MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2002

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 6: Due Mon. Nov. 4, Before 5PM in in 13-5114

Exercise 6.1

Methane CH$_4$ and carbon dioxide CO$_2$ are mixed in equal molar proportions in a reservoir fixed at $T=1750$K and 1 atmosphere pressure. Find the equilibrium concentrations of the gaseous components.

Assume that the only gaseous components are CH$_4$, CO$_2$, H$_2$, CO, O$_2$, and H$_2$O.

Free energies as a function of temperature for several reactions are tabulated below:

Reaction	Change in Molar Gibbs Free Energy (joules)
CH$_4$   (gas)$\ensuremath{\rightleftharpoons}C$   (graphite)$+ 2$   H$_2$   (gas)	$69120 - 22.25 T \log T + 65.35 T$
$2$CO   (gas)$\ensuremath{\rightleftharpoons}2 C$   (graphite)$+$   O$_2$   (gas)$$	$223400 + 175.3T$
CO$_2$   (gas)$\ensuremath{\rightleftharpoons}C$   (graphite)$+$   O$_2$   (gas)$$	$394100 + 0.8 T$
H$_2$   (gas)$+ \frac{1}{2}$   O$_2$   (gas)$\ensuremath{\rightleftharpoons}$H$_2$O   (gas)	$-246400 + 54.8T$

Exercise 6.2

Find an expression that relates the change in Gibbs free energy with temperature at constant volume, , in terms of the, entropy, volume, thermal expansivity $\alpha \equiv \frac{1}{V} \ensuremath{ \left( \frac{\partial{V}}{\partial{T}} \right)_{P} }$, and the isothermal compressibility $\kappa_T \equiv \frac{-1}{V} \ensuremath{ \left( \frac{\partial{V}}{\partial{P}} \right)_{T} }$.

Describe how you would measure $\kappa_T$ and $\alpha$ in an experiment.