rotational Kinetic Energy
KE = ½Iω²
Kinetic energy of rotating body
Total KE of Rolling Body
KE = ½Mv² + ½Iω²
Sum of translational and rotational kinetic energies
Condition for Pure Rolling
v = rω
Relation between linear and angular velocities in pure rolling
Acceleration of Rolling Body on Incline
a = g sin θ/(1 + I/(MR²))
Linear acceleration of body rolling down incline
Conservation of Angular Momentum
L = Iω = constant
Angular momentum conserved when net external torque is zero
Radius of Gyration
k = √(I/M)
Distance from axis where entire mass can be assumed concentrated
Universal Law of Gravitation
F = Gm₁m₂/r²
Gravitational force between two masses
Variables: G = 6.67 × 10⁻¹¹ N·m²/kg²
Gravitational Field Intensity
g = GM/r²
Gravitational field at distance r from mass M
Acceleration due to Gravity at Surface
g = GM/R²
Gravitational acceleration at Earth's surface
Variables: R = radius of Earth
Variation of g with Height
gₕ = g(1 - 2h/R)
Acceleration due to gravity at height h above surface (h << R)
Variation of g with Depth
gₐ = g(1 - d/R)
Acceleration due to gravity at depth d below surface
Gravitational Potential Energy
U = -GMm/r
Gravitational PE at distance r from center (taking U(∞) = 0)
Gravitational Potential
V = -GM/r
Gravitational potential at distance r
Orbital Velocity
vₒ = √(GM/r) = √(gr)
Velocity required for circular orbit at distance r from center
Orbital Velocity at Surface
vₒ = √(gR) ≈ 7.9 km/s
Orbital velocity at Earth's surface
Escape Velocity
vₑ = √(2GM/R) = √(2gR)
Minimum velocity to escape gravitational field
Relation between Escape and Orbital Velocity
vₑ = √2 vₒ
Escape velocity is √2 times orbital velocity
Time Period of Satellite
T = 2π√(r³/GM)
Period of satellite in circular orbit
Total Energy of Satellite
E = -GMm/(2r)
Total mechanical energy of satellite in orbit
Kepler's Third Law
T² ∝ r³
Square of period proportional to cube of semi-major axis
Binding Energy of Satellite
BE = GMm/(2r)
Energy required to remove satellite from orbit to infinity
Stress
Stress = F/A
Force per unit area
Strain
Strain = ΔL/L
Fractional change in dimension
Young's Modulus
Y = (F/A)/(ΔL/L) = Stress/Strain
Elastic modulus for longitudinal stress
Bulk Modulus
B = -ΔP/(ΔV/V)
Elastic modulus for volume stress
Shear Modulus
G = (F/A)/θ
Elastic modulus for shearing stress
Variables: θ = shear angle
Poisson's Ratio
σ = -lateral strain/longitudinal strain
Ratio of lateral to longitudinal strain
Energy Stored in Stretched Wire
U = ½ × Stress × Strain × Volume
Elastic potential energy in deformed body
Pressure
P = F/A
Force per unit area in fluid
Pressure at Depth
P = P₀ + ρgh
Pressure at depth h in fluid
Variables: P₀ = atmospheric pressure, ρ = density
Pascal's Law
P₁ = P₂
Pressure applied to enclosed fluid is transmitted undiminished
Buoyant Force
Fᵦ = ρVg
Upward force exerted by fluid (Archimedes' principle)
Variables: V = volume of displaced fluid
Condition for Floatation
Weight = Buoyant force
Body floats when its weight equals buoyant force
Equation of Continuity
A₁v₁ = A₂v₂
For incompressible fluid, product of area and velocity is constant
Bernoulli's Equation
P + ½ρv² + ρgh = constant
Energy conservation for ideal fluid flow
Torricelli's Theorem
v = √(2gh)
Velocity of efflux from hole at depth h
Stokes' Law
F = 6πηrv
Viscous drag force on sphere moving through fluid
Variables: η = coefficient of viscosity, r = radius, v = velocity
Terminal Velocity
vₜ = (2r²g(ρ-σ))/(9η)
Maximum velocity of falling sphere in viscous medium
Variables: ρ = density of sphere, σ = density of medium
Surface Tension
T = F/L
Force per unit length on liquid surface
Excess Pressure in Bubble
ΔP = 4T/r
Excess pressure inside soap bubble
Capillary Rise
h = (2T cos θ)/(rρg)
Height of liquid rise in capillary tube
Variables: θ = contact angle
Celsius-Fahrenheit Relation
F = (9/5)C + 32
Conversion between Celsius and Fahrenheit scales
Celsius-Kelvin Relation
K = C + 273.15
Conversion between Celsius and Kelvin scales
Linear Expansion
ΔL = αL₀ΔT
Change in length due to temperature change
Variables: α = coefficient of linear expansion
Volume Expansion
ΔV = γV₀ΔT
Change in volume due to temperature change
Variables: γ = coefficient of volume expansion = 3α
Heat Capacity
Q = mcΔT
Heat required to change temperature
Variables: c = specific heat capacity
Latent Heat
Q = mL
Heat required for phase change
Variables: L = latent heat of fusion/vaporization
Conduction Rate
Q/t = kA(T₁-T₂)/d
Rate of heat transfer by conduction
Variables: k = thermal conductivity, A = area, d = thickness
First Law of Thermodynamics
ΔQ = ΔU + ΔW
Heat supplied equals change in internal energy plus work done
Work Done by Gas
W = PΔV
Work done in constant pressure process
Isothermal Process
PV = constant, ΔU = 0
Process at constant temperature
Adiabatic Process
PVᵞ = constant
Process with no heat exchange
Variables: γ = Cₚ/Cᵥ
Carnot Engine Efficiency
η = 1 - T₂/T₁ = (T₁-T₂)/T₁
Maximum possible efficiency between two temperatures
Variables: T₁ = source temperature, T₂ = sink temperature (in Kelvin)
Heat Engine Efficiency
η = W/Q₁ = 1 - Q₂/Q₁
Ratio of work output to heat input
Coefficient of Performance
COP = Q₂/W = Q₂/(Q₁-Q₂)
Efficiency of refrigerator
Entropy Change
ΔS = Q/T
Change in entropy for reversible process
Ideal Gas Equation
PV = nRT
Equation of state for ideal gas
Variables: R = 8.314 J/(mol·K)
Ideal Gas Equation (Alternative Form)
PV = NkᵦT
Using Boltzmann constant
Variables: kᵦ = 1.38 × 10⁻²³ J/K, N = number of molecules
RMS Speed
vᵣₘₛ = √(3RT/M) = √(3kᵦT/m)
Root mean square speed of gas molecules
Variables: M = molar mass, m = molecular mass
Average Speed
v̄ = √(8RT/πM) = √(8kᵦT/πm)
Mean speed of gas molecules
Most Probable Speed
vₚ = √(2RT/M) = √(2kᵦT/m)
Most likely speed of gas molecules
Kinetic Energy per Molecule
KE = (3/2)kᵦT
Average translational kinetic energy of gas molecule
Internal Energy of Ideal Gas
U = (f/2)nRT
Internal energy depends on degrees of freedom
Variables: f = degrees of freedom
Mayer's Relation
Cₚ - Cᵥ = R
Relation between specific heats
Heat Capacity Ratio
γ = Cₚ/Cᵥ = 1 + 2/f
Ratio of specific heats
Kinetic Theory Pressure
P = (1/3)ρv²ᵣₘₛ
Pressure from molecular motion
Variables: ρ = density
Displacement in SHM
x = A sin(ωt + φ)
Displacement as function of time
Variables: A = amplitude, ω = angular frequency, φ = phase constant
Velocity in SHM
v = Aω cos(ωt + φ)
Velocity in simple harmonic motion
Acceleration in SHM
a = -Aω² sin(ωt + φ) = -ω²x
Acceleration proportional to negative displacement
Time Period and Frequency
T = 2π/ω, f = 1/T = ω/(2π)
Relations between period, frequency, and angular frequency
Maximum Velocity in SHM
vₘₐₓ = Aω
Maximum velocity at equilibrium position
Maximum Acceleration in SHM
aₘₐₓ = Aω²
Maximum acceleration at extreme positions
Kinetic Energy in SHM
KE = ½mω²(A² - x²)
Kinetic energy at displacement x
Potential Energy in SHM
PE = ½mω²x²
Potential energy at displacement x
Total Energy in SHM
E = ½mω²A² = ½kA²
Total mechanical energy is constant
Time Period of Spring-Mass System
T = 2π√(m/k)
Period of mass attached to spring
Variables: k = spring constant
Time Period of Simple Pendulum
T = 2π√(L/g)
Period of simple pendulum for small angles
Variables: L = length of pendulum
Time Period of Physical Pendulum
T = 2π√(I/(mgd))
Period of rigid body pendulum
Variables: I = moment of inertia, d = distance of CM from pivot
Damped Harmonic Motion
x = A₀e⁻ᵇᵗ sin(ωt + φ)
Displacement in damped oscillation
Variables: b = damping constant
Resonance Condition
ωₐᵣᵢᵥᵢₙ₉ = ω₀
Maximum amplitude when driving frequency equals natural frequency
Springs in Series
1/kₑ = 1/k₁ + 1/k₂
Effective spring constant for series combination
Wave Velocity
v = fλ = ω/k
Wave speed in terms of frequency and wavelength
Variables: λ = wavelength, k = wave number
Wave Equation
y = A sin(kx - ωt)
Displacement of wave traveling in +x direction
Wave Number
k = 2π/λ
Spatial frequency of wave
Speed of Wave on String
v = √(T/μ)
Wave velocity on stretched string
Variables: T = tension, μ = linear mass density
Speed of Sound in Gas
v = √(γRT/M) = √(γP/ρ)
Sound velocity in ideal gas
Speed of Sound in Solid
v = √(Y/ρ)
Sound velocity in solid rod
Variables: Y = Young's modulus
Intensity of Wave
I = P/A = ½ρω²A²v
Power per unit area
Intensity Level
β = 10 log₁₀(I/I₀) dB
Sound intensity in decibels
Variables: I₀ = 10⁻¹² W/m² (threshold of hearing)
Frequency of String (Both Ends Fixed)
fₙ = (n/2L)√(T/μ), n = 1,2,3...
Natural frequencies of vibrating string
Open Organ Pipe Frequencies
fₙ = nv/(2L), n = 1,2,3...
Frequencies for pipe open at both ends
Closed Organ Pipe Frequencies
fₙ = nv/(4L), n = 1,3,5...
Frequencies for pipe closed at one end (only odd harmonics)
Doppler Effect (General)
f' = f(v + vₒ)/(v - vₛ)
Apparent frequency when source and observer are moving
Variables: vₒ = observer velocity (+ towards), vₛ = source velocity (+ away)
Doppler Effect (Moving Observer)
f' = f(1 + vₒ/v)
Frequency when observer moves toward stationary source
Beat Frequency
fᵦ = |f₁ - f₂|
Number of beats per second
Constructive Interference
Δx = nλ
Path difference for constructive interference
Coulomb's Law
F = kq₁q₂/r² = (1/4πε₀)(q₁q₂/r²)
Force between two point charges
Variables: k = 9 × 10⁹ N·m²/C², ε₀ = 8.85 × 10⁻¹² C²/(N·m²)
Electric Field
E⃗ = F⃗/q = kQ/r²
Electric field due to point charge Q
Electric Field due to Dipole (Axial)
E = (1/4πε₀)(2p/r³)
Field on axis of dipole at distance r (r >> a)
Variables: p = dipole moment = q × 2a
Electric Field due to Dipole (Equatorial)
E = (1/4πε₀)(p/r³)
Field on equatorial line of dipole
Gauss's Law
∮E⃗·dA⃗ = qₑₙc/ε₀
Electric flux through closed surface
Electric Field due to Infinite Sheet
E = σ/(2ε₀)
Field near infinite charged sheet
Variables: σ = surface charge density
Electric Field due to Infinite Line
E = λ/(2πε₀r)
Field at distance r from infinite line charge
Variables: λ = linear charge density
Electric Potential
V = kQ/r = W/q
Electric potential due to point charge
Potential Difference
V = -∫E⃗·dl⃗
Work done per unit charge
Relation between Field and Potential
E = -dV/dr
Electric field is negative gradient of potential
Potential Energy of Two Charges
U = kq₁q₂/r
Electric potential energy of charge configuration
Capacitance
C = Q/V
Capacitance relates charge to potential
Parallel Plate Capacitor
C = ε₀A/d
Capacitance of parallel plate capacitor
Variables: A = area of plates, d = separation
Capacitor with Dielectric
C = Kε₀A/d
Capacitance with dielectric of constant K
Energy Stored in Capacitor
U = ½CV² = ½Q²/C = ½QV
Electrostatic energy in capacitor
Energy Density in Electric Field
u = ½ε₀E²
Energy per unit volume
Capacitors in Series
1/Cₑ = 1/C₁ + 1/C₂ + ...
