A system in its solid phase has a Helmholtz free energy per mole, \(a_{s}=B / T v^{3}\) and in its liquid phase it has a Helmholtz free energy per mole \(a_{1}=A / T v^{2}\), where \(\mathrm{A}\) and \(\mathrm{B}\) are constants, \(v\) is the volume per mole, and \(T\) is the temperature. (a) Compute the molar Gibbs free energy density of the liquid and solid phases. (b) How are the molar volumes, \(v\), of the liquid and solid related at the liquidsolid phase transition? (c) What is the slope of the coexistence curve in the \(P-T\) plane?

Helmholtz free energy, denoted as F, is a thermodynamic potential that measures the useful work obtainable from a system at constant temperature and volume. It's particularly useful in phase transitions. For the solid phase in the given exercise, the Helmholtz free energy per mole is represented as \(a_{s} = \frac{B}{Tv^{3}}\). Similarly, for the liquid phase, it is represented as \(a_{1} = \frac{A}{Tv^{2}}\). The equations suggest that as temperature (T) increases, the Helmholtz free energy decreases, making transitions feasible.

The relation between Helmholtz free energy and Gibbs free energy, G, helps in phase transition analysis: \(G = A + Pv\), where \(P\) is pressure and \(v\) is volume per mole. Knowing this aids in calculating the Gibbs free energies for different phases, as HFE directly influences phase stability at specific temperatures and volumes.

The Clapeyron equation is vital for understanding phase transitions. It describes the slope of the coexistence curve between two phases in the pressure-temperature (P-T) diagram. In the given exercise, the Clapeyron equation is used to determine the slope at which solid and liquid phases coexist:

\[ \frac{dP}{dT} = \frac{\Delta S}{\Delta V} \]
Here, \(\frac{dP}{dT}\) is the slope of the P-T curve, \(\Delta S\) is the change in entropy between phases, and \(\Delta V\) is the change in volume.
To find \(\Delta S\), we need the relation \(S = -\left(\frac{\partial a}{\partial T}\right)_{v}\). Calculating the partial derivatives of Helmholtz free energy for solid and liquid phases with respect to T helps in obtaining \(\Delta S\). Similarly, \(\Delta V\) can be determined from the molar volumes of solid and liquid phases.

Phase transition thermodynamics examines how substances change from one phase to another and the energy changes involved. The exercise highlights how Gibbs Free Energy (G) governs phase transitions. At equilibrium between solid and liquid phases, their Gibbs Free Energies are equal:

\[ \frac{B}{Tv^{3}} + Pv = \frac{A}{Tv^{2}} + Pv \]
This equality helps derive the relationship between molar volumes of the phases.
Understanding these transitions requires knowing:

• **Enthalpy (H)** - the heat content at constant pressure.
• **Entropy (S)** - the disorder or randomness in the system.
• **Temperature (T)** - driving factor which influences the phase stability.

The Clapeyron relation and entropy calculations further deepen our understanding of phase transition thermodynamics.

Molar volume, denoted as v, is the volume occupied by one mole of a substance. It significantly influences phase transitions. In the exercise, both the solid and liquid phases have different Helmholtz free energies due to their varying molar volumes:

Solid phase: \(a_{s}=\frac{B}{Tv^{3}}\)
Liquid phase: \(a_{1}=\frac{A}{Tv^{2}}\).
At the phase transition point, the Gibbs free energy equality influences how molar volumes of phases relate:

\[ \frac{B}{Tv^{3}} + Pv = \frac{A}{Tv^{2}} + Pv \]
Simplifying this relation offers insights into how solid and liquid molar volumes are connected during transitions. The change in molar volume (\(\frac{\Delta V}{\Delta T}\)) as a function of temperature helps determine the slope of the coexistence curve using the Clapeyron equation. Understanding molar volumes aids in predicting material behaviors under different pressure and temperature conditions.