PHYSICS 313
Problem set 5 2001
Given October 22
Due Oct 31

Problem 1: A volume $V_1=1$ m$^3$ of air is initially at $T_1=300$ K, and the pressure is $P_1=$1 bar. Assume air to be an ideal gas with $f=5$. The gas is compressed until the volume is $V_2=$ one half and the pressure is $P_2=$ twice the initial value, along a straight line in a $PV$ diagram. Along this line we can write for the pressure

$$P=P_1+(V-V_1)\frac{P_2-P_1}{V_2-V_1}$$

a: Show that if the volume changes by $dV$ along the straight line segment the heat provided to the gas will be

$${d\mkern-6mu\mathchar'26}Q=PdV+C_VdT$$

b: During which part of the compression will heat be provided to the gas? During which part will the gas expel heat?
c: Is the point along the straight line segment where ${d\mkern-6mu\mathchar'26}Q$ changes sign the same as where the temperature is a maximum?
Problem 2: In the "Sargent" ideal gas cycle n mols of the gas is initially at temperature $T_1$. The gas is then compressed adiabatically until the temperature is $T_2$. It is then heated at constant volume until the temperature is $T_3$. It is next expanded adiabatically until the temperature is $T_4$. Finally the cycle is completed by a compression at constant pressure until the temperature returns to $T_1$.
a: What is the heat $Q_H$ provided to the gas.
b: What is the waste heat $Q_C$ given up by the gas.
c: What is the net work done by the system
d: Assume that

$$\gamma=\frac{C_P}{C_V}=1.4$$

$T_1=300$ K, $T_2=554$ K, $T_3=754$ K, $T_4=360$ K. What is the thermodynamic efficiency $W/Q_H$?