Effective capacitance in series
Capacitors in Parallel
Cₑ = C₁ + C₂ + ...
Effective capacitance in parallel
Torque on Dipole
τ = pE sin θ
Torque on electric dipole in uniform field
Potential Energy of Dipole
U = -p⃗·E⃗ = -pE cos θ
Potential energy of dipole in electric field
Electric Current
I = Q/t = dQ/dt
Rate of flow of charge
Current Density
J = I/A = nev_d
Current per unit area
Variables: n = charge carrier density, e = charge, v_d = drift velocity
Ohm's Law
V = IR
Voltage across conductor is proportional to current
Resistance
R = ρL/A
Resistance of conductor
Variables: ρ = resistivity, L = length, A = area
Temperature Dependence of Resistance
R = R₀(1 + αΔT)
Resistance variation with temperature
Variables: α = temperature coefficient
Resistors in Series
Rₑ = R₁ + R₂ + ...
Effective resistance in series
Resistors in Parallel
1/Rₑ = 1/R₁ + 1/R₂ + ...
Effective resistance in parallel
Electric Power
P = VI = I²R = V²/R
Rate of electrical energy dissipation
Electrical Energy
E = Pt = VIt
Energy consumed in time t
EMF and Terminal Voltage
ε = V + Ir
EMF of cell with internal resistance r
Cells in Series
εₑ = ε₁ + ε₂ + ..., rₑ = r₁ + r₂ + ...
Effective EMF and internal resistance in series
Cells in Parallel
εₑ = ε (same), 1/rₑ = 1/r₁ + 1/r₂ + ...
Effective values for identical cells in parallel
Kirchhoff's Current Law (KCL)
ΣI_in = ΣI_out
Sum of currents entering equals sum leaving a junction
Kirchhoff's Voltage Law (KVL)
ΣV = 0
Sum of potential differences around closed loop is zero
Wheatstone Bridge Balance Condition
P/Q = R/S
Condition for null deflection in galvanometer
Meter Bridge Formula
X/R = l/(100-l)
Unknown resistance from balanced bridge
Variables: l = balancing length in cm
Potentiometer Principle
V ∝ l
Potential difference proportional to length
Joule's Law of Heating
H = I²Rt
Heat produced in conductor
Conductivity
σ = 1/ρ
Reciprocal of resistivity
Drift Velocity
v_d = eEτ/m = I/(neA)
Average velocity of charge carriers
Variables: τ = relaxation time
Biot-Savart Law
dB = (μ₀/4π)(Idl sin θ/r²)
Magnetic field due to current element
Variables: μ₀ = 4π × 10⁻⁷ T·m/A
Magnetic Field due to Straight Wire
B = (μ₀I)/(2πr)
Field at distance r from long straight wire
Magnetic Field at Center of Circular Loop
B = μ₀I/(2R)
Field at center of circular current loop
Magnetic Field on Axis of Circular Loop
B = (μ₀IR²)/(2(R² + x²)^(3/2))
Field at distance x on axis of loop
Magnetic Field inside Solenoid
B = μ₀nI
Field inside long solenoid
Variables: n = turns per unit length
Ampere's Circuital Law
∮B⃗·dl⃗ = μ₀Iₑₙc
Line integral of magnetic field equals enclosed current
Force on Current-Carrying Conductor
F⃗ = IL⃗ × B⃗ = ILB sin θ
Force on conductor in magnetic field
Force on Moving Charge
F⃗ = qv⃗ × B⃗ = qvB sin θ
Lorentz force on charge in magnetic field
Force between Parallel Wires
F/L = (μ₀I₁I₂)/(2πd)
Force per unit length between parallel current-carrying wires
Torque on Current Loop
τ = NIAB sin θ = MB sin θ
Torque on coil in magnetic field
Variables: M = NIA = magnetic moment
Radius of Circular Path in Magnetic Field
r = mv/(qB)
Radius of charged particle in uniform magnetic field
Time Period in Magnetic Field
T = 2πm/(qB)
Period of circular motion (independent of velocity)
Current Sensitivity of Galvanometer
Iₛ = θ/I = NAB/k
Deflection per unit current
Variables: k = torsional constant
Shunt Resistance
S = Ig G/(I - Ig)
Shunt to convert galvanometer to ammeter
Variables: Ig = full scale current, G = galvanometer resistance
Series Resistance for Voltmeter
R = (V/Ig) - G
Resistance to convert galvanometer to voltmeter
Faraday's Law of EMI
ε = -dΦ/dt
Induced EMF equals negative rate of change of flux
Magnetic Flux
Φ = B⃗·A⃗ = BA cos θ
Magnetic flux through area A
EMF in Coil
ε = -N dΦ/dt
Induced EMF in coil of N turns
Motional EMF
ε = Bvl
EMF induced in conductor moving perpendicular to field
Variables: l = length of conductor
Self Inductance
ε = -L dI/dt
EMF induced due to change in own current
Variables: L = self inductance (Henry)
Self Inductance of Solenoid
L = μ₀n²Al
Inductance of solenoid
Variables: n = turns per unit length, A = area, l = length
Energy Stored in Inductor
U = ½LI²
Magnetic energy in inductor
Mutual Inductance
ε₂ = -M dI₁/dt
EMF induced in coil 2 due to changing current in coil 1
AC Voltage
V = V₀ sin(ωt)
Instantaneous voltage in AC circuit
Variables: V₀ = peak voltage
RMS Voltage
Vᵣₘₛ = V₀/√2
Root mean square voltage
RMS Current
Iᵣₘₛ = I₀/√2
Root mean square current
Inductive Reactance
Xₗ = ωL = 2πfL
Opposition to AC by inductor
Capacitive Reactance
Xᴄ = 1/(ωC) = 1/(2πfC)
Opposition to AC by capacitor
Impedance in LCR Circuit
Z = √(R² + (Xₗ - Xᴄ)²)
Net opposition to AC in series LCR circuit
Power in AC Circuit
P = Vᵣₘₛ Iᵣₘₛ cos φ
Average power dissipated
Variables: cos φ = power factor
Power Factor
cos φ = R/Z
Ratio of resistance to impedance
Resonance Frequency
f₀ = 1/(2π√(LC))
Frequency at which Xₗ = Xᴄ and Z is minimum
Transformer Equation
Vₛ/Vₚ = Nₛ/Nₚ = Iₚ/Iₛ
Relation between primary and secondary
Variables: subscripts: p = primary, s = secondary
Transformer Efficiency
η = (Output power/Input power) × 100%
Efficiency of transformer
Phase Angle in LCR Circuit
tan φ = (Xₗ - Xᴄ)/R
Phase difference between voltage and current
Speed of EM Waves
c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s
Speed of electromagnetic waves in vacuum
Relation between E and B
E = cB
Relation between electric and magnetic field amplitudes in EM wave
Energy Density in EM Wave
u = ½ε₀E² + B²/(2μ₀)
Total energy density (electric + magnetic)
Intensity of EM Wave
I = ½ε₀cE₀² = cB₀²/(2μ₀)
Average power per unit area
EM Wave Frequency-Wavelength Relation
c = fλ
Relation for electromagnetic spectrum
Mirror Formula
1/f = 1/v + 1/u
Relation between object and image distances for mirrors
Variables: u = object distance (negative), v = image distance, f = focal length
Magnification by Mirror
m = -v/u = hᵢ/hₒ
Magnification produced by mirror
Focal Length and Radius Relation
f = R/2
Focal length is half the radius of curvature
Snell's Law
n₁ sin θ₁ = n₂ sin θ₂
Law of refraction at interface
Refractive Index
n = c/v
Refractive index as ratio of speeds
Critical Angle
sin θc = n₂/n₁ = 1/n
Angle for total internal reflection (n₁ > n₂)
Apparent Depth
d_apparent = d_real/n
Object in denser medium appears closer
Lens Formula
1/f = 1/v - 1/u
Relation for thin lens
Lens Maker's Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Focal length from radii of curvature
Magnification by Lens
m = v/u = hᵢ/hₒ
Magnification produced by lens
Power of Lens
P = 1/f (in metres)
Power measured in dioptre (D)
Power of Lenses in Contact
P = P₁ + P₂
Total power is sum of individual powers
Simple Microscope Magnification
m = 1 + D/f
Angular magnification of simple microscope
Variables: D = 25 cm (near point)
Compound Microscope Magnification
m = mₒ × mₑ = (v₀/u₀)(D/fₑ)
Total magnification = objective × eyepiece
Telescope Magnification
m = f₀/fₑ
Angular magnification of telescope
Variables: f₀ = focal length of objective, fₑ = focal length of eyepiece
Young's Double Slit Fringe Width
β = λD/d
Fringe width in interference pattern
Variables: λ = wavelength, D = screen distance, d = slit separation
Position of Bright Fringes
yₙ = nλD/d, n = 0,1,2,...
Distance of nth bright fringe from center
Position of Dark Fringes
yₙ = (n + ½)λD/d, n = 0,1,2,...
Distance of nth dark fringe from center
Path Difference for Constructive Interference
Δx = nλ
Path difference for bright fringe
Path Difference for Destructive Interference
Δx = (n + ½)λ
Path difference for dark fringe
Single Slit Diffraction Minima
a sin θ = nλ, n = 1,2,3,...
Position of minima in single slit diffraction
Variables: a = slit width
Rayleigh Criterion
θ = 1.22λ/D
Minimum angle of resolution
Variables: D = aperture diameter
Malus' Law
I = I₀ cos² θ
Intensity of polarized light through analyzer
Brewster's Law
tan θₚ = n
Angle of polarization
Dispersive Power
ω = (nᵥ - nᵣ)/(n - 1)
Measure of dispersion by prism
Variables: nᵥ, nᵣ = refractive indices for violet and red
Einstein's Photoelectric Equation
KEₘₐₓ = hf - φ
Maximum kinetic energy of photoelectrons
Variables: h = 6.63 × 10⁻³⁴ J·s, φ = work function
Photon Energy
E = hf = hc/λ
Energy of photon
Stopping Potential
eV₀ = KEₘₐₓ = hf - φ
Potential required to stop photoelectrons
Threshold Frequency
f₀ = φ/h
Minimum frequency for photoelectric effect
Threshold Wavelength
λ₀ = hc/φ
Maximum wavelength for photoelectric effect
de Broglie Wavelength
λ = h/p = h/(mv)
Wavelength associated with matter
de Broglie Wavelength (Charged Particle)
λ = h/√(2mqV)
Wavelength of charged particle accelerated through potential V
Minimum Wavelength of X-rays
λₘᵢₙ = hc/(eV)
Cut-off wavelength in X-ray tube
Photon Momentum
p = E/c = h/λ
Momentum of photon
Kinetic Energy and Wavelength
λ = h/√(2mKE)
de Broglie wavelength in terms of kinetic energy
Rydberg Formula
1/λ = R(1/n₁² - 1/n₂²)
Wavelength of spectral lines in hydrogen
Variables: R = 1.097 × 10⁷ m⁻¹ (Rydberg constant)
Radius of nth Bohr Orbit
rₙ = n²h²/(4π²mke²) = n²a₀
Radius of electron orbit in hydrogen
Variables: a₀ = 0.529 Å (Bohr radius)
Energy of nth Level
Eₙ = -13.6/n² eV
Energy of electron in nth orbit of hydrogen
Velocity of Electron in nth Orbit
vₙ = (2πke²)/(nh) = c/(137n)
Velocity in Bohr orbit
Angular Momentum of Electron
L = nh/(2π) = nℏ
Quantization of angular momentum
Energy of Photon Emitted
E = 13.6(1/n₁² - 1/n₂²) eV
Energy difference between levels
Nuclear Radius
R = R₀A^(1/3)
Radius depends on mass number
Variables: R₀ = 1.2 × 10⁻¹⁵ m, A = mass number
Radioactive Decay Law
N = N₀e^(-λt)
Number of nuclei remaining after time t
Variables: λ = decay constant
Activity of Radioactive Sample
A = λN = A₀e^(-λt)
Rate of decay
Half-Life
T₁/₂ = (ln 2)/λ = 0.693/λ
Time for half the sample to decay
Mean Life
τ = 1/λ = T₁/₂/ln 2
Average lifetime of nucleus
Mass-Energy Relation
E = mc²
Einstein's mass-energy equivalence
Mass Defect
Δm = [Zmₚ + Nmₙ - M]
Difference between constituent and actual mass
Variables: Z = protons, N = neutrons
Binding Energy
BE = Δmc²
Energy required to disassemble nucleus
Q-value of Nuclear Reaction
Q = (Mᵢ - Mf)c²
Energy released/absorbed in reaction
Variables: Mᵢ = initial mass, Mf = final mass
Diode Current Equation
I = I₀(e^(eV/kT) - 1)
Current-voltage relation for ideal diode
Dynamic Resistance of Diode
rₐ = ΔV/ΔI
AC resistance of forward-biased diode
Efficiency of Half-Wave Rectifier
η = 40.6%
Maximum efficiency of half-wave rectifier
Efficiency of Full-Wave Rectifier
η = 81.2%
Maximum efficiency of full-wave rectifier
Voltage Regulation
Vₒᵤₜ = Vᴢ (constant)
Output voltage equals Zener voltage in breakdown
Current Gain (β)
β = Iᴄ/Iᵦ
Ratio of collector to base current
Current Relation in Transistor
Iₑ = Iᵦ + Iᴄ
Emitter current equals sum of base and collector currents
Voltage Gain of Amplifier
Aᵥ = βRₗ/Rᵢ
Voltage amplification in CE amplifier
Variables: Rₗ = load resistance, Rᵢ = input resistance
Avogadro's Number
Nₐ = 6.022 × 10²³ mol⁻¹
Number of particles (atoms, molecules, ions) in one mole of substance.
