MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2000

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 6: Due Fri. Nov. 10, Before 5PM in 13-5026

Exercise 6.1

Show that the following relationships are true:

$$\begin{split}\ensuremath{\frac{\partial (u , v)}{\partial (x ... ...nsuremath{\frac{\partial (r , s)}{\partial (x , y)}}\end{split}$$ (1)

Write out all the Maxwell relations that do not involve changes in composition (i.e., the four that involve $P$, $V$, $S$, $T$) in terms of Jacobians.

Exercise 6.2

A reaction vessel that can be held at constant pressure and temperature initially has one mole of SO$_{2\mbox{(gas)}}$, one mole of H$_{2\mbox{(gas)}}$, and one mole of CO$_{\mbox{(gas)}}$.

The system can react as follows:

$$\begin{split}\mbox{H}_{2\mbox{(gas)}} + \mbox{SO}_{3\mbox{(ga... ...remath{\rightleftharpoons}\mbox{CO}_{2\mbox{(gas)}} \end{split}$$ (2)

Show how you can solve for the equilibrium concentrations. You will need to write a number of explicit equations in terms of an equal number of variables. Don't attempt to solve the problem, just set it up as clearly as possible.

Molar Gibbs Free Energy Changes for Gaseous Reactions
Reaction	$\Delta \ensuremath{\overline{G}}$ (Joules/mole)	Temperature range (Kelvin)
H$_{2\mbox{(gas)}} + \frac{1}{2} \mbox{O}_{2\mbox{(gas)}} \ensuremath{\rightleftharpoons} \mbox{H}_2 \mbox{O}_{\mbox{(gas)}}$	$-262600 + 54.8T$	298-2500
SO$_{2\mbox{(gas)}} + \frac{1}{2} \mbox{O}_{2\mbox{(gas)}} \ensuremath{\rightleftharpoons} \mbox{SO}_{3\mbox{(gas)}}$	$-94560 + 89.7T$	318-1800
CO$_{\mbox{(gas)}} + \frac{1}{2} \mbox{O}_{2\mbox{(gas)}} \ensuremath{\rightleftharpoons} \mbox{CO}_{2\mbox{(gas)}}$	$-282400 + 86.8T$	298-2500

Exercise 6.3

The isothermal compressibility, $\kappa_T$, is defined as:

$$\kappa_T \equiv -\frac{1}{V} \ensuremath{ \left( \frac{\partial{V}}{\partial{P}} \right)_{T} }$$ (3)

State in a clear sentence what the quantity $\kappa_T$ physically represents. Calculate $\kappa_T$ as a function of $n$, $P$, and $T$ for an ideal gas. Identify two other materials that would have a differing value of $\kappa_T$ and describe why those values differ.

Show that

$$\ensuremath{ \left( \frac{\partial{P}}{\partial{V}} \right)_{S} }= -\frac{C_P}{\kappa_T V C_V}$$ (4)

Exercise 6.4

Determine the slope and curvature of $\ensuremath{\overline{G}}(P =$   constant$, T)$.

Write down the Taylor series of $\ensuremath{\overline{G}}(\ensuremath{{P}_\circ} + \Delta P, \ensuremath{{T}_\circ} + \Delta T)$ to second order in $\Delta P$ and $\Delta T$, and where possible relate the coefficients to physical quantities.