The Molar Specific Heats of an Ideal Gas — Question 2

Question 2

Two calorimetric measurements determine both amount and molar capacity

Use $pV=nRT$, $R=8.314$ J/(mol K), and the sign convention $\Delta U=Q-W$, with $W$ the work done by the gas. Only pressure–volume work occurs. The same unknown ideal-gas sample requires 900 J at fixed volume and 1260 J at fixed pressure for an identical 40.0 K temperature rise. Capacities are constant, and apparatus heat capacity is negligible.

1. Infer the amount of gas, $C_V$ and $C_p$.

2. Find $\gamma=C_p/C_V$ and test a proposed second data set with $Q_p=800$ J and $Q_V=900$ J for the same positive rise.

Question 2 - Solution

Two calorimetric measurements determine both amount and molar capacity

(a) Method and calculation

$$Q_p-Q_V=nR\Delta T\ \Rightarrow\ n=\frac{360}{8.314(40)}\approx1.08251\,\mathrm{mol}.$$$$C_V=\frac{900}{n(40)}=2.50R=20.785\,\mathrm{J/(mol\,K)},\quad C_p=3.50R=29.099\,\mathrm{J/(mol\,K)}.$$

(b) Interpretation and further calculation

$$\gamma=1260/900=1.40.$$$$Q_p-Q_V=nR\Delta T>0\quad(n>0,\Delta T>0).$$The proposed negative difference would infer $n<0$ and is incompatible with the stated ideal-gas experiments. Apparatus effects or other work channels would require a revised model.

Physical check

The heat difference eliminates the unknown internal-energy change and isolates expansion work.