Equation Library
Explore a comprehensive collection of equations across Physics, Chemistry, and Engineering. Each equation is presented with a brief explanation to aid your understanding and application.
PHYSICS
Ohm's Law
V = IR
Relates voltage, current, and resistance.
Newton's Second Law
F = ma
Force equals mass times acceleration.
Gravitational Force
F = G(m₁m₂)/r²
Attractive force between two masses.
Kinetic Energy
KE = ½mv²
Energy due to motion.
Potential Energy
PE = mgh
Energy due to position in a gravitational field.
Work Done
W = Fd cos(θ)
Work done by a force over a distance.
Power
P = W/t
Rate of doing work or transferring energy.
Wave Speed
v = fλ
Speed of a wave based on frequency and wavelength.
Coulomb's Law
F = k(q₁q₂)/r²
Force between two charges.
Electric Field
E = F/q
Force experienced by a unit charge.
Magnetic Force
F = qvB sin(θ)
Force on a charge in a magnetic field.
Momentum
p = mv
Product of mass and velocity.
Impulse
J = Δp
Change in momentum.
Hooke's Law
F = kx
Force exerted by a spring.
Period of a Pendulum
T = 2π√(L/g)
Time for one oscillation of a pendulum.
Thermal Energy
Q = mcΔT
Energy transferred due to temperature change.
Entropy Change
ΔS = Q/T
Change in disorder of a system.
Einstein's Equation
E = mc²
Energy-mass equivalence.
Planck's Equation
E = hf
Energy of a photon.
Snell's Law
n₁sin(θ₁) = n₂sin(θ₂)
Refraction of light at an interface.
Torque
τ = rF sin(θ)
Rotational force applied to an object.
Angular Momentum
L = Iω
Momentum of a rotating object.
Capacitance
C = Q/V
Charge stored per unit voltage.
Energy Stored in a Capacitor
E = ½CV²
Energy in an electric field.
Thermal Conductivity
q = -kA(dT/dx)
Heat transfer through a material.
Lenz's Law
ε = -dΦ/dt
Induced EMF opposes the change in flux.
Bohr's Equation
E = -13.6/n² eV
Energy levels of a hydrogen atom.
Photoelectric Effect
Kmax = hf - Φ
Maximum kinetic energy of ejected electrons.
Relativistic Energy
E² = (pc)² + (mc²)²
Energy including momentum and rest mass.
Lorentz Force
F = q(E + v × B)
Force on a charge in electromagnetic fields.
Doppler Effect
f' = f(v±v₀)/(v±vs)
Frequency shift due to relative motion.
Escape Velocity
v = √(2GM/R)
Velocity needed to escape a gravitational field.
Angular Frequency
ω = 2πf
Frequency in radians per second.
Diffraction
dsinθ = nλ
Condition for constructive interference in a grating.
Intensity of Light
I = P/A
Power per unit area of a wave.
Refractive Index
n = c/v
Ratio of speed of light in a vacuum to a medium.
Mass-Energy Relation
E = mc²
Energy equivalent of mass in relativity.
Specific Heat Capacity
c = Q/mΔT
Heat energy required to raise temperature.
Fluid Pressure
P = ρgh
Pressure at depth in a fluid.
Stokes' Law
F = 6πηrv
Drag force on a spherical object in fluid.
Heat Transfer Rate
Q/t = kA(ΔT/L)
Rate of heat conduction.
Bragg's Law
nλ = 2dsinθ
Condition for X-ray diffraction.
Maxwell's First Equation
∇·E = ρ/ε₀
Gauss's law for electricity.
Maxwell's Second Equation
∇·B = 0
No magnetic monopoles exist.
Maxwell's Third Equation
∇ × E = -dB/dt
Faraday's law of induction.
Maxwell's Fourth Equation
∇ × B = μ₀J + μ₀ε₀dE/dt
Ampere's law with Maxwell's correction.
Planck's Radiation Law
Eλ = (8πhc/λ⁵) / (e^(hc/λkT)-1)
Spectral distribution of black body radiation.
Thermodynamic Efficiency
η = (T₁ - T₂)/T₁
Efficiency of a heat engine.
Radioactive Decay
N = N₀e^(-λt)
Number of atoms remaining after time t.
Hubble's Law
v = H₀d
Recession speed of galaxies proportional to distance.
