PHYSICS 313 Midterm October 1 2003.

Answer all questions. Allowed aids 1 double-sided formula sheet. Calculator.
Problem 1:
One mol of an ideal gas undergoes the following processes.
A Heating at constant volume from a state $P_0,V_0,T_0$ to one in which the pressure is $P=2P_0$.
B Expansion at constant temperature until $V=2V_0$.
C Cooling at constant volume until the temperature returns to $T_0$.
D Compression at constant temperature until $V=V_0$.
a: Draw a $P,V$ diagram for the process.
b: Using the sign convention for the fist law of thermodynamics

$$\Delta U=Q+W$$

list the sign of the heat $Q$ and work $W$ at the four stages of the process.
c: Calculate $Q$ and $W$ for each of the four steps.
Problem 2:
A hot air balloon has a volume of $500$ m$^3$. The temperature of the air inside is 450 K. The outside temperature is T=300 K. At ground level the pressure is 1 bar. The mass of one mol of air is $M=29\;10^{-3}$ kg mol$^{-1}$.
a: What is the mass of the air inside the balloon?
b: What is the maximum mass of the balloon plus payload, excluding the air inside, for the balloon to be able to rise. Neglect the volume of the "skin" of the balloon and its payload.
Problem 3:
Consider an Einstein solid at the Einstein temperature
a: How many vibrational quanta will there be per oscillator on the average?
b: What is the entropy per oscillator?
Some formulas: $R=8.315$ J mol$^{-1}$ K$^{-1}$
Ideal gas: $PV=nRT$ $, U=\frac{f}{2}nRT$
$N_A=6.022\times 10^{23}, k_B=1.381\times 10^{-23}$ J K$^{-1}$.
1 atm $=1.013$ bar $=1.013\times 10^5$ N m$^{-2}$.
For Einstein solid

$$S=k_B[(N+q)\ln(N+q)-N\ln N -q\ln q]$$

$$U=\frac{Nk_BT_E}{\exp\frac{T_E}{T}-1};\;\; T_E=\frac{\hbar\omega}{k_B}$$