Number of Moles
n = m / M = N / Nₐ
Calculate moles from mass, molar mass, or number of particles.
Variables: n = number of moles, m = mass (g), M = molar mass (g/mol), N = number of particles
Molarity
M = n / V = (m × 1000) / (M × V_mL)
Moles of solute per liter of solution.
Variables: M = molarity (mol/L), n = moles, V = volume (L), m = mass (g), M = molar mass (g/mol)
Molality
m = n_solute / m_solvent(kg) = (1000 × w_solute) / (M_solute × w_solvent)
Moles of solute per kilogram of solvent.
Variables: m = molality (mol/kg), n = moles, w = mass (g), M = molar mass (g/mol)
Mole Fraction
χₐ = nₐ / (nₐ + nᵦ + ...)
Ratio of moles of a component to total moles in the mixture.
Variables: χₐ = mole fraction of A, nₐ, nᵦ = moles of A, B respectively
Mass Percentage
% by mass = (mass of component / total mass) × 100
Percentage of mass of a component in a compound or mixture.
Parts Per Million (ppm)
ppm = (mass of solute / mass of solution) × 10⁶
Concentration in parts per million for very dilute solutions.
Mole Percentage
Mole % = (nᵢ / Σnᵢ) × 100
Percentage of moles of a component.
Percentage Yield
% Yield = (actual yield / theoretical yield) × 100
Ratio of actual to theoretical product in a reaction.
Limiting Reagent
Compare (moles given / stoichiometric coefficient) for all reactants
Reactant with smallest ratio determines the amount of product formed.
Variables: The reagent with minimum mole ratio is the limiting reagent.
Equivalent Weight
Equivalent weight = Molar mass / n-factor
Mass of substance that combines or displaces 8 g of oxygen.
Variables: n-factor = number of electrons exchanged in redox or H⁺/OH⁻ in acid-base
Normality
N = (n-factor × M) / (V in L) = (W × 1000 × n-factor) / (M × V_mL)
Equivalents of solute per liter of solution.
Variables: N = normality, M = molarity, W = mass (g), M = molar mass, V = volume
Dilution Formula
M₁V₁ = M₂V₂ (or N₁V₁ = N₂V₂)
Molarity and volume relationship for dilution of solutions.
Variables: M₁, V₁ = initial molarity and volume; M₂, V₂ = final molarity and volume
Empirical Formula Calculation
Mole ratio = (mass % / atomic mass)
Simplest whole-number ratio of atoms in a compound.
Molecular Formula
Molecular formula = (Empirical formula)ₙ where n = (Molecular mass / Empirical mass)
Actual formula showing all atoms in the molecule.
Oxidation State Rule
Σ(oxidation states) = overall charge of species
Sum of oxidation numbers equals the charge of the compound or ion.
Gas Density
d = PM / RT
Density of a gas in terms of pressure, molar mass, and temperature.
Variables: d = density, P = pressure (Pa), M = molar mass (g/mol), R = 8.314 J/(mol·K), T = temperature (K)
Molar Volume at STP
Vm = 22.4 L/mol (at STP: 0°C, 1 atm)
Volume occupied by one mole of ideal gas at standard temperature and pressure.
Bohr's Radius
rₙ = (n² × 0.529 × 10⁻¹⁰ m) / Z = (n² × a₀) / Z
Radius of the nth orbit in hydrogen-like atoms.
Variables: rₙ = radius of nth orbit, n = principal quantum number, a₀ = Bohr radius (0.529 Å), Z = atomic number
Bohr's Velocity
vₙ = (Z × 2.19 × 10⁶ m/s) / n = (Z × c × α) / n
Velocity of electron in the nth orbit (Bohr model).
Variables: vₙ = velocity in nth orbit, Z = atomic number, n = principal quantum number, c = speed of light, α = fine structure constant
Bohr's Energy
Eₙ = -(13.6 × Z² / n²) eV
Energy of electron in the nth orbit (negative indicates bound state).
Variables: Eₙ = energy of nth orbit, Z = atomic number, n = principal quantum number
Ionization Energy (Bohr Model)
IE = 13.6 × Z² eV
Energy required to remove electron from ground state to infinity.
Frequency of Orbital Revolution
ν = (2.07 × 10¹⁵ × Z²) / n³ Hz
Frequency of revolution of electron in nth orbit.
Rydberg Formula (Energy)
1/λ = R∞ × Z² × (1/n₁² - 1/n₂²)
Wavenumber of spectral lines for hydrogen-like atoms.
Variables: λ = wavelength, R∞ = Rydberg constant (1.097 × 10⁷ m⁻¹), Z = atomic number, n₁, n₂ = lower and upper energy levels
Energy Difference Between Orbits
ΔE = 13.6 × Z² × (1/n₁² - 1/n₂²) eV
Energy absorbed or emitted during transition between orbits.
de Broglie Wavelength
λ = h / (mv) = h / p
Wavelength associated with moving particle; shows matter-wave duality.
Variables: λ = wavelength, h = Planck's constant (6.626 × 10⁻³⁴ J·s), m = mass, v = velocity, p = momentum
Heisenberg's Uncertainty Principle
Δx × Δp ≥ h / (4π)
Cannot simultaneously determine position and momentum of particle with arbitrary precision.
Variables: Δx = uncertainty in position, Δp = uncertainty in momentum, h = Planck's constant
Quantum Numbers Definition
n (principal), l (azimuthal), m (magnetic), s (spin)
Define the orbital energy, shape, orientation, and electron spin respectively.
Variables: n = 1,2,3,...; l = 0,1,...,n-1; m = -l to +l; s = ±1/2
Maximum Electrons per Orbital/Subshell/Shell
Orbital: 2 electrons; Subshell: 2(2l+1); Shell: 2n²
Capacity of orbitals, subshells, and shells based on quantum numbers.
Variables: l = azimuthal quantum number, n = principal quantum number
Photoelectric Effect Equation
hν = Φ + KEmax
Energy of photon equals work function plus maximum kinetic energy of emitted electron.
Variables: h = Planck's constant, ν = frequency, Φ = work function, KEmax = maximum kinetic energy
Threshold Frequency
ν₀ = Φ / h
Minimum frequency of light required to eject electrons from metal surface.
Variables: ν₀ = threshold frequency, Φ = work function, h = Planck's constant
Lyman Series (UV)
1/λ = R∞ × (1/1² - 1/n²) where n = 2,3,4,...
Transitions from higher orbits to n=1 (ground state); in ultraviolet region.
Balmer Series (Visible)
1/λ = R∞ × (1/2² - 1/n²) where n = 3,4,5,...
Transitions to n=2; visible region; forms hydrogen emission spectrum.
Paschen Series (IR)
1/λ = R∞ × (1/3² - 1/n²) where n = 4,5,6,...
Transitions to n=3; in infrared region.
Orbital Angular Momentum
L = √[l(l+1)] × ℏ
Magnitude of angular momentum for an orbital.
Variables: l = azimuthal quantum number, ℏ = h/(2π)
Orbital Shape Classification
s-orbital (l=0, spherical); p-orbital (l=1, dumbbell); d-orbital (l=2, cloverleaf); f-orbital (l=3, complex)
Shape of atomic orbitals determined by azimuthal quantum number l.
Bond Order (Molecular Orbital Theory)
Bond order = (Nᵦ - Nₐ) / 2
Half the difference between bonding and antibonding electrons; indicates bond strength.
Variables: Nᵦ = number of bonding electrons, Nₐ = number of antibonding electrons
Formal Charge
FC = V - (N + B/2)
Charge assigned to atom in Lewis structure; helps determine most stable structure.
Variables: V = valence electrons, N = non-bonding electrons, B = bonding electrons
Born-Haber Cycle
ΔHf° = ΔHsublimation + IE + ΔHdissociation + EA - U
Energy cycle for formation of ionic compounds; relates lattice energy to atomization.
Variables: ΔHf° = enthalpy of formation, IE = ionization energy, EA = electron affinity, U = lattice energy
Lattice Energy (Born-Haber)
U = (1389 × |Z⁺| × |Z⁻|) / r₀ kJ/mol
Energy required to completely dissociate one mole of ionic solid.
Variables: Z⁺, Z⁻ = charges of ions, r₀ = interionic distance in pm
Electronegativity (Pauling Scale)
χₐ - χᵦ ≈ √[ΔH_dissociation(AB) - (ΔH_dissociation(A₂) + ΔH_dissociation(B₂))/2]
Measure of an atom's ability to attract electrons in a chemical bond.
Dipole Moment
μ = Q × d (Debye = 3.336 × 10⁻³⁰ C·m)
Measure of molecular polarity; product of charge and separation distance.
Variables: μ = dipole moment, Q = charge (esu), d = distance (Å)
sp Hybridization
Linear geometry; bond angle = 180°; e.g., BeCl₂, C≡C, C≡N
Two hybrid orbitals; forms 2 sigma bonds; maximum sp character.
sp² Hybridization
Trigonal planar geometry; bond angle = 120°; e.g., BF₃, C=C, NO₃⁻
Three hybrid orbitals; forms 3 sigma bonds; one unhybridized p orbital.
sp³ Hybridization
Tetrahedral geometry; bond angle = 109.5°; e.g., CH₄, NH₄⁺, CCl₄
Four hybrid orbitals; forms 4 sigma bonds; ideal for saturated compounds.
sp³d Hybridization
Trigonal bipyramidal geometry; axial angle = 180°, equatorial = 120°; e.g., PCl₅, SF₄
Five hybrid orbitals from d orbital involvement; forms 5 bonds.
sp³d² Hybridization
Octahedral geometry; bond angle = 90°; e.g., SF₆, [PtCl₆]²⁻
Six hybrid orbitals from d orbitals; forms 6 bonds in octahedral arrangement.
VSEPR Shape Prediction
Electron pairs (bonding + lone pairs) arrange to minimize repulsion
Valence Shell Electron Pair Repulsion theory predicts molecular geometry.
Resonance Energy
Resonance energy = Actual energy - Most stable single resonance form energy
Stabilization energy due to delocalization over multiple resonance forms.
