An ideal (but not necessarily perfect monatomic) gas undergoes the following cycle.
- The gas starts at pressure P_{0}, volume V_{0} and temperature T_{0}.
- The gas is heated at constant volume to a pressure \alpha P_{0}, where \alpha>1.
- The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure P_{0}
- The gas is cooled at constant pressure back to the original state.
The adiabatic constant \gamma is defined in terms of the specific heat at constant pressure C_{p} and the specific heat at constant volume C_{v} by the ratio \gamma=C_{p} / C_{v}.
a. Determine the efficiency of this cycle in terms of \alpha and the adiabatic constant \gamma. As a reminder, efficiency is defined as the ratio of work out divided by heat in.
b. A lab worker makes measurements of the temperature and pressure of the gas during the adiabatic process. The results, in terms of T_{0} and P_{0} are
\begin{array}{|l|c|c|c|c|c|c|} \hline
Pressure & \text{Units of } P_{0} & 1.21 & 1.41 & 1.59 & 1.73 & 2.14 \\ \hline
Temperature & \text{Units of } T_{0} & 2.11 & 2.21 & 2.28 & 2.34 & 2.49 \\ \hline
\end{array}
Plot an appropriate graph from this data that can be used to determine the adiabatic constant.
c. What is \gamma for this gas?