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Chapter 3: Q. 3.33 (page 114)

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 dHin terms of dSand dP.

Short Answer

Expert verified

The heat capacity expression is same for both at constant pressure and volume.

Step by step solution

01

Explanation of Solution

Given:

The thermodynamic identity for infinitesimal process is:

Internal energy, dU=TdS-PdV

Enthalpy,dH=dU+PdV

02

Calculation

At constant volume , the heat capacity is

CV=∂U∂TV

CV=TdS-PdV∂TV

CV=T∂S∂TV

At constant pressure the heat capacity is,

CP=∂H∂TP

CP=dU+PdV∂TP

CP=T∂S∂TP

The heat capacity expression is same for both at constant pressure and volume.

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Most popular questions from this chapter

Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle."

(a) Show that the chemical potential is

role="math" localid="1646995468663" μ=-kTlnN+qN

(b) Discuss this result in the limits N≫qand N≪q, concentrating on the question of how much Sincreases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?

In order to take a nice warm bath, you mix 50 liters of hot water at 55°C with 25 liters of cold water at 10°C. How much new entropy have you created by mixing the water?

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In Problem 2.32you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to U,A, and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense.
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