Hydrogen Bond Strength
H-bond energy typically 10-40 kJ/mol; much weaker than covalent bonds
Attractive force between H bonded to N, O, F and lone pair on N, O, F.
Delocalized Metallic Bonding
Metal atoms lose valence electrons to form cation lattice with delocalized electrons
Electrons move freely throughout structure; explains conductivity and malleability.
Ideal Gas Law
PV = nRT
Fundamental relationship between pressure, volume, moles, and temperature.
Variables: P = pressure (Pa or atm), V = volume (m³ or L), n = moles, R = 8.314 J/(mol·K) or 0.0821 L·atm/(mol·K), T = temperature (K)
Boyle's Law
P₁V₁ = P₂V₂ (at constant T and n)
Pressure inversely proportional to volume at constant temperature.
Charles's Law
V₁/T₁ = V₂/T₂ (at constant P and n)
Volume directly proportional to absolute temperature at constant pressure.
Gay-Lussac's Law
P₁/T₁ = P₂/T₂ (at constant V and n)
Pressure directly proportional to absolute temperature at constant volume.
Combined Gas Law
P₁V₁/T₁ = P₂V₂/T₂
Combines Boyle's, Charles's, and Gay-Lussac's laws.
Dalton's Law of Partial Pressures
Ptotal = P₁ + P₂ + P₃ + ...
Total pressure equals sum of partial pressures of individual gases.
Partial Pressure (Mole Fraction)
Pᵢ = χᵢ × Ptotal
Partial pressure of gas equals its mole fraction times total pressure.
Variables: Pᵢ = partial pressure of gas i, χᵢ = mole fraction of gas i
Graham's Law of Diffusion/Effusion
r₁/r₂ = √(M₂/M₁)
Rate of diffusion inversely proportional to square root of molar mass.
Variables: r₁, r₂ = rates of diffusion for gases 1 and 2, M₁, M₂ = molar masses
Root Mean Square Velocity
Vrms = √(3RT/M) = √(3kT/m)
Average speed of gas molecules; increases with temperature.
Variables: R = 8.314 J/(mol·K), T = temperature (K), M = molar mass (kg/mol), k = Boltzmann constant, m = molecular mass
Average Kinetic Energy
KE_avg = (3/2)RT per mole = (3/2)kT per molecule
Average kinetic energy directly proportional to absolute temperature.
Collision Frequency (Z)
Z = √2 × π × d² × n × v_avg
Number of collisions per molecule per unit time.
Variables: d = molecular diameter, n = number density, v_avg = average velocity
Van der Waals Equation
[P + a(n/V)²][V - nb] = nRT
Real gas equation correcting for intermolecular forces (a) and molecular volume (b).
Variables: a, b = Van der Waals constants (specific to each gas), P, V, n, R, T as in ideal gas law
Van der Waals Constants
a = 27R²Tc² / (64Pc), b = RTc / (8Pc)
Derived from critical point data; 'a' corrects for attractive forces, 'b' for volume.
Variables: Tc = critical temperature, Pc = critical pressure, R = gas constant
Compressibility Factor
Z = PV / (nRT)
Ratio of real to ideal gas; Z = 1 for ideal gas, Z < 1 (attractive forces dominate), Z > 1 (repulsive forces dominate).
Critical Constants Relationship
Z_c = (Pc × Vc) / (Rc × Tc) ≈ 0.27 for most real gases
Compressibility factor at critical point relates pressure, volume, and temperature.
Joule-Thomson Coefficient
μJT = (∂T/∂P)H = (1/Cp) × [T × (∂V/∂T)P - V]
Change in temperature with pressure during throttling; indicates cooling/warming on expansion.
Surface Tension
γ = Force / Length (N/m); also γ = Energy / Area
Tendency of liquid surface to minimize area due to cohesive forces.
Viscosity Temperature Dependence
η = η₀ × exp(E_v / RT)
Viscosity exponentially increases with activation energy; decreases with temperature.
Variables: η = viscosity, E_v = viscous flow energy, T = temperature
First Law of Thermodynamics
ΔU = q + w
Change in internal energy equals heat absorbed plus work done on system.
Variables: ΔU = change in internal energy, q = heat, w = work (w = -PΔV for expansion against constant pressure)
Internal Energy Change
ΔU = nCvΔT
Change in internal energy for ideal gas depends only on temperature change.
Variables: n = moles, Cv = molar heat capacity at constant volume, ΔT = temperature change
Work in Expansion (Constant Pressure)
w = -PΔV = -P(V₂ - V₁)
Work done by gas during expansion at constant external pressure.
Work in Isothermal Expansion
w = -nRT ln(V₂/V₁) = -2.303 nRT log(V₂/V₁)
Work done during reversible isothermal expansion of ideal gas.
Enthalpy and Internal Energy
ΔH = ΔU + Δ(PV) = ΔU + PΔV + VΔP
At constant pressure: ΔH = ΔU + PΔV
Variables: H = enthalpy, U = internal energy, P = pressure, V = volume
Heat at Constant Pressure
qp = nCpΔT = ΔH
Heat absorbed at constant pressure equals change in enthalpy.
Variables: qp = heat at constant pressure, Cp = molar heat capacity at constant pressure
Mayer's Relation
Cp - Cv = R
Difference between heat capacities for ideal gas equals gas constant.
Hess's Law of Constant Heat Summation
ΔH_reaction = Σ ΔH_products - Σ ΔH_reactants
Enthalpy change is path-independent; sum of reactions gives overall enthalpy change.
Standard Enthalpy of Formation
ΔHf° = Σ ΔHf°(products) - Σ ΔHf°(reactants)
Enthalpy change when one mole of compound forms from elements in standard state.
Enthalpy of Combustion
ΔHcomb = Σ ΔHf°(products) - Σ ΔHf°(reactants)
Enthalpy released when one mole of substance completely burns in oxygen.
Enthalpy of Neutralization
ΔHneutralization ≈ -57.3 kJ/mol (for strong acid-strong base in dilute aqueous solution)
Heat released when acid and base neutralize to form water.
Entropy Change (Heat at Constant Temperature)
ΔS = q_reversible / T
For reversible process, entropy change equals heat divided by temperature.
Gibbs Free Energy
ΔG = ΔH - TΔS
Determines spontaneity of reaction; ΔG < 0 is spontaneous at given T.
Variables: ΔG = Gibbs free energy, ΔH = enthalpy, T = temperature (K), ΔS = entropy
Gibbs-Helmholtz Equation
d(ΔG/T)/dT = -ΔH/T²
Temperature dependence of Gibbs free energy.
Kirchhoff's Equation
d(ΔH)/dT = ΔCp
Rate of change of enthalpy with temperature equals heat capacity change.
Spontaneity Criteria
ΔG < 0: spontaneous; ΔG > 0: non-spontaneous; ΔG = 0: equilibrium
At equilibrium, ΔG = 0 and system is at maximum stability.
Bond Enthalpy (Bond Energy)
ΔH_reaction ≈ Σ (bonds broken) - Σ (bonds formed)
Approximate method using average bond energies to calculate reaction enthalpy.
Heat Capacity of Substance
q = m × c × ΔT = n × C × ΔT
Heat required to change temperature of substance.
Variables: m = mass, c = specific heat capacity (J/g·K), n = moles, C = molar heat capacity (J/mol·K)
Equilibrium Constant (Kc)
Kc = [C]^c [D]^d / [A]^a [B]^b
For reaction aA + bB ⇌ cC + dD; ratio of product to reactant concentrations at equilibrium.
Variables: [X] = equilibrium concentration of species X
Equilibrium Constant (Kp)
Kp = (Pc)^c (Pd)^d / (Pa)^a (Pb)^b
Equilibrium constant in terms of partial pressures for gaseous systems.
Variables: Pi = partial pressure of species i
Relationship between Kp and Kc
Kp = Kc(RT)^Δn
Converts between concentration and pressure-based equilibrium constants.
Variables: Δn = moles of gas products - moles of gas reactants, R = 0.0821 L·atm/(mol·K), T = temperature (K)
Le Chatelier's Principle
System shifts to counteract any imposed change in temperature, pressure, or concentration
Qualitative prediction of equilibrium shift when conditions change.
Ionization Constant (Ka)
Ka = [H⁺][A⁻] / [HA]
Equilibrium constant for weak acid dissociation; smaller Ka means weaker acid.
Ionization Constant (Kb)
Kb = [BH⁺][OH⁻] / [B]
Equilibrium constant for weak base ionization.
Ion Product of Water
Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ (at 25°C)
Equilibrium constant for water autoionization.
Relationship between Ka and Kb
Ka × Kb = Kw
For conjugate acid-base pair, product of ionization constants equals Kw.
pH Definition
pH = -log[H⁺] = -log₁₀[H⁺]
Measure of acidity; lower pH means more acidic.
pOH Definition
pOH = -log[OH⁻]
Measure of basicity; lower pOH means more basic.
pH and pOH Relationship
pH + pOH = 14 (at 25°C)
Sum of pH and pOH equals 14 in neutral solution (pH = pOH = 7).
Henderson-Hasselbalch Equation
pH = pKa + log([A⁻]/[HA])
Gives pH of buffer solution in terms of Ka and acid-base ratio.
Variables: pKa = -log(Ka), [A⁻] = concentration of conjugate base, [HA] = concentration of weak acid
Buffer Capacity
Buffer capacity = Δn / ΔpH (moles of acid or base per pH unit change)
Ability of buffer to resist pH change; higher concentration = greater capacity.
Solubility Product (Ksp)
For AxBy: Ksp = [A^(+y)]^x [B^(-x)]^y
Product of ion concentrations at saturation; measure of solubility.
Common Ion Effect
Addition of common ion shifts equilibrium left, decreasing solubility (Le Chatelier)
Presence of ion from another source decreases salt solubility.
Degree of Dissociation
α = (initial moles - equilibrium moles) / initial moles = x / (c₀ - x)
Fraction of compound that has dissociated; 0 ≤ α ≤ 1.
Variables: x = moles dissociated, c₀ = initial concentration
van't Hoff Equation
ln(K₂/K₁) = -(ΔH°/R) × (1/T₂ - 1/T₁)
Effect of temperature on equilibrium constant; uses enthalpy of reaction.
Weak Acid Approximation (Ka)
Ka = [H⁺]² / [HA]₀ (if [H⁺] << [HA]₀)
Simplified calculation when dissociation is negligible; gives [H⁺] = √(Ka × [HA]₀)
Standard Cell Potential
E°cell = E°cathode - E°anode
EMF of cell under standard conditions; positive for spontaneous reaction.
Nernst Equation
E = E° - (RT/nF) × ln(Q) = E° - (0.0592/n) × log(Q) at 25°C
Cell potential under non-standard conditions; includes reaction quotient Q.
Variables: E = cell potential, E° = standard cell potential, R = 8.314 J/(mol·K), T = temperature (K), n = moles of electrons, F = Faraday's constant (96,485 C/mol), Q = reaction quotient
Gibbs Free Energy and Cell Potential
ΔG° = -nFE°cell
Relates standard cell potential to standard free energy change.
Variables: ΔG° = standard free energy change, n = moles of electrons transferred, F = Faraday's constant, E°cell = standard cell potential
Equilibrium Constant from Cell Potential
E° = (0.0592/n) × log(K) at 25°C
Relates standard cell potential to equilibrium constant.
Variables: K = equilibrium constant, n = moles of electrons
Faraday's First Law of Electrolysis
m = (M × Q) / (n × F) = (M × I × t) / (n × F)
Mass of substance deposited/liberated proportional to charge passed.
Variables: m = mass deposited (g), M = molar mass (g/mol), Q = charge (C), I = current (A), t = time (s), n = number of electrons, F = 96,485 C/mol
Faraday's Second Law
m₁/m₂ = (M₁/n₁) / (M₂/n₂) = E₁/E₂
For same charge, mass ratio equals equivalent weight ratio.
Variables: m₁, m₂ = masses, M₁, M₂ = molar masses, n₁, n₂ = electrons transferred, E₁, E₂ = equivalent weights
Charge (Coulombs)
Q = I × t
Total charge equals current multiplied by time.
