C.2 Shown is a portion of the phase diagram for CO$_2$.
Pressure (bar)
10
CO$_2$
Liquid
Solid
-90 -80 -70 -60 -50 -40 -30
Temperature (°C)
Gas
In a simplified model for solid CO$_2$ (dry ice), CO$_2$ molecules are spheres arranged in a
face centred cubic structure, interacting with neighbours through a Lennard-Jones pair
potential
$U_{pair} = 4\epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right]$,
where $\sigma = 3.6$ Ã… and $\epsilon = 0.044$ eV.
c. Calculate the equilibrium lattice parameter of solid face-centred cubic CO$_2$, and
hence calculate the molar volume of solid CO$_2$.
d. Show that the latent heat of vapourisation, $L_v$, of solid CO$_2$ is roughly 25 kJ/mol.
The Clausius-Clapeyron equation is given as
$\frac{dp}{dT} = \frac{L}{T\Delta V}$,
where $dp/dT$ is the slope of the phase boundary at pressure $p$ and temperature $T$, $\Delta V$ is
the difference in volume between phases at $p$ and $T$, and $L$ is the latent heat of transfor-
mation. You may assume 1 bar = 10$^5$ Pa
e. At ambient pressure, calculate the change in molar volume upon vapourisation in
CO$_2$.
f. By what factor does the volume increase upon vapourisation at ambient pressure?
Evaluate your answer for plausibility.
