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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

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 ℓ(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 ℓ.
(e) Show that when L << Nℓ, 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.)

Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgzin addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz:

μ(z)=-kTlnVN2πmkTh23/2+mgz.

(You can derive this result from either the definition μ=-T(∂S/∂N)U,Vor the formula μ=(∂U/∂N)S,V.

(b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

N(z)=N(0)e-mgz/kT

in agreement with the result of Problem 1.16.

Sketch a qualitatively accurate graph of the entropy of a substance (perhapsH2O ) as a function of temperature, at fixed pressure. Indicate where the substance is solid, liquid, and gas. Explain each feature of the graph briefly.

Use Table 3.1 to compute the temperatures of solid A and solid B when qA=1. Then compute both temperatures when qA=60. Express your answers in terms of ε/k, and then in kelvins assuming that ε=0.1eV.

Consider an ideal two-state electronic paramagnet such as DPPH, with μ=μB. In the experiment described above, the magnetic field strength was 2.06T and the minimum temperature was 2.2K. Calculate the energy, magnetization, and entropy of this system, expressing each quantity as a fraction of its maximum possible value. What would the experimenters have had to do to attain99% of the maximum possible magnetization?
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