Physics 4311: Thermal Physics – Homework 5

Problem 1: Exact differentials (25 points)

a)

Test whether the following differentials are exact.

$$\begin{array}{rcll}d{u}_a& =& 2x\mathit{dx}+\mathit{dy}& \\ d{u}_b& =& 2y\mathit{dx}+\mathit{dy}& \end{array}$$

b)

If the differential is exact, calculate the indefinite integral.

c)

Check the dependence of the integral on the path of integration by explicitly integrating both differentials from point $(x_i,y_i)=(0,0)$ to $(x_f,y_f)=(2,2)$ on two different path, $(0,0)\to (2,0)\to (2,2)$ and $(0,0)\to (0,2)\to (2,2)$. Compare the results of the two path and that of a calculation using the indefinite integral (if it exists).

Problem 2: Rubber elasticity (15 points)

The equation of state of a rubber band can be modeled by the so-called Guth-James equation

Here $F$ is the tension force, $L$ is the length of the rubber band (with $L_0=10$ cm being the unstretched length). $T$ is temperature, and $a=1.8\times 10^{-2}$ N/K is a constant.

a)

How much work is done if the rubber band is stretched isothermally from its original length of 5 cm to 15 cm? The temperature is kept at 293 K.

b)

The rubber band is held at fixed (stretched) length. Will the tension increase as the temperature increases?

c)

When the rubber band is heated at fixed tension, will its length increase or decrease with temperature?