CHEMISTRY
Ideal Gas Law
PV = nRT
Relationship between pressure, volume, and temperature of an ideal gas.
Boyle's Law
P₁V₁ = P₂V₂
Pressure and volume relationship at constant temperature.
Charles' Law
V₁/T₁ = V₂/T₂
Volume and temperature relationship at constant pressure.
Avogadro's Law
V/n = constant
Volume is proportional to the number of moles at constant pressure and temperature.
Dalton's Law of Partial Pressures
Ptotal = P₁ + P₂ + ... + Pn
Total pressure is the sum of the partial pressures.
Combined Gas Law
(P₁V₁)/T₁ = (P₂V₂)/T₂
Combines Boyle's, Charles', and Gay-Lussac's laws.
Van der Waals Equation
[P + a(n/V)²](V - nb) = nRT
Modification of the ideal gas law for real gases.
Mole Fraction
X = n₁/n_total
Ratio of moles of a substance to the total moles in a mixture.
Raoult's Law
Psolution = Xsolvent × Psolvent
Vapor pressure of a solution depends on mole fraction.
Henry's Law
C = kP
Solubility of gas is proportional to pressure.
Beer-Lambert Law
A = εlc
Absorbance of light is proportional to concentration.
Arrhenius Equation
k = Ae^(-Ea/RT)
Relationship between rate constant and activation energy.
Nernst Equation
E = E⁰ - (RT/nF)lnQ
Electrode potential depends on concentration and temperature.
Gibbs Free Energy
ΔG = ΔH - TΔS
Energy available to do work in a chemical reaction.
Equilibrium Constant
K = [products]/[reactants]
Ratio of concentrations of products and reactants at equilibrium.
Le Chatelier's Principle
Shift ∝ ΔConditions
System shifts to counteract a change in equilibrium conditions.
Entropy Change
ΔS = qrev/T
Change in disorder during a reversible process.
Enthalpy Change
ΔH = q at constant P
Heat exchanged at constant pressure.
Heat of Reaction
q = mcΔT
Heat absorbed or released during a reaction.
Specific Heat Capacity
c = q/mΔT
Heat required to change the temperature of a unit mass.
Hess's Law
ΔHreaction = ΣΔHproducts - ΣΔHreactants
Total enthalpy change is the sum of individual steps.
Faraday's Law of Electrolysis
m = (ItM)/(nF)
Mass of substance deposited depends on current and time.
Electrochemical Cell Potential
Ecell = Ered - Eox
Cell potential is the difference between reduction and oxidation potentials.
Colligative Properties
ΔT = Kb × m
Boiling point elevation depends on molality.
Osmotic Pressure
π = MRT
Pressure required to stop osmosis.
Rate Law
Rate = k[A]ᵐ[B]ⁿ
Rate of reaction depends on concentration of reactants.
Half-Life
t₁/₂ = 0.693/k
Time required for half the reactant to decay.
pH Formula
pH = -log[H⁺]
Measure of acidity.
pOH Formula
pOH = -log[OH⁻]
Measure of basicity.
Relationship between pH and pOH
pH + pOH = 14
Sum of pH and pOH in aqueous solution.
Ka and pKa
pKa = -logKa
Strength of an acid.
Kb and pKb
pKb = -logKb
Strength of a base.
Kw Formula
Kw = [H⁺][OH⁻]
Autoionization constant of water.
Buffers
pH = pKa + log([A⁻]/[HA])
pH of a buffer solution.
Redox Reaction Balancing
Oxidation = Reduction
Electrons lost equal electrons gained.
Molarity
M = moles/L
Concentration of a solution.
Molality
m = moles/kg
Moles of solute per kilogram of solvent.
Percent Composition
% = (mass of element/total mass) × 100
Percentage of each element in a compound.
Empirical Formula
Ratio of elements
Simplest ratio of elements in a compound.
Molecular Formula
n × empirical formula
Actual number of atoms in a molecule.
Avogadro's Number
6.022 × 10²³
Number of particles in a mole.
Atomic Mass
Weighted Average
Weighted average of isotopic masses.
Reaction Quotient
Q = [products]/[reactants]
Indicates reaction direction.
Lattice Energy
U = k(q₁q₂)/r
Energy required to separate ionic lattice.
Hydrolysis
AB + H₂O → AOH + HB
Reaction with water to form hydroxide and acid.
