A. Scardicchio, ICTP

This lesson starts by recalling some concepts introduced before like the **van der Waals equation of state,** the isotherms of a van der Waals gas in a Pressure-Volume plot for a temperature below the critical temperature in the case of the liquid-gas transformation and its differences with the experiment was remarked. Then it was introduced the **thermodynamics potentials** with the help of the **Clausius theorem. **From this theorem it can be shown that, in an isolated system that cannot interchange heat with the environment, the entropy increases in all spontaneous transformations. If the system is not isolated but is in contact with a thermal bath thus the temperature is constant, it is useful to define the **Helmholtz free energy (F=U-TS), **for this case can be demonstrated that the systems spontaneously evolve to a state that minimizes **F.** As an example of the application of the Helmholtz free energy it was calculated the difference in work done due to an infinitesimal change in temperature from *T* to *T+dT* when going from state A to state B and two cases were considered, the ideal gas and the van der Waals gas. Then some **Maxwell relations** were derived from **F**. Finally it was considered the case when the transformations occurs at constant temperature and pressure for which it is useful to define the **Gibbs free energy** **(G=F+pV). **As before it was shown that **G** can only decrease in an spontaneous transformation.

Thermodynamic potential (Wikipedia)

# <span style="font-size: large;">Thermodynamic potential</span>

Helmholtz free energy (Wikipedia)

# <span style="font-size: large;">Helmholtz free energy</span>

# <span>Gibbs free energy</span>

Maxwell relations

Derive the Maxwell relations using the internal energy and the thermodynamics potentials. Then watch the following videos.

Maxwell relations from internal energy.

Maxwell relations from internal energy.

Maxwell relations from Helmholtz free energy.

Maxwell relations from Helmholtz free energy.

Maxwell relations from Gibb's free energy.

Maxwell relations from Gibb's free energy.

Maxwell relations from Entalphy.Maxwell relations from Entalphy.