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Chapter 3: Q. 3.28 (page 113)

A liter of air, initially at room temperature and atmospheric pressure, is heated at constant pressure until it doubles in volume. Calculate the increase in its entropy during this process.

Short Answer

Expert verified

The increase in entropy is816.51JK-1.

Step by step solution

01

Given Information

Pressure =P=101325Pa

Temperature =T=300K

Volume =V=1L

Heat capacity at constant pressure=Cp=29JK-1

02

Calculation

For an ideal gas, as the volume doubles, the temperature must also double.

The change in entropy is given as:

ΔS=∫TiTfQTΔS=∫TiTfCpdTTΔS=Cpln[T]TiTfΔS=CplnTfTiΔS=Cpln2TiTiΔS=Cpln2

For the number of molecules in one liter of air,

n=PVRTn=101325×18.314×300n=40.62mol

Hence, change in entropy for one liter of air can be calculated as:

ΔS=nCpln2ΔS=40.62×29×ln2ΔS=816.51JK-1

03

Final answer

Hence, the entropy of air will be 816.51JK-1when air is heated at constant pressure and its volume doubles.

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Most popular questions from this chapter

Use a computer to reproduce Table 3.2 and the associated graphs of entropy, temperature, heat capacity, and magnetization. (The graphs in this section are actually drawn from the analytic formulas derived below, so your numerical graphs won't be quite as smooth.)

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.

Experimental measurements of heat capacities are often represented in reference works as empirical formulas. For graphite, a formula that works well over a fairly wide range of temperatures is (for one mole)

CP=a+bT-cT2

where a=16.86J/K,b=4.77×10-3J/K2, and c=8.54×105J·K. Suppose, then, that a mole of graphite is heated at constant pressure from 298Kto 500K. Calculate the increase in its entropy during this process. Add on the tabulated value of S(298K)(from the back of this book) to obtain S(500K).

A cylinder contains one liter of air at room temperature ( 300K) and atmospheric pressure 105N/m2. At one end of the cylinder is a massless piston, whose surface area is 0.01m2. Suppose that you push the piston in very suddenly, exerting a force of 2000N. The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort.

(a) How much work have you done on this system?

(b) How much heat has been added to the gas?

(c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase?

(d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).

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.)
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