PHYSICS 313
Sessional exam, December 13 2001

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:
a: Which of the following statements is most likely to be correct? The heat capacity $C_P$ at constant pressure for a mole of $O_2$ is

1

$C_P=29.38\; J K^{-1}$

2

$C_P=21.57\; J K^{-1}$

3

$C_P=35.31\; J K^{-1}$

Justify your answers!
b:For an ideal gas $C_P-C_V=nR$ where $n$ is the number of moles of gas. This implies that $C_P>C_V$. Explain qualitatively why $C_P$ must be greater than $C_V$ for an ideal gas.

Problem 2:
The Free energy of Black body radiation is

$$F=-{{\pi ^2V}\over{45\hbar ^3c^3\beta ^4}}$$

where $\beta=1/kT$, $c$ is the speed of light and $\hbar$ is Planck's constant divided by $2\pi$. Derive formulas for
a: The entropy and pressure of black-body radiation.
b: The internal energy and enthalpy.
c: Check that the Gibbs free energy $G=N\mu=0$ for black body radiation. Give a physical reason that the chemical potential has to be zero in this case.

Problem 3:
a: An imperfect gas obeys the equation of state

$$(P+\frac{an^2}{V^2})(V-nb)=nRT$$

where $n$ is the number of moles of gas. For this gas $a=0.5$ J m$^3$ mole$^{-2}$ and $b=3\times10^{-5}$ m$^3$ mole$^{-1}$. What is the work required to compress 0.2 mole of this gas from $10^{-3}$ m$^3$ to $10^{-4}$ m$^3$ at $500$ K. Express your result as a formula before substituting numbers.
b: If the same process had taken place for an ideal gas the work done in compressing the gas would equal to the heat expelled to keep the temperature constant. For the imperfect gas this will not be true. Explain why.

Problem 4:
When lead is melted at the pressure of 1 bar the melting point is $327.0^oC$, the density decreases from $1.101$ to $1.065\times10^4$ kgm$^{-3}$ The latent heat of melting is $24.5$ kJ kg$^{-1}$. How much will the melting temperature change if the pressure is increased to 100 bar. Derive a formula for the temperature change before substituting numbers. Is the melting temperature increasing or decreasing?

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:A building is heated by a heat pump which may be assumed to have an ideal performance. The pump consumes a constant power $W$ and looses heat to its surroundings at the rate $\alpha(T_h-T_c)$. What temperature will the building reach if the outside temperature is constant at $T_c$?
-------------------
Some formulas:
$G=\mu N=U+PV-TS$ Gibbs free energy
$F=U-TS$ Helmholtz free energy
$dU=TdS-PdV+\mu dN$ Differential form of second law
Heat capacity/mole at constant volume of ideal gas

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

$f=$ number of thermodynamic degrees of freedom/molecule.
Clausius Clapeyron equation

$$\frac{dP}{dT}=\frac{L}{T\Delta V}$$

Some constants

$$R=8.315\; {\rm J mole}^{-1}{\rm K}^{-1}$$

$$1 \;{\rm bar}=10^5 \;{\rm N\;m}^{-2}$$

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

$$0^o{\rm C}=273.16\;{\rm K}$$