Maxwell Relations Generator
Maxwell Relations Generator derives thermodynamic relations from potentials (U, H, F, G), like (∂T/∂V)_S, to show variable links in a system.
Maxwell Relations Overview
Maxwell relations are derived from the exact differentials of thermodynamic potentials:
Internal Energy (\\(U\\)): \\( dU = T dS – P dV \\)
Relation:
\\[ \left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V \\]Enthalpy (\\(H\\)): \\( dH = T dS + V dP \\)
Relation:
\\[ \left( \frac{\partial T}{\partial P} \right)_S = \left( \frac{\partial V}{\partial S} \right)_P \\]Helmholtz Free Energy (\\(F\\)): \\( dF = -S dT – P dV \\)
Relation:
\\[ \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \\]Gibbs Free Energy (\\(G\\)): \\( dG = -S dT + V dP \\)
Relation:
\\[ \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P \\]Where:
- \\(T\\): Temperature (K)
- \\(S\\): Entropy (kJ/(mol·K))
- \\(P\\): Pressure (bar)
- \\(V\\): Volume (L/mol)
Example Calculation for Internal Energy
Example: Internal Energy (\\(U\\)) with \\(n = 1 \, \text{mol}, R = 0.008314 \, \text{kJ/(mol·K)}, V = 22.4 \, \text{L/mol}\\)
Differential: \\( dU = T dS – P dV \\)
Maxwell Relation: \\(\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V\\)
For an ideal gas: \\(\left( \frac{\partial P}{\partial T} \right)_V = \frac{nR}{V} = \frac{1 \times 0.008314}{22.4} \approx 0.000371 \, \text{bar/K}\\)
Thus: \\(\left( \frac{\partial S}{\partial V} \right)_T = \frac{nR}{V} \approx 0.000371 \, \text{kJ/(mol·K·L)}\\)