Variables: Q = charge (C), I = current (A), t = time (s)
Electrical Conductivity
κ (kappa) = (A / l) × (1 / R) = σ / l (where A = area, l = length)
Measure of a solution's ability to conduct electricity.
Molar Conductivity
Λm = κ / c = 1000κ / M
Conductivity per unit molar concentration.
Variables: Λm = molar conductivity (S·cm²/mol), κ = conductivity, c = molarity (mol/L), M = molarity (mol/mL)
Cell Constant
Cell constant (G*) = l / A = R × κ
Geometric factor of cell relating resistance to conductivity.
Variables: l = distance between electrodes, A = area of electrodes, R = resistance
Kohlrausch's Law of Independent Ion Migration
Λm = Λm° - K√c
Molar conductivity decreases with concentration due to ion interactions.
Variables: Λm° = molar conductivity at infinite dilution, K = Kohlrausch constant, c = concentration
Degree of Dissociation from Conductivity
α = Λm / Λm°
Fraction of electrolyte dissociated; approaches 1 for strong electrolytes.
Ka from Conductivity
Ka = α² × c / (1 - α) = (Λm / Λm°)² × c / (1 - Λm/Λm°)
Calculate ionization constant from conductivity measurements.
Cathode (Reduction) vs Anode (Oxidation)
Cathode: reduction (gain of electrons); Anode: oxidation (loss of electrons)
In galvanic cell, anode is negative and cathode is positive.
Lead-Acid Battery (Pb-PbO₂)
Cell potential ≈ 2 V; 6 cells in series = 12 V battery
Most common rechargeable battery; uses lead electrodes and H₂SO₄.
Overpotential (Overvoltage)
Applied potential = E°cell + η_cathode + η_anode + iR
Extra voltage needed due to resistance to ion/electron transfer.
Variables: η = overpotential, i = current, R = resistance
Average Rate of Reaction
Average rate = -Δ[A]/Δt = Δ[P]/Δt
Change in concentration per unit time; instantaneous rate is the derivative.
Rate Law
Rate = k[A]^m [B]^n
Rate depends on concentration with experimentally determined order m and n.
Variables: k = rate constant, [A], [B] = concentrations, m, n = orders of reaction
Zero-Order Reaction
[A]t = [A]₀ - kt; t₁/₂ = [A]₀ / 2k
Rate independent of concentration; half-life increases with initial concentration.
First-Order Reaction
ln[A]t = ln[A]₀ - kt; [A]t = [A]₀e^(-kt); t₁/₂ = 0.693/k = ln(2)/k
Rate proportional to concentration; half-life independent of concentration.
Second-Order Reaction
1/[A]t = 1/[A]₀ + kt; t₁/₂ = 1/(k[A]₀)
Rate proportional to square of concentration; half-life inversely proportional to initial concentration.
Arrhenius Equation
k = A × e^(-Ea/RT) = A × exp(-Ea/RT)
Temperature dependence of rate constant; A is pre-exponential factor.
Variables: k = rate constant, A = frequency factor (Arrhenius constant), Ea = activation energy, R = 8.314 J/(mol·K), T = temperature (K)
Arrhenius Equation (Logarithmic Form)
ln(k) = ln(A) - Ea/(RT); log(k) = log(A) - Ea/(2.303RT)
Linear form used for graphical determination of Ea.
Activation Energy (Two Temperature Method)
ln(k₂/k₁) = (Ea/R) × (1/T₁ - 1/T₂)
Calculate activation energy from rate constants at two temperatures.
Effect of Catalyst
Catalyst lowers Ea; Ea(catalyzed) < Ea(uncatalyzed); equilibrium constant K unchanged
Catalyst increases reaction rate without being consumed.
Temperature Coefficient
k increases by factor 2-4 for every 10°C increase (rule of thumb)
Qualitative measure of temperature sensitivity of reaction rate.
Pseudo-First-Order Kinetics
If [B] >> [A], then Rate = k[A]^m becomes approximately first-order in [A]
When one reactant concentration is so high it remains essentially constant.
Collision Theory: Effective Collision
Rate ∝ Z × P × f (where Z = collision frequency, P = steric factor, f = fraction with E ≥ Ea)
Molecules must collide with proper orientation and sufficient energy.
Elementary Reaction
Molecularity = number of molecules reacting; order = sum of exponents in rate law
Elementary reactions follow rate law directly from stoichiometry.
Rate-Determining Step
Overall rate law matches rate law of slowest (rate-determining) elementary step
Slowest step controls overall reaction rate.
Enzyme Catalysis (Michaelis-Menten)
v = (Vmax[S]) / (Km + [S]); at [S] >> Km, v ≈ Vmax
Enzyme velocity depends on substrate concentration and affinity constant Km.
Raoult's Law
Pₐ = χₐ × Pₐ°; PA° - PA = χB × Pₐ°
Partial pressure of volatile solvent proportional to its mole fraction.
Variables: Pₐ = partial pressure of component A, χₐ = mole fraction, Pₐ° = vapor pressure of pure A
Relative Lowering of Vapor Pressure
(Pₐ° - Pₐ) / Pₐ° = χB = nB / (nₐ + nB)
Fractional decrease in vapor pressure equals mole fraction of solute.
Boiling Point Elevation
ΔTb = Kb × m
Boiling point increase proportional to molality of non-volatile solute.
Variables: ΔTb = boiling point elevation, Kb = ebullioscopic constant (for water, 0.512 K·kg/mol), m = molality
Freezing Point Depression
ΔTf = Kf × m
Freezing point decrease proportional to molality of non-volatile solute.
Variables: ΔTf = freezing point depression, Kf = cryoscopic constant (for water, 1.86 K·kg/mol), m = molality
Osmotic Pressure
π = CRT = (n/V)RT = i × C × R × T
Pressure needed to prevent osmosis; depends on solute concentration.
Variables: π = osmotic pressure (Pa), C = molarity, R = 8.314 J/(mol·K), T = temperature (K), i = van't Hoff factor
Van't Hoff Factor
i = (observed number of particles) / (number of formula units dissolved)
Accounts for ion dissociation; i = 1 for non-electrolytes, up to 4 for salts with strong dissociation.
General Colligative Property
ΔT = i × K × m; π = i × C × R × T
All colligative properties proportional to number of particles and i factor.
Henry's Law
P = KH × x = KH × (n_gas / n_total)
Pressure of dissolved gas proportional to its mole fraction in solution.
Variables: P = partial pressure, KH = Henry's law constant, x = mole fraction
Solubility Definition
Solubility (s) = mass of solute dissolved per 100g solvent at given temperature
Typically expressed in g/100g solvent or mol/L.
Abnormal Molar Mass
Observed molar mass = (theoretical molar mass) / i
Measured molar mass lower than expected for dissociating solutes.
Degree of Dissociation from Molar Mass
α = (M_theoretical - M_observed) / (M_theoretical × (n - 1))
Calculate dissociation from anomalous molar mass; n = number of particles formed.
Isotonic Solutions
π₁ = π₂ ⟹ i₁C₁T₁ = i₂C₂T₂
Solutions with same osmotic pressure have same solute particle concentration.
Distribution Law (Partition Coefficient)
K = [A]_organic / [A]_aqueous
Ratio of solute concentrations in two immiscible liquids at equilibrium.
Determination of Molar Mass (Ebullioscopy)
M = (Kb × m_solute × 1000) / (m_solvent × ΔTb)
Measure boiling point elevation to determine molar mass.
Determination of Molar Mass (Cryoscopy)
M = (Kf × m_solute × 1000) / (m_solvent × ΔTf)
Measure freezing point depression to determine molar mass.
Freundlich Adsorption Isotherm
x/m = k × P^(1/n); or log(x/m) = log(k) + (1/n)log(P)
Empirical relationship between amount adsorbed and pressure at constant temperature.
Variables: x = mass of adsorbate, m = mass of adsorbent, P = pressure, k, n = constants
Langmuir Adsorption Isotherm
x/m = (a × b × P) / (1 + b × P); or 1/(x/m) = 1/(ab) + (1/a)(1/P)
Monolayer adsorption model; linear plot for data verification.
Variables: x/m = surface coverage, a, b = constants, P = pressure
Frumkin Adsorption Isotherm
Extended form accounting for lateral interactions between adsorbate molecules
More complex model than Langmuir for strongly interacting adsorbates.
Catalyst Types
Homogeneous (same phase as reactants); Heterogeneous (different phase); Enzyme (biological)
Catalysts increase reaction rate by providing alternative pathway with lower Ea.
Autocatalysis
Products catalyze the reaction; autocatalytic curve shows induction period then acceleration
Reaction accelerates as products accumulate and catalyze further reaction.
Zeta Potential
ζ (zeta potential) determines colloidal stability; larger |ζ| means better stability
Electric potential at the shear plane; affects charge and electrostatic repulsion.
Critical Coagulation Concentration (CCC)
CCC inversely proportional to (charge on ion)^6; CCC ∝ 1/z⁶
Minimum ion concentration needed to precipitate colloid (Schulze-Hardy rule).
Gold Number
Gold number = minimum mass of protective colloid to prevent precipitation of 10 mL gold sol
Lower gold number = better protective power.
Ionization Energy (IE)
IE (eV) = 13.6 × Z_eff² / n²
Energy needed to remove electron; increases across period, decreases down group.
Variables: Z_eff = effective nuclear charge, n = principal quantum number
Electron Affinity (EA)
Negative of ionization energy of anion formation
Energy released when electron added; generally increases across period.
Electronegativity (Pauling Scale)
χ (unitless scale, 0-4); F = 4.0 (highest), Cs = 0.79 (lowest)
Ability to attract electrons; increases across period, decreases down group.
Atomic Radius
Metallic radius for metals; covalent radius for non-metals; ionic radius for ions
Decreases across period (increased Z_eff), increases down group (new shell).
Ionic Radius
Cation radius < neutral atom < anion radius; increases with negative charge
Depends on number of protons vs electrons.
Effective Nuclear Charge (Slater's Rules)
Z_eff = Z - S; where S = screening constant calculated by Slater's rules
Nuclear charge 'felt' by outer electrons after shielding by inner electrons.
Variables: Z = atomic number, S = screening constant
Mulliken's Electronegativity
χM = (IE + EA) / 2; conversion: χM = 0.187(IE + EA) - 0.17
Average of ionization energy and electron affinity.
Oxidation State Trends
Main group: max = group number; min = group number - 18; variable oxidation states for transition metals
Shows typical oxidation states for elements by group.
Block Classification
s-block: Groups 1, 2; p-block: Groups 13-18; d-block: Groups 3-12; f-block: Lanthanides and Actinides
Classification based on outermost orbital type.
Diagonal Relationships
Li≈Mg, Be≈Al, B≈Si similarities due to similar electronegativity and charge density
Elements diagonally adjacent show anomalous similarities.
Boron Hydrides
B₂H₆ (diborane); Higher boranes: B₁₀H₁₄, B₁₂H₁₂²⁻
Electron-deficient compounds with three-center two-electron bonds.
Nitrogen Oxides
NO, NO₂, N₂O₃, N₂O₄, N₂O₅; NO + O₂ → NO₂; 2NO₂ + F₂ → 2NO₂F
Important nitrogen oxides and their reactions.
Phosphoric Acid Formation
4P + 5O₂ → P₄O₁₀; P₄O₁₀ + 6H₂O → 4H₃PO₄
White phosphorus oxidizes to phosphorus pentoxide, then hydrates to phosphoric acid.
Allotropes of Sulfur
S₈ (rhombic - stable), S₈ (monoclinic), plastic sulfur, S₂, S₆, S₇ (gaseous)
Multiple forms depending on temperature and conditions.
Ozone Formation
3O₂ → 2O₃; ΔH = +285 kJ/mol
Endothermic oxidation of oxygen; powerful oxidizing agent.
Interhalogen Compounds
XY, XY₃, XY₅, XY₇ (where X = halogen with more electrons)
Binary halides with unusual bonding; more reactive than parent halogens.
Disproportionation of Halogens
X₂ + H₂O ⇌ H⁺ + X⁻ + HXO (e.g., Cl₂ + H₂O → H⁺ + Cl⁻ + HClO)
Halogen undergoes simultaneous oxidation and reduction.
Diagonal Relationship: N and P vs O and S
N and O: strong triple/double bonds; P and S: single/double bonds (weaker)
Explains why P≠O and N≠S despite diagonal position.
