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Chapter 3: Q. 3.33 (page 114)

Use the thermodynamic identity to derive the heat capacity formula

CV=T鈭�S鈭�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=鈭�U鈭�TV

CV=TdS-PdV鈭�TV

CV=T鈭�S鈭�TV

At constant pressure the heat capacity is,

CP=鈭�H鈭�TP

CP=dU+PdV鈭�TP

CP=T鈭�S鈭�TP

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

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Most popular questions from this chapter

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."

Use the definition of temperature to prove the zeroth law of thermodynamics, which says that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system C. (If this exercise seems totally pointless to you, you're in good company: Everyone considered this "law" to be completely obvious until 1931, when Ralph Fowler pointed out that it was an unstated assumption of classical thermodynamics.)

In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is 渭=-系(where 系is the size of an energy unit and we treat each oscillator as a "particle"). Suppose instead that the solid has three oscillators and four units of energy. How does the chemical potential then compare to -系 ? (Don't try to get an actual value for the chemical potential; just explain whether it is more or less than -系.)

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.

Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude model of a rubber band, consider a chain of N links, each of length 鈩�(see Figure 3.17). Imagine that each link has only two possible states, pointing either left or right. The total length L of the rubber band is the net displacement from the beginning of the first link to the end of the last link.

(a) Find an expression for the entropy of this system in terms of N and NR, the number of links pointing to the right.
(b) Write down a formula for L in terms of N and NR.
(c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure P is replaced by the tension force F. Taking F to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system.
(d) Using the thermodynamic identity, you can now express the tension force F in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of L, T, N, and 鈩�.
(e) Show that when L << N鈩�, the tension force is directly proportional to L (Hooke's law).
(f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behavior make sense?
(g) Suppose that you hold a relaxed rubber band in both hands and suddenly stretch it. Would you expect its temperature to increase or decrease? Explain. Test your prediction with a real rubber band (preferably a fairly heavy one with lots of stretch), using your lips or forehead as a thermometer. (Hint: The entropy you computed in part (a) is not the total entropy of the rubber band. There is additional entropy associated with the vibrational energy of the molecules; this entropy depends on U but is approximately independent of L.)
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