Thermodynamics: Maxwell Equations Significance and understanding

Enthalpy and Internal energy for an ideal gas is a function of temperature only, but for real gases these also depends on one other parameter i.e. volume or pressure. In order to find change in internal energy and enthalpy for a real gas we need to get the expression in terms of measuring parameters like pressure, temperature and volume. Entropy is one important parameter in determining the change in internal energy and enthalpy for real gases. So it is necessary to first find change in entropy with pressure, temperature and volume keeping one other parameter constant. Maxwell equations tell the change in entropy w.r.t. pressure and volume.


 

These equations look somewhat difficult but there is an easier way to understand these equations:

every relation hold the change in entropy with volume at constant T & P and change in entropy with pressure at constant T & V. 

if we see the dimensional consistency, the terms W=-PdV and Q_Rev = TdS have the units of energy. So, these appear in opposite positions in the equations on left and right sides to make the equations dimensionally consistent. so like dS/dV=dP/dT

Now we have to see the constraint, like temperature. Temperature is in denominator on the right, so if we want it on left, then the constraint on the right will be V, (i.e. denominator on the left). 

Now the sign convention, when the volume increases of the system, the entropy increases, so the change in entropy with volume is positive, and when the pressure increases, the entropy decreases so the change in entropy with pressure is negative, and there will be a negative sign with the equations of change in entropy with pressure.