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Thermodynamics Formula Sheet

All thermodynamics equations: gas laws, heat transfer, entropy, Carnot efficiency, and statistical mechanics.

Gas Laws

Ideal Gas Law
PV = nRT
P=pressure (Pa), V=volume (m³), n=moles, R=8.314 J/(mol·K), T in K
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Boyle's Law (const T)
P₁V₁ = P₂V₂
Isothermal: pressure and volume are inversely proportional
Charles's Law (const P)
V₁/T₁ = V₂/T₂
Isobaric: volume proportional to temperature (K)
Combined Gas Law
P₁V₁/T₁ = P₂V₂/T₂
Applies when all three variables change
RMS Molecular Speed
v_rms = √(3RT/M)
M=molar mass (kg/mol), T=temperature (K)
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Average KE per Molecule
KE_avg = (3/2)k_BT
k_B=1.381×10⁻²³ J/K; temperature is KE
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Pressure from KT
P = (1/3)(N/V)mv²_rms
N=number of molecules, V=volume
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First Law of Thermodynamics

First Law
ΔU = Q − W
ΔU=internal energy change, Q=heat added to system, W=work BY system
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Work by Gas (isobaric)
W = PΔV
P=constant pressure, ΔV=volume change (m³)
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Isothermal (ΔT=0)
ΔU=0 → Q=W
All heat input converts to work
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Adiabatic (Q=0)
ΔU = −W
Internal energy decreases as gas does work
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Isochoric (ΔV=0)
W=0 → ΔU=Q
All heat goes to internal energy
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Internal Energy (ideal)
ΔU = nCᵥΔT
Cᵥ=(3/2)R monatomic; (5/2)R diatomic
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Second Law and Entropy

Entropy Change
ΔS = Q_rev/T
For reversible heat transfer at constant T (in Kelvin)
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Second Law
ΔS_universe ≥ 0
Entropy never decreases for isolated systems
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Statistical Entropy
S = k_B ln(Ω)
Ω=number of microstates; k_B=Boltzmann constant
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Carnot Efficiency
η = 1 − T_c/T_h
T_c=cold temp (K), T_h=hot temp (K); max possible efficiency
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COP (Refrigerator)
COP = T_c/(T_h−T_c)
Coefficient of performance; heat moved per work input
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Heat Engine Efficiency
η = 1 − Q_c/Q_h = W/Q_h
η<η_Carnot for any real engine
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Heat Transfer

Specific Heat Capacity
Q = mcΔT
m=mass (kg), c=specific heat (J/kg·K), ΔT in K or °C
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Latent Heat
Q = mL
L=latent heat (J/kg); no temperature change during phase transition
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Thermal Conduction
P = kAΔT/d
k=thermal conductivity (W/m·K), A=area, d=thickness
Stefan-Boltzmann
P = εσAT⁴
σ=5.67×10⁻⁸ W/(m²·K⁴), ε=emissivity