The tension of a rubber band in equilibrium is given by t=A T((x)/(l0)-(l0^2)/(x^2)), where t= t

Step 1: u000ATo find u005C(u005Cleft(u005Cfrac{u005Cpartial E}{u005Cpartial x}u005Cright)_Tu005C), we start with the expression

Question

The tension of a rubber band in equilibrium is given by $$ t=A T\left(\frac{x}{l_0}-\frac{l_0^2}{x^2}\right), $$ where $t=$ tension, $T=$ absolute temperature, $x=$ length of the band, $l_0$ $=$ length of the band when $t=0, A=$ constant. When $x$ is the constant length $l_0$, the thermal capacity $c_x(x, T)$ is observed to be a constant $K$. (a) Find as functions of $T$ and $x$ : (1) $\left(\frac{\partial E}{\partial x}\right)_T$ where $E=$ internal energy, (2) $\left(\frac{\partial c_x}{\partial x}\right)_T$, (3) $c_x(x, T)$, (4) $E(x, T)$, (5) $S(x, T)$, where $S=$ entropy. (b) The band is stretched adiabatically from $x=l_0$ to $x=1.5 l_0$. Its initial temperature was $T_0$. What is its final temperature?

   The tension of a rubber band in equilibrium is given by
$$
t=A T\left(\frac{x}{l_0}-\frac{l_0^2}{x^2}\right),
$$
where $t=$ tension, $T=$ absolute temperature, $x=$ length of the band, $l_0$ $=$ length of the band when $t=0, A=$ constant.

When $x$ is the constant length $l_0$, the thermal capacity $c_x(x, T)$ is observed to be a constant $K$.
(a) Find as functions of $T$ and $x$ :
(1) $\left(\frac{\partial E}{\partial x}\right)_T$ where $E=$ internal energy, (2) $\left(\frac{\partial c_x}{\partial x}\right)_T$, (3) $c_x(x, T)$, (4) $E(x, T)$, (5) $S(x, T)$, where $S=$ entropy.
(b) The band is stretched adiabatically from $x=l_0$ to $x=1.5 l_0$. Its initial temperature was $T_0$. What is its final temperature?

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 1, Problem 83

Instant Answer

Step 1

\] The internal energy $E$ can be related to the tension and the length of the rubber band. The work done on the rubber band when it is stretched is given by: \[ dE = t \, dx. \] Thus, we can express the change in internal energy with respect to $x$ at Show more…

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The tension of a rubber band in equilibrium is given by $$ t=A T\left(\frac{x}{l_0}-\frac{l_0^2}{x^2}\right), $$ where $t=$ tension, $T=$ absolute temperature, $x=$ length of the band, $l_0$ $=$ length of the band when $t=0, A=$ constant. When $x$ is the constant length $l_0$, the thermal capacity $c_x(x, T)$ is observed to be a constant $K$. (a) Find as functions of $T$ and $x$ : (1) $\left(\frac{\partial E}{\partial x}\right)_T$ where $E=$ internal energy, (2) $\left(\frac{\partial c_x}{\partial x}\right)_T$, (3) $c_x(x, T)$, (4) $E(x, T)$, (5) $S(x, T)$, where $S=$ entropy. (b) The band is stretched adiabatically from $x=l_0$ to $x=1.5 l_0$. Its initial temperature was $T_0$. What is its final temperature?

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