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Q. 3.8

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An Introduction to Thermal Physics
Found in: Page 93
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

Starting with the result of Problem 3.5, calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature.

The required expression is Cv=Nε2kT2e-εkT.

The graph of the predicted heat capacity as a function of temperature can be sketched as follows:

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given

The equation for Einstein solid at low temperature is calculated as:

U=Nεe-εkT ............(1)

Here, N is number of oscillator, ε is the amount of energy quanta, k is Boltzmann constant and T is temperature.

Step 2: Calculation of heat capacity

Heat capacity at constant volume is given as:

Cv=∂U∂TN,V

Where, U is internal energy.

By substututing the value of U in the above equation, we get,

Cv=∂∂TNεe-εkTCv=Nε2kT2e-εkT

Step 3: Graph of the heat capacity as a function of temperature

Consider the equation which gives the relation of the heat capacity as a function of temperature:

Cv=Nε2kT2e-εkT

Now, by considering the rest other factors as a constant, heat capacity as a function of temperature can be given as:

Cv∝1T2e-1T

Based on the above relation, the graph can be plotted as below:

Step 4: Final answer

The required expression is derived as Cv=Nε2kT2e-εkT and the graph showing the heat capacity as a function of temperature can be made as follows:

Most popular questions for Physics Textbooks

Figure 3.3 shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word "temperature."

Starting with the result of Problem 2.17, find a formula for the temperature of an Einstein solid in the limit q≪N. Solve for the energy as a function of temperature to obtain U=Nϵe-ϵ/kT (where ϵ is the size of an energy unit).

Use the thermodynamic identity to derive the heat capacity formula

CV=T∂S∂TV

which is occasionally more convenient than the more familiar expression in terms of U. Then derive a similar formula for CP, by first writing dH in terms of dS and dP.

Use a computer to reproduce Table 3.2 and the associated graphs of entropy, temperature, heat capacity, and magnetization. (The graphs in this section are actually drawn from the analytic formulas derived below, so your numerical graphs won't be quite as smooth.)

Sketch a qualitatively accurate graph of the entropy of a substance (perhaps H2O ) as a function of temperature, at fixed pressure. Indicate where the substance is solid, liquid, and gas. Explain each feature of the graph briefly.
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