The Adiabatic Expansion of an Ideal Gas — Question 4

Question 4

Endpoint measurements infer a constant adiabatic exponent

Use $pV=nRT$, $R=8.314$ J/(mol K), and $\Delta U=Q-W$ with $W$ positive for work done by the gas. A gas follows a reversible adiabatic path with constant capacities. Its volume increases by a factor of 2.50 while pressure decreases to one-quarter of its initial value.

1. Infer $\gamma$, $C_V/R$ and the temperature ratio.

2. If equipartition is used to define $f=2C_V/R$, calculate the effective count and discuss whether these endpoint data identify a unique pure molecular species.

Question 4 - Solution

Endpoint measurements infer a constant adiabatic exponent

(a) Method and calculation

$$p_iV_i^\gamma=p_fV_f^\gamma\ \Rightarrow\ 4=(2.50)^\gamma\ \Rightarrow\gamma=\frac{\ln4}{\ln2.50}\approx1.51294.$$$$C_V/R=\frac1{\gamma-1}\approx1.94954,\quad T_f/T_i=(p_f/p_i)(V_f/V_i)=.625.$$

(b) Interpretation and further calculation

$$f=\frac2{\gamma-1}\approx3.89908.$$$$T_f/T_i=(2.50)^{1-\gamma}=.625.$$A noninteger effective count can reflect a mixture or internal-mode response. Identification also depends on the assumptions of reversibility and constant capacities; endpoint ratios alone do not establish them.

Physical check

The log–log slope identifies a path exponent only within the stated model.