Fundamental Thermodynamic Relation - Derivation From The First and Second Laws of Thermodynamics

Derivation From The First and Second Laws of Thermodynamics

The first law of thermodynamics states that:

According to the second law of thermodynamics we have for a reversible process:

Hence:

By substituting this into the first law, we have:

Letting dW be reversible pressure-volume work, we have:

This equation has been derived in the case of reversible changes. However, since, and are thermodynamic functions of state, the above relation holds also for non-reversible changes. If the system has more external parameters than just the volume that can change and if the numbers of particles in the system can also change, the fundamental thermodynamic relation generalizes to:

Here the are the generalized forces corresponding to the external parameters . The are the chemical potentials corresponding to particles of type .