The Molar Specific Heats of an Ideal Gas — Question 10

Question 10

A controlled cooling transient identifies heat capacity from its time scale

A rigid vessel contains .500 mol of ideal gas with constant $C_V$ and negligible vessel heat capacity. Heat loss follows $\dot Q=-G(T-T_a)$ with $G=.250$ W/K and room temperature $T_a=300$ K. A small fan keeps the gas uniform without significant energy input. Starting at 360 K, the gas reaches 330 K after $40.0\ln2$ seconds.

1. Derive the temperature history and infer $C_V$ from the measurement.

2. How long after the start does temperature reach 315 K? If the gas amount doubles with unchanged $G$ and $C_V$, how does that time change?

Question 10 - Solution

A controlled cooling transient identifies heat capacity from its time scale

(a) Method and calculation

$$nC_V\frac{dT}{dt}=-G(T-T_a)\ \Rightarrow\ T-T_a=(T_i-T_a)e^{-t/\tau},\quad\tau=\frac{nC_V}{G}.$$$$\frac{330-300}{360-300}=\frac12=e^{-(40\ln2)/\tau}\ \Rightarrow\tau=40.0\,\mathrm s,\quad C_V=\frac{G\tau}{n} =20.0\,\mathrm{J/(mol\,K)}.$$

(b) Interpretation and further calculation

$$\frac{315-300}{360-300}=\frac14\ \Rightarrow\ t=\tau\ln4=80.0\ln2\,\mathrm s\approx55.4518\,\mathrm s.$$$$n\to2n\ \Rightarrow\tau\to2\tau,\quad t\to110.904\,\mathrm s.$$The capacity is inferred from the decay curve without assigning a molecular model.

Physical check

The time scale determines total thermal capacity divided by conductance; a nonnegligible vessel would add to the inferred total.