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 .
The graph of the predicted heat capacity as a function of temperature can be sketched as follows:
The equation for Einstein solid at low temperature is calculated as:
Here, is number of oscillator, is the amount of energy quanta, is Boltzmann constant and is temperature.
Heat capacity at constant volume is given as:
Where, is internal energy.
By substututing the value of in the above equation, we get,
Consider the equation which gives the relation of the heat capacity as a function of temperature:
Now, by considering the rest other factors as a constant, heat capacity as a function of temperature can be given as:
Based on the above relation, the graph can be plotted as below:
The required expression is derived as and the graph showing the heat capacity as a function of temperature can be made as follows:
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."
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