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 4: Due Mon. Oct. 15, Before 5PM in in 13-5114

Exercise 4.1

Let the molar internal energy and molar entropy of an ideal gas at standard temperature and pressure be $\ensuremath{\overline{\ensuremath{{U}_\circ}}}(\ensuremath{{P}_\circ},\ensuremath{{T}_\circ})$ and $\ensuremath{\overline{\ensuremath{{S}_\circ}}}(\ensuremath{{P}_\circ},\ensuremath{{T}_\circ})$.

Derive expressions for the state functions $\ensuremath{\overline{U}}(P,T)$, $\ensuremath{\overline{U}}(V,T)$, $\ensuremath{\overline{U}}(P,V)$ and $\ensuremath{\overline{S}}(P,T)$, $\ensuremath{\overline{S}}(V,T)$, $\ensuremath{\overline{S}}(P,V)$.

Exercise 4.2

The molar heat capacity of solid aluminum is a weak function of temperature
$\ensuremath{\overline{C_p}} = 20.7 + \ensuremath{{12.4} \times 10^{-3}} T \left(\frac{\mbox{joule}}{\mbox{degree mole}}\right)$ for $300$   K$< T < 900$   K.

Suppose $c$ grams of aluminum at temperature $T_c = 600(1 - \frac{\theta}{2})$    (kelvin) are put in thermal contact with $h$ grams of aluminum at $T_h = 600(1 + \frac{\theta}{2})$    (kelvin).

1. Calculate an expression for the final temperature as a function of the ratio $c/h$ and $\theta$ $(0 < \theta < 1)$.

2. Plot the change in total entropy per mole of aluminum as a function of $\theta$ for values of $c/h$ = (1/16, 1/8, 1/4, 1/2, 1).

Hints: Regard the two masses of aluminum together as comprising an isolated system. Work out an algebraic expression for the first part--using a program like Mathematica or Matlab on Athena will make life much much easier. Don't try to calculate and write out an expression for the second part--it is too messy. Find a team member that knows how to plot numbers and make sure that the plot that he or she produces makes sense.