Entropy in the Real World: Engines — Question 4

Question 4

The adiabatic legs of a Carnot gas cycle enforce matching isothermal ratios

A .200 mol ideal gas executes a reversible Carnot cycle with $T_h=500$ K, $T_c=300$ K and $\gamma=1.40$. Hot isothermal expansion A to B has $V_B/V_A=2.00$. B to C is adiabatic expansion, C to D cold isothermal compression, and D to A adiabatic compression. Use $TV^{\gamma-1}=\text{constant}$ and $R=8.314$ J/(mol K).

1. Find $V_C/V_B$, $V_D/V_A$ and $V_C/V_D$.

2. Calculate hot and cold heat magnitudes and cycle work, verifying efficiency without explicitly integrating the adiabatic work.

Question 4 - Solution

The adiabatic legs of a Carnot gas cycle enforce matching isothermal ratios

(a) Method and calculation

$$V_C/V_B=V_D/V_A=(T_h/T_c)^{1/(\gamma-1)}=(5/3)^{5/2}\approx3.58610.$$$$\frac{V_C}{V_D}=\frac{V_B}{V_A}=2.00.$$

(b) Interpretation and further calculation

$$Q_h=nRT_h\ln2\approx576.283\,\mathrm J,\quad Q_c=nRT_c\ln2\approx345.770\,\mathrm J.$$$$W=Q_h-Q_c\approx230.513\,\mathrm J,\quad\eta=1-300/500=.400.$$The adiabatic legs exchange work but no heat; their internal-energy changes cancel over the whole cycle.

Physical check

Equal isothermal volume ratios follow from both adiabatic constraints, not from an assumed symmetric drawing.