PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2000

Quiz 2 Thursday, April 13, 2000

NAME: ______SOLUTIONS______

To receive credit for a problem, you must show your work, or explain how you arrived at your answer.

1. (30%) Enthalpy is defined as H = E + pV, where E is internal energy, p is pressure, and V is volume.

a) Show that: dH = TdS + Vdp + dN

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

dH = TdS – pdV + dN + pdV +Vdp = TdS + Vdp + dN

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

H is a function of S, p,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.

2. (20%) For a certain system, there is only one accessible state and it has energy,

where V0 is a constant.

a)What is the partition function for this system?

b) Use the result of part a) to find the average pressure for this system as a function of temperature and volume.

or, as above.

3. (40%) One mole of an ideal momatomic gas (three degrees of freedom per molecule) goes through the cycle shown in the diagram.

The cycle begins at point 1, where the pressure is p0, the volume is V0, and the temperature is T0. It proceeds to point 2, then to 3, to 4, and back to 1. Heat is put into the gas as it goes from 1 to 2 to 3, and heat is removed as it goes from 3 to 4 and back to 1.

a) Express T0in terms of p0 and V0.

For one mole, p0V0 = RT0, so

b) What is the temperature of the gas at points 2, 3, and 4 in terms of T0?

2p0V0 = RT22p02V0 = RT3 p02V0 = RT4

c) What is the net work done by the gas as it goes through the cycle, in terms of p0 and V0?

The net work equals the area of the cycle which equals p0V0.

d) How much heat is added to the gas as it goes from 1 to 2, in terms of p0 and V0?

Q12 = W12 +E12 . Since W12 = 0, Q12 = E12

For one mole of a monatomic ideal gas, , so

e) How much heat is added to the gas as it goes from 2 to 3, in terms of p0 and V0?

Q23 = W23 +E23 . Here, W23 = 2 p0V0.

Then, Q23 = 2 p0V0 +3 p0V0 = 5 p0V0.

f) Use the results of parts c), d) and e) to calculate the efficiency of the cycle if used as a heat engine. The answer should be a number. (You can get credit even if c), d) and e) are not all correct.)

. In this case, Q1 = Q12 + Q23 = , and W = p0V0.

Then,

g) Find the efficiency of a Carnot cycle, if it operated between reservoirs at temperatures of T3 and T0, the highest and lowest temperatures reached by the gas in the cycle shown in the diagram. The answer should be a number.

4. (10%) The flux of heat that flows down a temperature gradient in the x direction is given by,

,

where K is the thermal conductivity which is given by,

.

In addition, , so K is a function of temperature.

Suppose heat flows through a gas from a high temperature reservoir at temperature T1 to a low temperature reservoir at T2. The two reservoirs are a distance L apart. Show that Qxis given by,

.

Find the constant a in terms of , cm, l, m, L, and fundamental constants.

Clearly, K and will vary with position, but Qx will not. The heat that goes in at the high temperature end must come out at the low temperature end, in steady state. If we substitute for K and in the thermal conductivity equation, we can integrate it to get T as a function of x.

, where

. Then, .

Now integrate the left side from x = 0 to x = L, and integrate the right side from T1 to T2.

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