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Chapter 3: Q. 3.29 (page 113)

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.

Short Answer

Expert verified

The graph of the entropy of a substance as a function of temperature, at fixed pressure can be sketched as below:

Step by step solution

01

Given Information

A graph of a substance is to be made showing the entropy of a substance as a function of temperature.

It is also given that the pressure should be constant.

02

Explanation

It is given that the pressure should be fixed.

Hence, the entropy at constant pressure is given as:

dS=CpdTTdSdTP=CpT

The graph can be sketched as below:

From the graph, it can be seen that at low temperatures, the slope of the solid is zero. As the temperature rises, so does the slope, causing the solid to melt. At the same temperature, this increases entropy. The curve grows shallower as the solid melts, showing the liquid-gas phase.

03

Final answer

The required graph can be sketched as below:

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Most popular questions from this chapter

An ice cube (mass 30g)0掳Cis left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is 25掳C.

(a) Calculate the change in the entropy of the ice cube as it melts into water at 0掳C. (Don't worry about the fact that the volume changes somewhat.)

(b) Calculate the change in the entropy of the water (from the melted ice) as its temperature rises from 0掳Cto 25掳C.

(c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water.

(d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect?

Verify every entry in the third line of Table 3.2 (starting with N鈫�=98.

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.)

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

Consider an ideal two-state electronic paramagnet such as DPPH, with 渭=渭B. 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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