Oxoacid Strength Order
For P: H₃PO₄ < H₃PO₃ < H₃PO₂; Ka increases with number of O atoms
More terminal oxygens = stronger acid.
Tautomerism
Example: H₃PO₃ exists as HPO(OH)₂ and H₂P(=O)OH forms
Oscillation between different structural forms.
Effective Atomic Number (EAN)
EAN = Atomic number - Oxidation state - (number of ligands)/2
Represents total electrons around metal; helps predict stability and geometry.
Magnetic Moment (Spin-Only)
μ = √[n(n+2)] BM; where n = number of unpaired electrons
Measure of paramagnetism; only considers spin, not orbital angular momentum.
Variables: μ = magnetic moment in Bohr magnetons (BM), BM = 9.284 × 10⁻²⁴ J/T
Effective Magnetic Moment (Including Orbital)
μeff = √[L(L+1) + S(S+1)] BM × g
More accurate formula including orbital angular momentum; g ≈ 2 for most cases.
d-d Transition Absorption
ΔE = hν = energy gap between d orbitals; determines color
Visible light absorption causes characteristic colors; ΔE typically 10-40 kJ/mol.
Common Oxidation States
+2 (most stable for most d-block); +3; variable higher states for Cr, Mn, Fe
Transition metals show multiple oxidation states due to similar 3d and 4s energies.
Lanthanide Contraction
Ionic radius of Ln³⁺ decreases regularly; from La (1.06 Å) to Lu (0.84 Å)
Poor shielding of 4f electrons causes unusually rapid size decrease.
Actinide Series
5f orbitals being filled (U: [Rn]5f³6d¹7s²; Th: [Rn]6d²7s²)
All are radioactive; uranium is the heaviest naturally occurring element.
Crystal Field Stabilization Energy (CFSE)
CFSE = sum of (number of electrons) × (orbital energy relative to barycenter)
Energy lowering due to d-orbital splitting in ligand field.
Coordination Number
Number of ligands attached to central metal atom; usually 4 or 6
Determines geometry; 4 = tetrahedral/square planar, 6 = octahedral.
Crystal Field Theory (CFT) Splitting
Octahedral: t₂g (lower) and eg (upper); Δo = Eg - E(t₂g)
d-orbitals split into higher and lower energy sets due to ligand field.
Variables: Δo = octahedral crystal field splitting parameter
Spectrochemical Series
I⁻ < Br⁻ < Cl⁻ < NO₃⁻ < F⁻ < OH⁻ < H₂O < NCS⁻ < NH₃ < en < NO₂⁻ < CN⁻ < CO
Order of increasing crystal field splitting; weak to strong field ligands.
Tetrahedral vs Square Planar
Tetrahedral: 4 ligands at corners; Square planar: 4 ligands in plane
d⁸ complexes can be either; square planar usually favored for strong field ligands.
Coordination Isomerism
[Co(NH₃)₆][Cr(CN)₆] ≠ [Cr(NH₃)₆][Co(CN)₆]
Different distribution of ligands between metal centers.
Linkage Isomerism
[Co(NO₂)(NH₃)₅]²⁺ vs [Co(ONO)(NH₃)₅]²⁺
Ligand coordinates through different atoms (NO₂ vs ONO).
Stereoisomerism: cis-trans
[M(NH₃)₄Cl₂]⁺ can form cis (adjacent Cl) or trans (opposite Cl)
Different spatial arrangements of ligands.
Optical Isomerism
[M(en)₃]³⁺ forms Δ (dextrorotatory) and Λ (levorotatory) isomers
Non-superimposable mirror images; rotate plane of polarized light.
IUPAC Naming Rule (Charge)
[M(ligand)ₓ]^(n±); anionic complex ends in -ate
Include charge; name ligands then metal; if anionic, metal name becomes -ate form.
Chelate Effect
Bidentate/polydentate ligands (e.g., en) form more stable complexes than monodentate
Entropy gain from chelation increases complex stability.
Combustion of Alkanes
CₙH₂ₙ₊₂ + (3n+1)/2 O₂ → n CO₂ + (n+1) H₂O
Complete oxidation of saturated hydrocarbon to CO₂ and H₂O.
Degree of Unsaturation
DBE = (2C + 2 + N - H - X) / 2
Counts total number of π bonds and rings.
Variables: C = carbons, N = nitrogens, H = hydrogens, X = halogens
Markovnikov's Rule
In addition to unsymmetrical alkene, H adds to C with more H; halogen to C with fewer H
Predicts regioselectivity in HX addition to alkenes.
Acidity of Terminal Alkynes
R-C≡C-H; pKa ≈ 25 (acidic like phenol); C≡C-⁻ (acetylide anion)
Terminal alkynes are acidic; form nucleophilic carbanions.
Aromatic Criterion (Hückel's Rule)
(4n + 2) π electrons = aromatic (n = 0, 1, 2, 3...)
Benzene (6π), naphthalene (10π), anthracene (14π) are aromatic.
Variables: n = any non-negative integer
Friedel-Crafts Alkylation
C₆H₆ + RCl ⎯(AlCl₃)→ C₆H₅R + HCl
Substitution of aromatic H with alkyl group; requires Lewis acid catalyst.
Friedel-Crafts Acylation
C₆H₆ + RCOCl ⎯(AlCl₃)→ C₆H₅COR + HCl
Substitution with acyl group; forms ketone; no rearrangement.
Directing Effects: Ortho-Para vs Meta
Activating (−I, +R): −OH, −OR, −NR₂, −R = ortho/para directors
Electron-donating groups activate benzene; electron-withdrawing deactivate.
Wurtz Reaction
2R-X + 2Na ⎯(ether)→ R-R + 2NaX
Couple two alkyl halides using sodium; useful for symmetric products.
Cannizzaro Reaction
2 C₆H₅CHO + NaOH → C₆H₅CH₂OH + C₆H₅COONa
Disproportionation of aldehyde without α-H in presence of strong base.
Williamson Ether Synthesis
R-O⁻ + R'-X → R-O-R' + X⁻
SN2 reaction of alkoxide with alkyl halide; forms ether.
Aldol Condensation
2 R-CHO ⎯(base)→ R-CH(OH)-CH₂-CHO ⎯(heat)→ R-CH=CH-CHO + H₂O
Nucleophilic addition of enolate to aldehyde followed by elimination.
Clemmensen Reduction
R-CO-R' ⎯(Zn-Hg, HCl)→ R-CH₂-R'
Reduction of carbonyl to methylene; converts ketone/aldehyde to alkane.
Wolff-Kishner Reduction
R-CO-R' ⎯(NH₂NH₂, KOH, heat)→ R-CH₂-R'
Alternative to Clemmensen; reduction via hydrazone intermediate.
Sandmeyer Reaction
Ar-N₂⁺ Cl⁻ + CuCl ⎯(heat)→ Ar-Cl + N₂
Converts aryl diazonium to aryl halide using Cu(I) salt.
Nitration of Benzene
C₆H₆ + HNO₃ ⎯(H₂SO₄)→ C₆H₅NO₂ + H₂O
Electrophilic aromatic substitution; NO₂⁺ is electrophile.
Reduction with LiAlH₄
4 R-CHO + LiAlH₄ → 4 R-CH₂OH + Li[Al(OCH₂R)₄]
Powerful reducing agent; reduces aldehydes, ketones, carboxylic acids, esters.
Oxidation with KMnO₄
R-CH=CH-R ⎯(KMnO₄)→ R-CHOH-CHOH-R (cold); further oxidation (hot)
Oxidizes C=C bonds; cold = dihydroxylation, hot = cleavage.
Chromic Acid (Jones) Oxidation
R₂CH-OH ⎯(H₂CrO₄, H₂SO₄)→ R₂C=O; primary alcohols → carboxylic acids
Selective oxidation of alcohols; secondary → ketone, primary → carboxylic acid.
Reimer-Tiemann Reaction
Ar-OH + CHCl₃ + KOH → Ar-CHO (salicylaldehyde if ortho-directing OH)
Formylation of phenol; carbene intermediate.
Kolbe Reaction
Ar-O⁻ + CO₂ ⎯(heat, pressure)→ Ar-COOH
Carboxylation of phenoxide; high temperature and pressure required.
SN1 Mechanism (Unimolecular)
Rate = k[RX]; rate ∝ stability of carbocation; tertiary > secondary > primary
Two-step: formation of carbocation, then nucleophile attack; racemization.
SN2 Mechanism (Bimolecular)
Rate = k[RX][Nu]; primary > secondary >> tertiary; inversion of configuration
One-step: simultaneous bond breaking and forming; back-side attack.
E1 vs E2 Elimination
E1: two-step (carbocation formation); E2: one-step (anti-periplanar)
Competition with SN1/SN2; E1 at high temperature, SN1 at low temperature.
Carbohydrate General Formula
CₙH₂ₙOₙ; monosaccharide (n=3-7), disaccharide (two monosaccharides)
Polyhydroxy aldehydes or ketones; glucose is most common monosaccharide.
Mutarotation
α-D-glucose (36%) ⇌ open-chain form ⇌ β-D-glucose (64%)
Equilibration between anomers through open-chain hemiacetal.
Isoelectric Point (pI)
pI = (pKa1 + pKa2) / 2 (for amino acids with only COOH and NH₃⁺)
pH at which amino acid has zero net charge; varies by amino acid.
Zwitterion Form
⁺NH₃-CHR-COO⁻ (at physiological pH ≈ 7)
Amino acid exists with both positive and negative charges; dipolar ion.
Peptide Bond
R₁-CHNH-CO-CHR₂ + H₂O ⇌ R₁-COOH + H₂N-CHR₂
Amide linkage between amino acids; hydrolyzed by strong acid/base or enzymes.
Addition Polymerization
n (C=C) → polymer chain (-C-C-)
Monomers with C=C double bonds open up and link via radical or ionic mechanisms.
Condensation Polymerization
n (monomer with -OH and -COOH) → polyester + n H₂O
Monomers with two functional groups; each linkage releases small molecule.
R-S Nomenclature (Cahn-Ingold-Prelog)
Assign priorities based on atomic number (higher Z = higher priority) at chiral center
R/S designation independent of stereochemistry (D/L).
Relative Acidity Order
Carboxylic acid (pKa ≈ 4.7) > Phenol (pKa ≈ 10) > Alcohol (pKa ≈ 15)
Acidity depends on stability of conjugate base.
Free Radical Halogenation (Initiation-Propagation)
Initiation: X₂ → 2 X•; Propagation: R• + X₂ → RX + X•
Chain reaction producing alkyl halides; selectivity based on C-H bond strength.
Grignard Reaction
R-Mg-X + C=O → R-CH(OH)-R' (if aldehyde/ketone); R-COOH (if CO₂)
Nucleophilic addition of Grignard to carbonyl; Mg increases carbanion character.
Unit Cell Types
Primitive (SC): 1 atom; Body-centered (BCC): 2 atoms; Face-centered (FCC): 4 atoms; Hexagonal (HCP): 2 atoms
Basic repeating unit that defines crystal structure.
Packing Fraction (SC)
SF = 52.4% (Simple Cubic)
Fraction of space occupied by atoms in simple cubic.
Packing Fraction (BCC)
SF = 68% (Body-centered Cubic)
Fraction of space occupied by atoms in BCC.
Packing Fraction (FCC/HCP)
SF = 74% (Face-centered Cubic and Hexagonal Close Packing)
Highest packing efficiency for sphere packing.
Density of Crystal
ρ = (Z × M) / (a³ × Nₐ)
Z = number of formula units per unit cell, M = molar mass, a = lattice parameter.
Variables: Z = formula units per cell, M = molar mass (g/mol), a = edge length (cm), Nₐ = Avogadro's number
Coordination Numbers
SC = 6; BCC = 8; FCC = 12; HCP = 12
Number of nearest neighboring atoms.
Radius Ratio Rules (Ionic Compounds)
r⁺/r⁻: 0.225-0.414 (tetrahedral); 0.414-0.732 (octahedral); > 0.732 (cubic)
Predicts coordination number and geometry in ionic crystals.
Variables: r⁺ = cation radius, r⁻ = anion radius
Rock Salt (NaCl) Structure
Both Na⁺ and Cl⁻ in FCC arrangement; coordination number = 6
Each ion surrounded by 6 of opposite charge; cubic close packing.
