PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 1999

Quiz 2 Thursday, April 15, 1999

NAME: ______SOLUTIONS______

To receive credit for a problem, you must show your work, or explain how you arrived at your answer (except for problems 1 and 2).

1. (12%) Consider a cylinder with a moveable piston which contains a fixed amount of an idealgas of monatomic atoms. Initially, the ideal gas is at a temperature of 100 K, occupies a volume of 1.0 m3, has a pressure of 1 atmosphere (=1.0 105 N/m2), and has an entropy of 2,000 J/K.

On the following axes sketch p vs. V and T vs. S when: (i) the gas expands adiabatically to a volume of 2.0 m3; and (ii) when the gas expands isothermally to a volume of 2.0 m3. There should be two curves on each plot; be sure to label which is which. Each curve should be marked with an arrow to indicate which way the thermodynamic variables are changing as the gas expands. Also be sure to carefully show the qualitative features of each curve on these plots. However, there is no need to calculate the final temperatures, pressures or entropies for this problem.

2. (15%) Consider whether each of the following ensembles is microcanonical, canonical, or grand canonical (Circle the correct choice.):

a) The contents of a large number of identical, sealed, perfectly insulating thermos jugs, which contain a mixture of ice and water would constitute which kind of ensemble?

microcanonicalcanonicalgrand canonical

b) One of the ice cubes in any one of these thermos bottles would be a member of which kind of ensemble?

microcanonical canonicalgrand canonical

c) One water molecule in any one of these thermos bottles would be a member of which kind of ensemble?

microcanonical canonicalgrand canonical

d) A large number of identical copies of the Pacific ocean, as it actually exists (use your imagination), would constitute which kind of ensemble?

microcanonical canonicalgrand canonical

e) A large number of identical copies of the entire universe, as it actually exists (use more of your imagination), would constitute which kind of ensemble?

microcanonicalcanonicalgrand canonical

3. (20%) An ideal Carnot engine is being used as a refrigerator to cool a room where some students are working with computers. The students plus computers are generating 10,000 watts of heat (10,000 joules/second). The Carnot refrigerator removes heat from the room, where the temperature is 20°C, and delivers it outside the building, where the temperature is 30°C. How much power must be put into the Carnot refrigerator to maintain the room at 20°C?

and . Then, and . Also, , so . Finally,

4. (40%) The Helmholtz Free Energy is defined as F = E – TS, where E is internal energy, T is temperature, and S is entropy.

a) Show that: dF = – SdT – pdV + dN

dF = dE – TdS – SdT, but the First Law states, dE = TdS – pdV + dN. Then dF becomes,

dF = TdS – pdV + dN – TdS – SdT = – SdT – pdV + dN

b) Use the equation given in part a) to show that:

F is a function of T, V, and N. Therefore,

A term by term comparison with the equation given in part a) shows that

c) Use the equation given in part a) to show that:

(This is one of the Maxwell relations.)

Using the method of part b), we also see that . Then,

, and . Clearly, these are equal.

d) Consider a system for which the partition function is, , where 0 and V0 are constants. Find the Helmholtz Free Energy for this system.

e) Use the results of parts b) and d) to find an expression for the pressure of the system introduced in part d).

5. (13%) A micrometeorite punctures a hole of area A in the side of a spacecraft. The spacecraft has a volume V, and contains N molecules of air at temperature T. As the molecules leak out, the temperature of the air remaining inside is kept constant at temperature T.

a) Find an expression for the rate at which molecules leave the spacecraft. Express your answer in terms of A, V, N, and , the average speed of an air molecule. (Hint: the flux of molecules moving in the + x direction can be written, , where  is the number of molecules per volume.)

b) Find an expression for the time it will take for the number of air molecules inside the spacecraft to reduce to N/2, one half of the starting number. Express your answer in terms of A, V, and .

The rate of change of the number of molecules in the spacecraft is the negative of the rate at which molecules are leaving the spacecraft.

. Therefore, , and . That gives,

, Where N0 is the number of molecules at t = 0. Now, find when N = ½ N0.

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