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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

In the experiment of Purcell and Pound, the maximum magnetic field strength was 0.63Tand the initial temperature was 300K. Pretending that the lithium nuclei have only two possible spin states (in fact they have four), calculate the magnetization per particle, M/N, for this system. Take the constant μto be 5×10-8eV/T. To detect such a tiny magnetization, the experimenters used resonant absorption and emission of radio waves. Calculate the energy that a radio wave photon should have, in order to flip a single nucleus from one magnetic state to the other. What is the wavelength of such a photon?

Sketch (or use a computer to plot) a graph of the entropy of a two-state paramagnet as a function of temperature. Describe how this graph would change if you varied the magnetic field strength.

In Problem 2.18 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately

Ω(N,q)≈q+Nqqq+NNN

(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large.

(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is U=qϵ, where ϵis a constant.) Be sure to simplify your result as much as possible.

(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.

(d) Show that, in the limit T→∞, the heat capacity is C=Nk. (Hint: When x is very small, ex≈1+x.) Is this the result you would expect? Explain.

(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot C/Nkvs. the dimensionless variable t=kT/ϵ, for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the value of ϵ, in electron-volts, for each of those real solids.

(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x3 in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than(ϵ/kT)2 in the final answer. When the smoke clears, you should find C=Nk1-112(ϵ/kT)2.

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?

Fill in the missing algebraic steps to derive equations 3.30, 3.31, and 3.33.
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