CsCl Structure
Cs⁺ in BCC position, Cl⁻ at corners (or vice versa); coordination number = 8
Each ion coordinated by 8 oppositely charged ions.
Point Defects: Schottky and Frenkel
Schottky: paired cation-anion vacancy; Frenkel: cation moves to interstitial
Deviation from perfect lattice structure.
Semiconductors
Intrinsic: Si, Ge with Band gap 1-3 eV; Extrinsic: n-type (e⁻ donor), p-type (hole donor)
Partially filled/empty bands; electrical conductivity between insulator and conductor.
Types of Magnetism
Diamagnetic (↓ in B field); Paramagnetic (↑ in B field); Ferromagnetic (↑↑ persistent)
Behavior of materials in magnetic field; depends on electron configuration.
Taxonomic Hierarchy Acronym
KPCOFGS
Mnemonic for taxonomic ranks: Kingdom, Phylum, Class, Order, Family, Genus, Species
Variables: K = Kingdom, P = Phylum, C = Class, O = Order, F = Family, G = Genus, S = Species
Five Kingdom Classification
Monera | Protista | Fungi | Plantae | Animalia
Five kingdoms differentiated by cell type, cell wall presence, nutrition mode, and mobility
Binomial Nomenclature Rules
Genus species (italicized)
Genus capitalized, species lowercase. Example: Felis catus (cat)
TMV Structure
Length: 300 nm, Diameter: 15 nm, 2130 protein molecules
Tobacco Mosaic Virus dimensions and structure
Prokaryotic Cell Structures
Cell Wall | Membrane | 70S Ribosomes | Flagella | Pili | Nucleoid
Key structures in prokaryotes
Plant Kingdom Divisions
Algae | Bryophytes | Pteridophytes | Gymnosperms | Angiosperms
Major plant divisions
Plant Examples
Algae: Chlamydomonas | Bryophytes: Funaria | Pteridophytes: Fern | Gymnosperms: Pine | Angiosperms: Mango
Representative examples for each plant division
Major Animal Phyla
Porifera | Cnidaria | Platyhelminthes | Nematoda | Annelida | Arthropoda | Mollusca | Echinodermata | Chordata
Nine major animal phyla with increasing complexity
Chordata Characteristics
Notochord | Dorsal Hollow Nerve Cord | Pharyngeal Gill Slits | Segmented Body | Postanal Tail
Five defining features of Phylum Chordata
Five Vertebrate Classes
Pisces | Amphibia | Reptilia | Aves | Mammalia
Vertebrate classes with vertebral column
Viral Genome Types
DNA (ds/ss) | RNA (ds/ss) | Retroviruses (RNA→DNA)
Virus classification by genetic material
Bacterial Morphology
Cocci | Bacilli | Spirilla | Vibrios
Four basic bacterial cell shapes
Root Modifications
Tap Root: Carrot | Fibrous: Grass | Adventitious: Banyan
Root system variations for different functions
Stem Modifications
Rhizome: Ginger | Tuber: Potato | Bulb: Onion | Corm: Gladiolus | Stolon: Strawberry
Underground and above-ground stem modifications
Leaf Modifications
Tendril: Pea | Spines: Cactus | Scales: Asparagus | Pitcher: Nepenthes | Flytrap: Venus
Leaves modified for climbing, protection, nutrition
Plant Tissue Types
Meristematic | Dermal | Ground | Vascular
Four tissue types in plants
Dicot vs Monocot Root
Dicot: Solid Xylem + Pith | Monocot: Scattered Xylem, No Pith
Root cross-section differences
Dicot vs Monocot Stem
Dicot: Ring Arrangement | Monocot: Scattered
Stem vascular bundle organization
Floral Formula Format
K(n) C(n) A(n) G(n)
K=Calyx, C=Corolla, A=Androecium, G=Gynoecium
Malvaceae Formula
K5 C5 A(∞) G(5)
Hibiscus, Cotton floral formula
Brassicaceae Formula
K4 C4 A4+2 G2
Mustard, Radish floral formula
Fabaceae Formula
K5 C5 A(9)+1 G1
Pea, Bean floral formula
Asteraceae Formula
K0/C5 A5 G2 (Disc) | K(hair) C0 A0 G2 (Ray)
Sunflower, Dahlia composite flowers
Poaceae Formula
Lemma+Palea, A3, G1
Grass, Wheat floral formula
Animal Tissue Types
Epithelial | Connective | Muscular | Nervous
Four primary animal tissue types
Epithelial Tissues
Simple: Squamous, Cuboidal, Columnar | Stratified
Epithelial tissue classification by cell arrangement
Connective Tissues
Fibrous | Elastic | Adipose | Cartilage | Bone | Blood
Connective tissue types with extracellular matrix
Muscle Types
Skeletal (striated/voluntary) | Cardiac (striated/involuntary) | Smooth (non-striated/involuntary)
Three muscle tissue types
Nervous Tissue
Neurons | Glial Cells
Nervous tissue components
Leaf Anatomy
Dicot: Dorsiventral | Monocot: Isobilateral
Leaf cross-section organization
Ribosome Comparison
Prokaryotic: 70S (50S+30S) | Eukaryotic: 80S (60S+40S)
Ribosome size differences
Fluid Mosaic Model
Phospholipid Bilayer + Proteins + Cholesterol + Carbs
Cell membrane structure composition
Prokaryote vs Eukaryote
Prokaryote: No Nucleus | Circular DNA | 70S Ribosomes | No Organelles
Prokaryotic vs eukaryotic cell differences
Cell Cycle Phases
G1 (6-12 hrs) | S (6-8 hrs) | G2 (3-4 hrs) | M (1 hr)
Cell cycle phases and duration
Mitosis Stages
Prophase → Metaphase → Anaphase → Telophase
Four stages of mitotic division
Prophase Events
Condensation → Envelope breaks → Centrioles → Spindle
Events during prophase
Metaphase Events
Chromosomes align at metaphase plate
Chromosome alignment at cell equator
Meiosis I
Prophase I → Metaphase I → Anaphase I → Telophase I
First meiotic division reduces chromosome number
Meiosis II
Prophase II → Metaphase II → Anaphase II → Telophase II
Second meiotic division separates sister chromatids
Diploid Chromosome Numbers
Human: 46 | Fruit fly: 8 | Pea: 14 | Onion: 16 | Wheat: 42 | Chicken: 78
Chromosome numbers in various organisms
Mitochondria
ATP Production via Krebs + ETC
Powerhouse of cell; produces 36-38 ATP per glucose
Chloroplast
Light Reactions (Thylakoids) + Dark Reactions (Stroma)
Photosynthesis organelle in plant cells
Rough ER
Ribosomes attached → Protein synthesis
Rough ER synthesizes secretory proteins
Golgi Apparatus
Modification → Sorting → Packaging
Golgi modifies and packages proteins
Lysosome
Single membrane + ~40 hydrolytic enzymes
Digestive organelle with acid hydrolases
Nucleus Components
Nuclear Envelope | Nucleoplasm | Nucleolus | Chromatin
Nuclear structure and components
Cytokinesis
Animal: Cleavage furrow | Plant: Cell plate
Cytoplasmic division differences
Cell Dimensions
Prokaryotic: 1-10 μm | Eukaryotic: 10-100 μm | Animal: 20-30 μm
Typical cell sizes
EC Classification
EC1: Oxidoreductases | EC2: Transferases | EC3: Hydrolases | EC4: Lyases | EC5: Isomerases | EC6: Ligases
Six enzyme classes based on reaction type
Oxidoreductases
Dehydrogenases | Oxidases | Peroxidases
EC1 enzyme examples
Hydrolases
Proteases | Lipases | Amylases | Nucleases
EC3 hydrolytic enzyme examples
Amino Acid Structure
H2N-CHR-COOH
General amino acid structure
Peptide Bond
R1-CHN-H + H-O-CO-R2 → R1-CHN-CO-O-R2 + H2O
Peptide bond formation between amino acids
Reducing vs Non-Reducing
Reducing: Monosaccharides + most Disaccharides | Non-reducing: Sucrose
Sugar classification by reducing power
Glycosidic Bond
C1-OH + HO-C4' → C1-O-C4' + H2O
Glycosidic bond formation in disaccharides
Nucleotide Structure
Pentose Sugar + Nitrogenous Base + Phosphate
Building block of nucleic acids
DNA vs RNA
DNA: Deoxyribose+Thymine+Double helix | RNA: Ribose+Uracil+Single strand
Key differences between DNA and RNA
Chargaff Rules
A=T | G=C | Purines=Pyrimidines
Base pairing ratios in DNA
Protein Structure
Primary (sequence) → Secondary (helix/sheet) → Tertiary (3D) → Quaternary (subunits)
Four levels of protein organization
Michaelis-Menten
v = (Vmax[S])/(Km+[S])
Enzyme kinetics equation
Enzyme Inhibition
Competitive | Non-competitive | Irreversible
Types of enzyme inhibitors
Storage Polysaccharides
Starch (plants) | Glycogen (animals)
Energy storage molecules
Lipid Types
Fats/Oils | Phospholipids | Steroids | Waxes
Lipid classification
RNA Types
mRNA (messenger) | tRNA (transfer) | rRNA (ribosomal)
Functional RNA types
DNA Packaging
DNA → Nucleosome → 30nm fiber → Chromatin → Chromosome
DNA compaction levels
Nucleosome
147 bp DNA + histone octamer
Basic chromatin repeat unit
Photosynthesis Equation
6CO2 + 6H2O + light → C6H12O6 + 6O2
Overall photosynthetic reaction
Light Reactions
H2O + NADP+ + ADP → O2 + NADPH + ATP
Light-dependent reactions in thylakoids
Calvin Cycle
3CO2 + 9ATP + 6NADPH → Glucose + 9ADP + 8Pi + 6NADP+
Dark reactions in stroma
C3 vs C4 Plants
C3: Direct Calvin cycle | C4: Hatch-Slack pathway
Photosynthetic pathway differences
Photorespiration
RuBisCO+O2 → 3-PG+2-phosphoglycolate
Wasteful oxygenation reaction
Respiration Equation
C6H12O6 + 6O2 → 6CO2 + 6H2O + 2870 kJ
Aerobic respiration overall equation
Glycolysis ATP
Glucose → 2 Pyruvate + 2 ATP (net) + 2 NADH
Glycolysis ATP yield
Krebs Cycle ATP
2 Acetyl-CoA → 2 ATP + 6 NADH + 2 FADH2 + 4 CO2
Citric acid cycle ATP and electron carrier yield
ETC ATP Yield
NADH → 2.5 ATP | FADH2 → 1.5 ATP
Electron transport chain ATP production
Total ATP Yield
36-38 ATP per glucose (~30-32 modern estimate)
Total ATP from aerobic respiration
Respiratory Quotient
RQ = CO2 released / O2 consumed
RQ values: carbs=1.0, protein=0.8-0.9, fat=0.7
Blackman Law
Rate = function of least available factor
Law of limiting factors in photosynthesis
Compensation Point
Photosynthesis rate = Respiration rate
Net gas exchange point in plants
Absorption Spectrum
Wavelengths absorbed by pigments
vs Action spectrum effectiveness
Auxin
IAA - promotes cell elongation, apical dominance
Indole-3-acetic acid functions
Gibberellin
~136 GAs - promotes elongation, seed germination, flowering
Gibberellic acid functions
Cytokinin
Promotes cell division, shoot development
Cytokinin functions
Ethylene
C2H4 - promotes ripening, abscission, senescence
Ethylene hormone functions
Abscisic Acid
ABA - stress hormone, stomatal closure, seed dormancy
ABA stress response functions
Transpiration
Water pulls up via evaporation + root pressure
Transpiration pull mechanism
Photosynthetic Pigments
Chlorophyll a | Chlorophyll b | Xanthophyll | Carotenoid
Pigment types in photosynthesis
Nitrogen Fixation
N2 + 8H+ + 8e- → 2NH3 (16 ATP)
Bacterial nitrogen fixation process
Tidal Volume
TV = 500 mL
Normal breath volume at rest
Inspiratory Reserve Volume
IRV = 2500-3100 mL
Maximum inhalation after normal inspiration
Expiratory Reserve Volume
ERV = 1000-1100 mL
Maximum exhalation after normal expiration
Residual Volume
RV = 1100-1200 mL
Air remaining in lungs after maximal expiration
Vital Capacity
VC = TV+IRV+ERV = 3400-4800 mL
Maximum air in/out of lungs
Total Lung Capacity
TLC = VC+RV = 6000 mL approx
Total air-holding capacity
