This page is intended to generate and display the thermodynamic equations.

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State parameters Type Parameter Meaning Intensive \( p \) Pressure Intensive \( T \) Temperature Extensive \( V \) Volume Extensive \( S \) Entropy Extensive \( E \) Internal energy Extensive \( N \) Number of particles Extensive \( H \overset{\underset{\mathrm{def}}{}}{=} E + pV \) Enthaply Extensive \( A \overset{\underset{\mathrm{def}}{}}{=} E - TS \) Helmholtz energy Extensive \( G \overset{\underset{\mathrm{def}}{}}{=} E + pV - TS \) Gibbs energy Exact \( N_A \) Avogradro's number of particles Extensive \( n \overset{\underset{\mathrm{def}}{}}{=} \frac{N}{N_A} \) Amount of substance (mols) Intensive \( V_m \overset{\underset{\mathrm{def}}{}}{=} \frac{V}{n}\) Molar volume Intensive \( S_m \overset{\underset{\mathrm{def}}{}}{=} \frac{S}{n}\) Molar entropy Intensive \( E_m \overset{\underset{\mathrm{def}}{}}{=} \frac{E}{n}\) Molar internal energy Intensive \( H_m \overset{\underset{\mathrm{def}}{}}{=} \frac{H}{n}\) Molar enthalpy Intensive \( A_m \overset{\underset{\mathrm{def}}{}}{=} \frac{A}{n}\) Molar Helmholtz energy Intensive \( G_m \overset{\underset{\mathrm{def}}{}}{=} \frac{G}{n}\) Molar Gibbs energy Intensive \( J \overset{\underset{\mathrm{def}}{}}{=} \frac{2E}{pV} \) Number of energy components

Process differentials Type Parameter Meaning Extensive \( dQ \) Heat Extensive \( dw \) Work Extensive \( dQ_r \overset{\underset{\mathrm{def}}{}}{=} T \cdot dS \) Reversible heat Extensive \( dw_c \overset{\underset{\mathrm{def}}{}}{=} -p \cdot dV \) Quasistatic work Consequence A reversible process is quasistatic Equations Equation Comments \( dE = dQ + dw \) for a system with no chemical changes \( dE = dQ + dw + \sum_i \mu_{N,i} dN_i \) general law (\( \mu_{i} \overset{\underset{\mathrm{def}}{}}{=} N_A \mu_{N,i} \)) \( dE = dQ + dw + \sum_i \mu_i dn_i \) \( E = Q + w + \sum_i \int \mu_i dn_i \) Integral general law (assumes null integration constant) \( dE = dQ - pdV + \sum_i \mu_i dn_i \) quasistatic (\(dw=-pdV\)) processess \( dH = dQ + Vdp + \sum_i \mu_i dn_i \) \( dA = dQ - pdV - TdS -SdT + \sum_i \mu_i dn_i \) \( dG = dQ + Vdp - TdS -SdT + \sum_i \mu_i dn_i \) \( dE = TdS - pdV + \sum_i \mu_i dn_i \) reversible (\(dQ=TdS\)) & quasistatic (\(dw=-pdV\)) processess \( dH = TdS + Vdp + \sum_i \mu_i dn_i \) \( dA = -pdV - SdT + \sum_i \mu_i dn_i \) \( dG = Vdp - SdT + \sum_i \mu_i dn_i \)

Gradients

Gradient	Formula
Isothermal compresibility	\( \beta _T \overset{\underset{\mathrm{def}}{}}{=} -\frac{1}{V} \cdot \left. \frac{\partial V}{\partial p} \right | _{T=ct.} \)
Volumetric coefficient of thermal expansion	\( \alpha _V \overset{\underset{\mathrm{def}}{}}{=} \frac{1}{V} \cdot \left. \frac{\partial V}{\partial T} \right | _{p=ct.} \)
Heat capacity at constant pressure	\( C_p \overset{\underset{\mathrm{def}}{}}{=}\left. \frac{\partial H}{\partial T} \right | _{p=ct.} \)
Joule-Thomson isothermal coefficient	\( \mu_T \overset{\underset{\mathrm{def}}{}}{=}\left. \frac{\partial H}{\partial p} \right | _{T=ct.} \)
Heat capacity at constant volume	\( C_V \overset{\underset{\mathrm{def}}{}}{=}\left. \frac{\partial E}{\partial T} \right | _{V=ct.} \)
Internal pressure	\( \pi_T \overset{\underset{\mathrm{def}}{}}{=}\left. \frac{\partial E}{\partial V} \right | _{T=ct.} \)

First partial derivatives