Book edition 1st

Author(s) Daniel V. Schroeder

Pages 356 pages

ISBN 9780201380279

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Short Answer

In Problem 1.55 you used the virial theorem to estimate the heat capacity of a star. Starting with that result, calculate the entropy of a star, first in terms of its average temperature and then in terms of its total energy. Sketch the entropy as a function of energy, and comment on the shape of the graph.

The required expression for the entropy of a star is $S = - \frac{3}{2} N K \ln (\frac{2 U}{3 N K}) + f (N, V)$ and the graph can be sketched as below.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

The heat capacity of a star that was estimated using the virial theorem is given as:

$C_{V} = - \frac{3}{2} N K$

Where,

$N$ is the number of particles (typically dissociated protons and electrons).

The negative sign symbolizes that it is a gravitational bound system.

Step 2: Calculation

The change in entropy is given as:

$S = \int \frac{C_{V} (T)}{T} d T$

Where,

$C_{V}$ = specific heat

$T$ = Temperature in Kelvin

By substituting the value of $C_{V}$ in the above equation, we get,

$S = \int - \frac{3}{2} (\frac{N K}{T}) d T S = - \frac{3}{2} N K \int (\frac{1}{T}) d T S = - \frac{3}{2} N K T \ln (T) + f (N, V) . . . . . . . . . . (1)$

In this equation, $f$ is the function of $N$ and volume $V$ .

Total energy of gravitationally bound system is negative and from the virial theorem, it is found that:

$U = - K = - \frac{3}{2} N K T$

By rearranging the terms, we get,

$T = - \frac{2 U}{3 N K}$

By substituting this value in equation (1), we get,

$S = - \frac{3}{2} N K \ln (\frac{2 U}{3 N K}) + f (N, V)$

For plotting the graph, let us further simplify the above equation,

$S = - \frac{3}{2} N K \ln (U) - \frac{3}{2} N K \ln (\frac{3 N K}{2}) + f (N, V) S = - \frac{3}{2} N K \ln (U) + g (N, V)$

From the above equation, the graph can be plotted as below:

Step 3: Final answer

Hence, the required expression is: $S = - \frac{3}{2} N K \ln (\frac{2 U}{3 N K}) + f (N, V)$

The graph of entropy as a function can be sketched as follow:

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