MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2001

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 7 Part 1: Due Fri. Nov. 30, Before 5PM in 4-047

Exercise 7.1

Recall from Problem 6.2, that the bulk modulus, $B$, is a material property that relates the change in volume with a change in pressure.

$$B \equiv -V \ensuremath{ \left( \frac{\partial{P}}{\partial{V}} ... ...math{ \left( \frac{\partial{P}}{\partial{\log V}} \right)_{T=\mbox{constant}} }$$

To be explicit, let's call this bulk modulus the ``isothermal bulk modulus,'' $B_{\Delta T = 0}$

Show that $B_{\Delta T = 0}$ is related to the ``adiabatic bulk modulus,'' $B_{\Delta S = 0}$, where

$$B_{\Delta S = 0} \equiv -V \ensuremath{ \left( \frac{\partial{P}... ...math{ \left( \frac{\partial{P}}{\partial{\log V}} \right)_{S=\mbox{constant}} }$$

by

$$\frac{B_{\Delta T = 0}}{B_{\Delta S = 0}} = \frac{C_V}{C_P}$$

and state whether you expect a solid material to be ``stiffer'' if it is reversibly squeezed at constant temperature or with no heat being transfered to it.

Exercise 7.2

Also recall from Problem 6.2 that, for a polymer with isothermal bulk modulus of 0.14 GPa and a density of 0.93 relative to water, there was a critical ocean depth where the work rate to sink (or float) that material switches from positive to negative.

Show that the critical depth is an unstable equilibrium.