PHYSICS 313
Sessional Examination, December 18 2000

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 two sheets of notes (4pages), calculator.

Problem 1:
a: The heat capacity at constant pressure for a mole of $O_2$ is $C_P=29.38\; J K^{-1}$ at room temperature. What is the thermodynamic number of degrees of freedom $f$ at this temperature?
b: Give a physical interpretation of this result.
c: Answer the same questions for a mole of $CH_4$ for which $C_P$ is $35.31\; J K^{-1}$ at room temperature.

Problem 2:
a: A container contains a monatomic ideal gas with $N$ atoms at temperature $T$ in a volume $V$. We now halve the number of particles keeping $V$ and $T$ constant. What is the change in the chemical potential? (A formula for the chemical potential of a monatomic ideal gas is given at the end of the exam).
b: We didn't derive a formula for the chemical potential when the gas molecules have internal degrees of freedom, but let us assume that the chemical potential of a diatomic gas is

$$\mu=-kT\ln\left[\frac{cVT^{5/2}}{N}\right]$$

where c is a constant. A diatomic ideal gas, originally with N molecules, temperature T and volume V, is heated at constant N,V so that the temperature is doubled. What is the change in the chemical potential?
c: The inner surface of the container contains absorption sites to which gas molecules can stick. Assume that the probability that a given absorption site is occupied is

$$p=\frac{1}{e^{(\epsilon-\mu)/kT}+1}$$

where $-\epsilon$ is the binding energy to the site and $\mu$ is the chemical potential of the gas. Assume that at a given pressure the probability that a site is occupied is 0.5. What is the probability if one half of the molecules are pumped out of the container keeping $V$ and $T$ constant? Does it matter if the gas is monatomic or diatomic?
Problem 3:
A van der Waals fluid satisfies the equation of state

$$(P+\frac{aN^2}{V^2})(V-Nb)=NkT$$

and the critical points occurs at

$$V_c=3Nb;\;\;P_c=\frac{a}{27b^2};\;\;kT_c=\frac{8a}{27b}$$

The compression factor of a fluid is defined as the ratio

$$\frac{PV}{NkT}$$

a: Find the compression factor for a van der Waals fluid at the critical point. Show that this quantity is independent of the constants $a$ and $b$.
b: For water at the critical point $T_c=647K$, $P_c=220.6 bar$ and the compression ratio is 0.274. Find $V_c$ for 1 mole of water in $m^3$.
c: You can fit the constants $a$ and $b$ to any of two of the three quantities $T_c, P_c$ and $V_c$. The third quantity will then deviate from its true value. What are the values of $a$ and $b$ in SI units if you fit to $T_c, P_c$? to $P_c,V_c$?

Problem 4:
A car battery contains 6 lead acid cells. Each cell runs on the reaction

$$Pb+PbO_2+4H^++2SO_4^{2-}->2PbSO_4+2H_2O$$

We have $\Delta G=-394\;kJ$, $\Delta H=-316\; kJ$ per mole for the reaction at $T=298K$. At -electrode 2 electrons are given up to the solution while 2 electrons are picked up at the +electrode for each time the reaction occurs.
a: What is the voltage of the battery?
b: Is excess heat produced when the battery is charged or when it is discharged? Justify your answer.
c: Estimate how the voltage changes if the temperature is increased to 320 K.

Problem 5:
A heat pump is a device which heats a volume, e.g. a building, by pumping in heat from the cold outside. Let $T_h\;(T_c)$ be the inside (outside) temperature, $Q_h$ the heat provided and $W$ the work needed to provide the heat. The coefficient of performance is $COP=Q_h/W$.
a: Use the second law of thermodynamics to derive an upper limit to the achievable COP.
b:It is $0^oC$ outside and $20^oC$ inside. How many kilowatt hours of electric energy would be the minimum required to produce a kW hour of heat, using a heat pump? What would be the answer if it is $-40^oC$ outside?
c: In the summer a heat pump be run in reverse to perform as an air conditioner. How many kW hours of work would be the minimum required to remove one kW hour of heat from the building, if it is $20^oC$ inside and $35^oC$ outside.
-------------------
Some formulas:
Chemical potential of monatomic ideal gas:

$$\mu=-kT\ln\left[\frac{V}{N}\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\right]$$

Gibbs free energy:

$$G=\mu N=U+PV-TS$$

Helmholtz free energy:

$$F=U-TS$$

Differential form of second law:

$$dU=TdS-PdV+\mu dN$$

Heat capacity/mole at constant volume of ideal gas

$$C_V=\frac{fR}{2}$$

$f=$ number of thermodynamic degrees of freedom/molecule.
Specific heat at constant pressure of ideal gas

$$C_P=C_V+R$$

Some constants

$$R=8.315 J mol^{-1}K^{-1}$$

$$1 bar=10^5 Nm^{-2}$$

$$N_A=6.022\times10^{23}$$

$$e=1.602\times 10^{-19}C$$

$$0^oC=273.16K$$

Return to title page