Thermodynamic Potentials, the Gibbs Function and the Clausius-Clapeyron Equation

Vapour pressure, latent heat, Trouton's rule and why ice melts under pressure.

Objectives

On completing this experiment you should be able to:

Measure the saturation vapour pressure of a liquid over a range of temperatures with an isoteniscope.
Obtain the molar enthalpy of vaporisation from a plot of ln P against $\frac{1}{T}$ , and state the approximations involved.
Compare that value with the one obtained from the exact Clapeyron equation and account for the difference.
Test Trouton's rule across several liquids and explain the exceptions.
Identify an unknown liquid from its saturation curve.
Measure the depression of the melting point of ice with pressure and show that it follows from the Clapeyron equation with a negative volume change.

Theory

Two phases in equilibrium have equal molar Gibbs functions. Requiring that equality to persist along the coexistence curve gives the Clapeyron equation

\frac{dP}{dT} = \frac{s_2 - s_1}{v_2 - v_1} = \frac{L}{T (v_2 - v_1)} ,

which is exact. For a liquid in equilibrium with its vapour, neglecting $v_l$ against $v_g$ and treating the vapour as ideal turns it into the Clausius-Clapeyron equation

\frac{d(\ln P)}{d(1/T)} = -\frac{L}{R} ,

so a plot of ln P against $\frac{1}{T}$ is a straight line of gradient -L/R. Both approximations are good but not perfect, and both bias the result in a predictable direction.

For the melting of ice, $v_{liquid}$ < $v_{solid}$ , so $\frac{dP}{dT}$ is negative and the melting temperature falls as the pressure rises.

Key relations

$\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P$
The Gibbs function.
$g_1(T,P) = g_2(T,P)$
Condition for phase coexistence.
$\frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T(v_2-v_1)}$
Clapeyron equation (exact).
$\ln P = -\frac{L}{RT} + \text{constant}$
Clausius-Clapeyron equation (ideal vapour, $v_l$ neglected).

Procedure by part

Part A - Saturation vapour pressure

Set the thermostat and read the absolute pressure in the isoteniscope once equilibrium is reached.

Select the liquid and set the bath temperature.
Take at least twelve readings spanning the recommended window for that liquid.
Plot ln P against $\frac{1}{T}$ , fit a straight line and obtain L from the gradient.
Compute the boiling point at 101.325 kPa from your fit and compare with the accepted value.
Repeat for at least three liquids, and for the unlabelled one.

Part B - Trouton's rule

Determine the normal boiling point of each liquid from your fits and test whether the entropy of vaporisation is universal.

Use the apparatus to locate the temperature at which the saturation pressure equals 101.325 kPa, by bisection.
Compute the entropy of vaporisation $\frac{L}{T_b}$ for each liquid.
Tabulate the results and identify which liquids obey Trouton's rule and which do not.

Part C - The melting curve of ice

Compress a mixture of ice and water in the high-pressure cell and record the equilibrium temperature.

Set the ram pressure and take the reading once the cell has equilibrated.
Record at least eight pressures up to 150 bar.
Plot the melting temperature against pressure and obtain the gradient.
Use the Clapeyron equation with your gradient to obtain the latent heat of fusion, using the accepted densities of ice and water.
Explain the sign of the gradient and its consequences.

Questions to answer in your report

Derive the Clapeyron equation from the equality of the molar Gibbs functions of two coexisting phases, and then state precisely the two approximations that turn it into the Clausius-Clapeyron equation. [8]
Compare your two values of L for water. Which is larger, by how much, and which of the approximations is responsible? [6]
Trouton's rule holds for benzene and propanone but fails badly for water and ethanol. Explain, and say what the failure tells you about the liquid state. [6]
Ice melts at a lower temperature under pressure. Derive the magnitude of the effect from your data and comment critically on the common claim that this is why ice skates work. [6]