A Closer Look at Heat and Work — Question 8

Question 8

Equal volume ratios give equal work along an isotherm

A supplied ideal-gas model has $pV=K=600\,\mathrm J$ throughout a quasistatic constant-temperature expansion. It moves from 2.00 L to 8.00 L. Use $W=\int p\,dV$.

1. Find the work and the intermediate volume at which half the total work has been done.

2. Compare the work from 2 to 4 L with the work from 4 to 8 L. Explain why the halfway volume is not 5 L.

Question 8 - Solution

Equal volume ratios give equal work along an isotherm

(a) Method and calculation

$$W=K\int_{V_i}^{V_f}\frac{dV}{V}=K\ln\frac{V_f}{V_i}=600\ln4\approx831.777\,\mathrm J.$$$$\ln(V_m/V_i)=\tfrac12\ln(V_f/V_i)\ \Rightarrow\ V_m=\sqrt{V_iV_f}=4.00\,\mathrm L.$$

(b) Interpretation and further calculation

$$W_{2\to4}=600\ln2=W_{4\to8}\approx415.888\,\mathrm J.$$$$\frac{W_{2\to5}}{W_{2\to8}}=\frac{\ln(5/2)}{\ln4}\approx.66096.$$At 5 L more than half the work is already done because earlier volume increments occur at higher pressure.

Physical check

The logarithm acts on a dimensionless volume ratio; using litres or cubic metres gives identical work.