Blood Composition
Plasma: 54-55% | RBC: 40-45% | WBC: ~1% | Platelets: <1%
Blood cell and plasma percentages
Plasma Components
Water: 90% | Proteins: 7% | Salts: 0.9% | Glucose: 0.1%
Plasma composition breakdown
RBC Count
Male: 4.5-5.5M/mm3 | Female: 4.0-5.0M/mm3
Normal red blood cell count
Cardiac Output
CO = SV × HR
Stroke volume times heart rate
Blood Pressure
Normal: 120/80 mmHg (Systolic/Diastolic)
Normal blood pressure values
ECG Waves
P wave (atrial) | QRS (ventricular) | T wave (repolarization)
Electrocardiogram waves
Glomerular Filtration Rate
GFR = 125 mL/min
Rate of kidney blood filtration
Urine Composition
Water: 95% | Urea: 2% | Salts: 2% | Others
Normal urine components
Resting Potential
-70 mV (inside negative)
Resting membrane potential
Action Potential Peak
+30 mV (depolarization peak)
Action potential maximum value
Sarcomere
Z disc to Z disc unit with myosin/actin/H zone
Functional muscle contraction unit
Pancreatic Hormones
Insulin (lowers glucose) | Glucagon (raises glucose)
Blood glucose regulation hormones
Thyroid Hormones
T3 (Triiodothyronine) + T4 (Thyroxine)
Metabolic rate regulation hormones
Adrenal Hormones
Cortex: Cortisol, Aldosterone | Medulla: Adrenaline, Noradrenaline
Adrenal gland hormones
Digestive Enzymes
Amylase (carbs) | Pepsin (proteins) | Lipase (fats)
Major digestive enzyme examples
Enzyme Specificity
Amylase→Starch | Protease→Peptides | Lipase→Triglycerides
Enzyme-substrate specificity
Liver Functions
Glycogen storage | Protein synthesis | Detoxification | Bile
Major liver metabolic functions
Lymphocytes
T cells (cell-mediated) | B cells (antibodies) | NK cells
White blood cell types
Hemoglobin
Hb + 4O2 ⇌ Hb(O2)4
Hemoglobin oxygen binding
Blood pH Buffer
pH 7.35-7.45 | H2CO3/HCO3- buffer
Blood pH regulation system
Double Fertilization
Pollen(n)+Egg(n)→Zygote(2n) | Polar(2n)+Sperm(n)→Endosperm(3n)
Double fertilization in angiosperms
Menstrual Cycle
Menses(1-5) | Follicular(1-13) | Ovulation(14) | Luteal(15-28)
28-day menstrual cycle phases
Spermatogenesis
Primary(2n)→Secondary(n)→Spermatids(n)→Spermatozoa
Male gamete formation in 74 days
Oogenesis
Oogonia(2n)→Primary(2n, Prophase I arrested)
Female gamete formation
Spermatogenesis vs Oogenesis
Sperm: Continuous, 4 gametes, 74 days | Oocyte: Cyclic, 1 gamete
Comparison of male and female gametogenesis
Embryonic Stages
Fertilization→Zygote→Cleavage→Blastula→Gastrulation→Neurulation
Early embryonic development stages
Germ Layers
Ectoderm (skin, nervous) | Mesoderm (muscles, bones) | Endoderm (GI, respiratory)
Three germ layers and derivatives
Placental Hormones
hCG (maintain CL) | Progesterone (maintain pregnancy) | Estrogen (labor)
Pregnancy-sustaining hormones
Placental Structure
Trophoblast→Syncytiotrophoblast | Blood barrier
Placenta structure and function
Barrier Methods
Condom | Diaphragm | Cervical cap | Spermicide
Physical barrier contraceptive methods
Hormonal Methods
Pills | Patches | Injections | Implants | Hormonal IUD
Hormonal contraceptive methods
Assisted Reproduction
IVF | ZIFT | GIFT | Surrogacy
Assisted reproductive technologies
Sex Determination
XX=Female | XY=Male
Human sex determination system
Lactation
Prolactin (production) | Oxytocin (letdown)
Milk production and secretion hormones
Pregnancy Duration
~280 days (40 weeks LMP) | ~266 days (38 weeks fertilization)
Human pregnancy length
Labor Stages
Stage 1: Dilation | Stage 2: Delivery | Stage 3: Placenta
Stages of labor in childbirth
Law of Segregation
Alleles separate during gamete formation
Mendel's first law
Law of Independent Assortment
Genes assort independently
Mendel's second law
Monohybrid Cross
Aa × Aa → 1 AA : 2 Aa : 1 aa (3:1 phenotypic)
Single gene cross results
Dihybrid Cross
AaBb × AaBb → 9:3:3:1 ratio
Two gene cross results
Test Cross
Dominant phenotype × homozygous recessive → 1:1:1:1
Test cross for linkage detection
Incomplete Dominance
Aa shows intermediate phenotype (1:2:1)
Neither allele fully dominant
Codominance
IAIB = AB blood type (both expressed)
Both alleles fully expressed
ABO Blood Groups
IAIA/IAi=A | IBIB/IBi=B | IAIB=AB | ii=O
Blood group genetics
Sex-Linked Inheritance
XH/Xh/XY males and females different ratios
X-linked trait inheritance
X-linked Recessive
XBXh × XBY → 1XBXh : 1XBY : 1XhY carriers/affected
Carrier female cross results
Hardy-Weinberg
p2 + 2pq + q2 = 1
Population genetic equilibrium
HW Conditions
No mutation | No selection | Random mating | Large pop | No migration
Conditions for genetic equilibrium
Genetic Code
64 codons (triplet, universal, degenerate)
Genetic code properties
Start/Stop Codons
Start: AUG | Stop: UAA, UAG, UGA
Translation initiation and termination
Semiconservative Replication
Original + new strand in each daughter DNA
DNA replication mechanism
Base Pairing
A-T (2 bonds) | G-C (3 bonds)
DNA base pairing specificity
Griffith Experiment
S strain + R strain → R transformed to S
DNA as genetic material proof
Hershey-Chase
32P (DNA) inside bacteria, 35S (protein) outside
DNA confirmed as genetic material
Lac Operon Repressed
No lactose → repressor on operator → no transcription
Lac operon OFF state
Lac Operon Induced
Lactose present → repressor off → transcription
Lac operon ON state
Mutation Types
Point | Insertion | Deletion | Inversion | Translocation
Types of DNA mutations
Genetic Linkage
Same chromosome genes inherit together (<50% recombination)
Linked gene inheritance
Miller-Urey Experiment
CH4+NH3+H2O+H2+lightning → amino acids
Abiogenesis simulation experiment
Geological Time
Hadean→Archean→Proterozoic→Phanerozoic
Geological eons and major events
Homologous vs Analogous
Homologous: similar origin, different function | Analogous: different origin, similar function
Structural comparison evidence
Vestigial Structures
Appendix, coccyx, wisdom teeth remnants
Evolutionary history evidence
Natural Selection Types
Directional | Stabilizing | Disruptive
Types of natural selection
Human Evolution
Australopithecus(400-500 cc) → Homo habilis(500-800 cc) → Homo erectus(800-1200 cc) → Neanderthal/Sapiens(1200-1400 cc)
Human brain size increase
Reproductive Isolation
Prezygotic (prevent mating) | Postzygotic (zygote fails)
Speciation isolation mechanisms
Adaptive Radiation
One species → multiple species with different adaptations
Evolution in new environments (Darwin's finches)
Pathogens
Bacteria: Cholera, TB | Virus: Polio, COVID-19, HIV
Disease-causing organisms
Human Parasites
Plasmodium (malaria, mosquito) | Leishmania (kala-azar) | Trypanosoma (sleeping sickness)
Parasitic diseases and vectors
Immunity Types
Innate (non-specific) | Acquired (specific)
Immune system classification
Antibodies
IgG (secondary) | IgM (primary) | IgA (secretory) | IgE (allergy) | IgD (activation)
Five immunoglobulin classes
Humoral Response
Antigen → Th → B cells → plasma + memory
Antibody-mediated immune response
Cell-Mediated
MHC I → Tc → destroy infected cells
T-cell mediated immune response
Cancer Features
Unlimited division | Loss of differentiation | Metastasis
Cancer cell characteristics
HIV Structure
Lipid bilayer + gp120/gp41 + reverse transcriptase
Retrovirus structure
AIDS Stage
CD4+ count <200 cells/mm3
AIDS definition
Drug Types
Analgesics | Antipyretics | Antibiotics | Antivirals
Drug classification by effect
Vaccination
Attenuated/Inactivated pathogen → immune memory
Vaccination mechanism
Allergic Response
Allergen → IgE → mast cell → histamine
Type I hypersensitivity
Restriction Enzyme Naming
EcoRI: E(genus) + co(species) + R(strain) + I(number)
Restriction enzyme nomenclature
Sticky vs Blunt Ends
Sticky: overhangs for pairing | Blunt: no overhangs
DNA end types from restriction enzymes
PCR Cycle
Denaturation(94-95C) | Annealing(50-65C) | Extension(72C)
Polymerase Chain Reaction steps
pBR322 Features
ori | ampR | tetR | MCS
Plasmid vector characteristics
pBR322 Cloning
Insert at MCS → cut tetR → select ampR, screen tet
Insertional inactivation cloning
Bt Cotton
CryIAc (Lepidoptera) | CryIIAb (Diptera)
Cry proteins in genetically engineered cotton
ADA Gene Therapy
ADA deficiency → SCID → first human gene therapy (1990)
First successful human gene therapy
Insulin Production
Gene → Plasmid → E.coli → Inclusion bodies → Purification
Recombinant insulin production
DNA Fingerprinting
Extract → Restrict → Gel → Southern → Probe → Compare
DNA profiling technique steps
VNTR Analysis
VNTRs = tandem repeats at loci, length varies
Variable Number Tandem Repeats
Plant Tissue Culture
Explant → Dedifferentiation → Callus → Organogenesis → Plant
In vitro plant propagation
Exponential Growth
dN/dt = rN
Unlimited population growth model
Logistic Growth
dN/dt = rN[(K-N)/K]
Population growth limited by carrying capacity
Growth Rate
λ=1 (stable) | >1 (growing) | <1 (declining)
Finite rate of increase
10% Law
Each level retains ~10%, ~90% lost as heat
Energy transfer efficiency
Energy Pyramid
Always upright: producers > herbivores > carnivores
Pyramid of energy shape
Biomass Pyramid
Usually upright, inverted in ocean (phytoplankton)
Pyramid of biomass shape
Numbers Pyramid
Upright or inverted depending on organism size
Pyramid of numbers shape variability
Species-Area
log S = log C + Z log A
Species diversity vs area relationship
Hotspots
>1500 endemic plants + <30% original vegetation
Biodiversity hotspot criteria
Food Chain
Producer → Primary → Secondary → Tertiary
Linear energy transfer
Food Web
Multiple interconnected food chains
Complex feeding relationships
Carbon Cycle
CO2 → photosynthesis → respiration/decomposition → CO2
Carbon cycling through ecosystems
Nitrogen Cycle
N2 → fixation → NO3- → assimilation → organic N → denitrification
Nitrogen cycling in ecosystems
Phosphorus Cycle
Rocks → Soil → Plants → Animals → Decomposition
Phosphorus cycling (no atmospheric phase)
Primary Productivity
GPP (total) | NPP = GPP - Respiration
Ecosystem energy production
Biome Types
Tropical forest → Savanna → Grassland → Desert → Temperate → Taiga → Tundra
Major terrestrial biomes
Species Interactions
Predation(+/-) | Competition(-/-) | Mutualism(+/+) | Commensalism(+/0) | Parasitism(+/-) | Amensalism(-/0)
Types of ecological interactions
Ecological Succession
Primary: bare rock → lichens → moss → grass → shrubs → forest
Primary succession stages