Physics 4311: Thermal Physics – Homework 9

Problem 1: Thermodynamic potentials of a surface (10 points)

The work required to change the area $A$ of a surface by an infinitesimal amount $\mathit{dA}$ is given by $\mathit{\delta W}=\sigma \mathit{dA}$ where $\sigma$ is the surface tension. Start from the first law $\mathit{dU}=T\mathit{dS}+\sigma \mathit{dA}$ and derive the formulas for the thermodynamic potentials and their total differentials in terms of the natural variables.

a)

enthalpy,

b)

Helmholtz free energy,

c)

Gibbs free energy.

Problem 2: Maxwell relations for a surface (10 points)

Using the thermodynamic potentials that you found in homework 9.1, derive the four Maxwell relations for a surface of area $A$ under surface tension $\sigma$.

Problem 3: Response functions of the ideal gas (20 points)

An ideal gas obeys the equation of state $\mathit{pV}=N{k}_{B}T$ with $p$ being pressure, $V$ volume, $N$ the number of particles, $k_B$ the Boltzmann constant, and T the temperature. The internal energy is $U=(3∕2)N{k}_{B}T$. Calculate

a)

the specific heat at constant volume $C_V$,

b)

the specific heat at constant pressure $C_p$,

c)

the isothermal compressibility ${\kappa }_T$,

d)

the adiabatic compressibility ${\kappa }_S$, and

e)

the volume thermal expansion coefficient $\alpha$.