f2002p313

PHYSICS 313
Sessional exam, December 10 2002

Answer 4 out of the 5 problems. All problems have equal value. If all 5 problems are attempted credit will be given for the best 4 answers. A number of potentially useful formulas are given at the end of the paper. Time 2 $\frac{1}{2}$ hours. Allowed aids three sheets of notes (6 pages), calculator.

Problem 1:
An imperfect gas satisfies the van der Waals equation of state. The Helmholtz free energy can be written

$\begin{displaymath}F=-Nk_BT(\ln (\frac{V-Nb}{Nv_q})+1)-\frac{a N^2}{V}\end{displaymath}$

where

and

are constants and

$\begin{displaymath}v_q=(\frac{2\pi\hbar^2}{mk_BT})^{3/2}\end{displaymath}$

a: Find the entropy of such a gas with

particle at temperature

, volume

.
b: What is the heat capacity

, under the conditions of a: ?
c: The gas is expanded from volume

to 2

, keeping the temperature constant.

What is the change in internal energy during the expansion?
How much work is done by the gas during the expansion?

Problem 2:
In a hydrogen fuel cell the chemical reactions are

At negative electrode $H_2+2OH^-\Rightarrow 2H_20+2e^-$
At positive electrode $\frac{1}{2}O_2+H_20+2e^-\Rightarrow 2OH^-$
The net result is $H_2+\frac{1}{2}O_2\Rightarrow H_2O$ .

Some thermodynamic properties per mol of the constituents at 298 K, 1 bar are listed below

Substance	$\Delta H$ (kJ)	$\Delta G$ (kJ)	(JK $^{-1})$	(JK $^{-1})$	(cm)
	0	0	130.68	28.82
	-285.83	-237.13	69.91	75.29	18.068
	0	0	205.14	29.38

a: What is the voltage of the cell?
b: What is the change in the Gibbs free energy for the constituents if the pressure of the oxygen, hydrogen and water each are increased from 1 bar to 10 bar. The temperature is kept constant at 298 K. Neglect any volume change of the liquid water with pressure.
c: How much will the voltage of the cell increase or decrease after the pressure has been increased?
Problem 3:
You have a system in which particles can occupy any one of two energy levels with energy $\epsilon_1$ and $\epsilon_2$ respectively. Assume that there are two particles in the system and that the temperature is given by $\beta=1/k_BT$ . Find the probability that any one particle is in state 1 and the other in 2 if
a: The particles are distinguishable.
b: The particles are identical and two particles are forbidden from occupying the same state (fermions).
c: The particles are identical and several particles are allowed to occupy the same state (bosons).
Problem 4:
One mole of a monatomic ideal gas is initially at 300 K and 1 bar. The system then undergoes the following cyclic sequence of processes:

Heating at constant volume until the temperature is 400 K.
Adiabatic expansion until the temperature returns to 300 K.
Isothermal compression until the volume returns to its initial value.

a: Draw a PV diagram for the cycle?
b: Will the system work as an engine or a heat pump? Justify your answer!
c: Work out the heat and work for each step of the cycle. Specify if the heat and work is added to the system or taken from it.
Problem 5:
The temperature and of air at ground level is 293 K. The saturation vapor pressure at that temperature is 0.023 bar and the actual partial pressure 50% of that (50% relative humidity). The latent heat of evaporation of water around that temperature is 44 KJ/mol. Assume that the temperature drops at a rate of 1 K per 100 m.
a: Neglect the pressure drop with altitude and estimate at what altitude will the saturation water pressure be exceeded, and a cloud forms.
b: Not only the temperature, but also the pressure will drop with altitude. Estimate how much the vapor pressure has dropped at the altitude you calculated under a:.
-------------------
Some formulas:
$G=\mu N=U+PV-TS=\mu N$ Gibbs free energy