Thermodynamics relies on a core set of mathematical equations that describe energy conservation, entropy, heat transfer, gas behavior, and the efficiency limits of thermal machines. Mastering these formulas is essential for solving problems in physics, chemistry, and engineering. This guide organizes the most important thermodynamics formulas by topic, explains what each variable represents, and describes when each formula applies. Keeping these equations accessible and understanding their conditions of validity will make thermodynamic problem solving systematic rather than haphazard.
First Law and Energy Formulas
dU = Q - W
This is the first law of thermodynamics. dU is the change in internal energy, Q is heat added to the system, and W is work done by the system. For a cyclic process, dU = 0, so Q{sub}net{/sub} = W{sub}net{/sub}. The sign convention requires care: Q is positive for heat flowing in, W is positive for work flowing out.
W = integral P dV
This gives the work done by a gas during expansion or compression. For an isobaric process: W = P delta V. For an isothermal ideal gas: W = nRT ln(V{sub}2{/sub}/V{sub}1{/sub}). For an adiabatic ideal gas: W = (P{sub}1{/sub}V{sub}1{/sub} - P{sub}2{/sub}V{sub}2{/sub})/(gamma - 1). For an isochoric process: W = 0.
q = mc delta T
This is the basic heat transfer equation for temperature changes. q is the heat transferred, m is mass, c is specific heat capacity, and delta T is the temperature change. For phase changes at constant temperature: q = mL, where L is the latent heat (of fusion, vaporization, or sublimation).
Ideal Gas and Process Formulas
PV = nRT
The ideal gas law relates pressure (P), volume (V), moles (n), and temperature (T) through the gas constant R = 8.314 J/(mol K). For comparing two states of the same gas: P{sub}1{/sub}V{sub}1{/sub}/T{sub}1{/sub} = P{sub}2{/sub}V{sub}2{/sub}/T{sub}2{/sub}. Temperature must be in Kelvin.
PVgamma = constant
This describes a reversible adiabatic process for an ideal gas. gamma = C{sub}p{/sub}/C{sub}v{/sub} is the heat capacity ratio (1.4 for diatomic gases like air, 5/3 for monatomic gases). Equivalent forms: TV
gamma-1 = constant, and T P
(1-gamma)/gamma = constant.
For ideal gas heat capacities: monatomic C{sub}v{/sub} = (3/2)R, C{sub}p{/sub} = (5/2)R. Diatomic (moderate T): C{sub}v{/sub} = (5/2)R, C{sub}p{/sub} = (7/2)R. The general relation C{sub}p{/sub} - C{sub}v{/sub} = R holds for all ideal gases. Internal energy of an ideal gas: U = nC{sub}v{/sub}T. Enthalpy: H = nC{sub}p{/sub}T.
Entropy and Second Law Formulas
dS = dQ{sub}rev{/sub} / T
This defines the entropy change for a reversible process. For irreversible processes: dS > dQ/T. Boltzmann statistical definition: S = k ln W, where k = 1.38 x 10
-23 J/K and W is the number of microstates.
For an ideal gas changing from state 1 to state 2: delta S = nC{sub}v{/sub} ln(T{sub}2{/sub}/T{sub}1{/sub}) + nR ln(V{sub}2{/sub}/V{sub}1{/sub}), or equivalently delta S = nC{sub}p{/sub} ln(T{sub}2{/sub}/T{sub}1{/sub}) - nR ln(P{sub}2{/sub}/P{sub}1{/sub}). For isothermal expansion of an ideal gas: delta S = nR ln(V{sub}2{/sub}/V{sub}1{/sub}). For heating at constant volume: delta S = nC{sub}v{/sub} ln(T{sub}2{/sub}/T{sub}1{/sub}).
Entropy change for phase transitions: delta S = delta H / T, where delta H is the latent heat and T is the transition temperature. This applies at the equilibrium transition temperature where the process is reversible.
Efficiency and Free Energy Formulas
eta = W/Q{sub}H{/sub} = 1 - Q{sub}C{/sub}/Q{sub}H{/sub}
This defines thermal efficiency for a heat engine. The Carnot limit is eta{sub}max{/sub} = 1 - T{sub}C{/sub}/T{sub}H{/sub}. For refrigerators: COP = Q{sub}C{/sub}/W = T{sub}C{/sub}/(T{sub}H{/sub} - T{sub}C{/sub}) at the Carnot limit. For heat pumps: COP = Q{sub}H{/sub}/W = T{sub}H{/sub}/(T{sub}H{/sub} - T{sub}C{/sub}) at the Carnot limit.
G = H - TS
Gibbs free energy determines spontaneity at constant T and P. A process is spontaneous when delta G < 0. At equilibrium: delta G = 0. The relationship to equilibrium constant: delta G degrees = -RT ln K. Under non-standard conditions: delta G = delta G degrees + RT ln Q.
A = U - TS
Helmholtz free energy determines spontaneity at constant T and V. delta A < 0 for spontaneous processes at constant T and V. For electrochemical cells: delta G = -nFE, where n is moles of electrons, F = 96,485 C/mol (Faraday constant), and E is cell potential.
Key Takeaway
These formulas form the complete toolkit for thermodynamic calculations. The key to using them correctly is identifying which variables are held constant and which units are appropriate.
Heat Transfer and Material Property Formulas
Q = kA(delta T / L)
Fourier law for steady-state conduction through a flat slab. k is thermal conductivity (W/(m K)), A is cross-sectional area, delta T is temperature difference, and L is thickness. For cylindrical geometry: Q = 2 pi k L delta T / ln(r{sub}2{/sub}/r{sub}1{/sub}).
Q = hA(T{sub}s{/sub} - T{sub}f{/sub})
Newton law of cooling for convective heat transfer. h is the convective heat transfer coefficient (W/(m
2 K)), T{sub}s{/sub} is surface temperature, and T{sub}f{/sub} is fluid temperature. The Stefan-Boltzmann law for radiation: P = epsilon sigma A T
4, where epsilon is emissivity and sigma = 5.67 x 10
-8 W/(m
2 K
4).
Thermal expansion: delta L = alpha L{sub}0{/sub} delta T (linear), delta V = beta V{sub}0{/sub} delta T (volumetric). For solids, beta is approximately 3 alpha. The Clausius-Clapeyron equation for phase boundaries: dP/dT = delta H / (T delta V), where delta H and delta V are the enthalpy and volume changes of the phase transition.