Bond Energy
ΔH = ΣBond broken - ΣBond formed
Energy required to break or form bonds.
Limiting Reagent
Reactant consumed first
Determines maximum product formed.
Percentage Yield
Yield = (Actual/Theoretical) × 100
Efficiency of a reaction.
Energy of a Photon
E = hf
Energy associated with a photon.
Thermodynamic Efficiency
η = (W/Qh) × 100
Efficiency of a heat engine.
ENGINEERING
Stress
σ = F/A
Force per unit area on a material.
Strain
ε = ΔL/L₀
Deformation per unit original length.
Young's Modulus
E = σ/ε
Material's stiffness or elasticity.
Hooke's Law (Engineering)
σ = Eε
Stress is proportional to strain.
Moment of Inertia
I = ∫y²dA
Resistance to bending or rotation.
Beam Deflection
δ = FL³/3EI
Deflection of a beam under load.
Shear Stress
τ = VQ/It
Stress due to shear force.
Thermal Expansion
ΔL = αL₀ΔT
Change in length due to temperature.
Torque
T = Fd
Rotational force times distance.
Power (Mechanical)
P = τω
Rotational power in a system.
Fluid Flow Rate
Q = Av
Volume of fluid flowing per unit time.
Bernoulli’s Equation
P + ½ρv² + ρgh = constant
Energy conservation in fluid flow.
Reynolds Number
Re = ρvd/μ
Dimensionless number for flow type.
Frictional Force
F = μN
Force opposing motion due to friction.
Heat Transfer
Q = kAΔT/L
Conduction of heat through a material.
Fourier's Law
q = -k∇T
Rate of heat conduction.
Stress Concentration
K = σmax/σnominal
Ratio of maximum to nominal stress.
Safety Factor
n = σfailure/σworking
Design safety measure.
Efficiency of a Machine
η = (Pout/Pin) × 100
Ratio of output to input power.
Electric Power
P = VI
Power in an electrical circuit.
Heat Exchanger Effectiveness
ε = Qactual/Qmax
Efficiency of heat transfer.
Hydraulic Power
P = QΔP
Power in a hydraulic system.
Gear Ratio
GR = Nout/Nin
Ratio of gear teeth.
Specific Energy
SE = E/m
Energy per unit mass.
Buckling Load
Pcr = π²EI/L²
Critical load causing buckling.
Thermal Efficiency
η = (Qin - Qout)/Qin
Ratio of useful energy to input energy.
Compressive Strength
σ = F/A
Material strength under compression.
Fluid Drag Force
F = ½ρv²CdA
Force exerted by fluid flow.
Heat Capacity
C = Q/ΔT
Heat energy required to raise temperature.
Pipe Flow Velocity
v = Q/A
Velocity of fluid in a pipe.
Thermodynamic Work
W = -PΔV
Work done by a system.
Boiling Heat Transfer
q = hA(Tsurface - Tsaturation)
Rate of heat transfer during boiling.
Stress-Strain Relationship
σ = Eε
Stress proportional to strain within elastic limit.
Truss Force
F = Tcosθ
Force in a truss member.
Voltage Drop
V = IR
Voltage loss across a resistor.
Thermal Resistance
R = L/kA
Resistance to heat flow in a material.
Critical Reynolds Number
Re = 2300
Value determining laminar to turbulent flow.
Tensile Stress
σ = F/A
Stress due to tension.
Shear Modulus
G = τ/γ
Material's resistance to shear.
Specific Weight
γ = ρg
Weight per unit volume of a fluid.
Hydraulic Head
H = P/γ
Height equivalent of fluid pressure.
Moment of a Force
M = Fd
Rotational effect of a force.
Pipe Head Loss
hf = f(L/D)(v²/2g)
Energy loss due to friction in pipes.
Rotational Kinetic Energy
KErot = ½Iω²
Energy due to rotation.
Fluid Power
P = ρgQH
Power in fluid systems.
Axial Load
P = σA
Load applied along the axis of a member.
Welding Heat Input
H = VI/S
Heat transferred during welding.
Circuit Resistance
Rtotal = R₁ + R₂ + ...
Total resistance in a series circuit.
Inductor Energy
E = ½LI²
Energy stored in an inductor.
Capacitor Energy
E = ½CV²
Energy stored in a capacitor.