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Chapter 3: Q. 3.27 (page 112)

What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what you already knew? Explain.

Short Answer

Expert verified

The relation that can be derived from the thermodynamic identity is P=-dUdVN,S.

The derived equation agrees with the first law of thermodynamics.

Step by step solution

01

Given Information

The thermodynamic identity is:

dU=TdS-PdV

The process is taking place at constant entropy.

Hence,dS=0

02

Calculation

Since the process is taking place at constant entropy, the thermodynamic identity can be written as:

dU=TdS-PdVdU=-PdVP=-dUdVN,S

This is the derived relation.

From the relation, it can be seen that the volume change is fully responsible for the energy shift. As a result, there is no heat transfer into or out of the system. Heat flows until equilibrium is reached, resulting in a rise in entropy. As a result, the equation follows the first law of thermodynamics.

03

Final answer

The derived relation is P=-dUdVN,S. It agrees with the first law of thermodynamics.

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Most popular questions from this chapter

Estimate the change in the entropy of the universe due to heat escaping from your home on a cold winter day.

When the sun is high in the sky, it delivers approximately 1000 watts of power to each square meter of earth's surface. The temperature of the surface of the sun is about 6000K, while that of the earth is about 300K.

(a) Estimate the entropy created in one year by the flow of solar heat onto a square meter of the earth.

(b) Suppose you plant grass on this square meter of earth. Some people might argue that the growth of the grass (or of any other living thing) violates the second law of thermodynamics, because disorderly nutrients are converted into an orderly life form. How would you respond?

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.)

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 the result of Problem 2.42 to calculate the temperature of a black hole, in terms of its mass M. (The energy is Mc2. ) Evaluate the resulting expression for a one-solar-mass black hole. Also sketch the entropy as a function of energy, and discuss the implications of the shape of the graph.
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