A Closer Look at Heat and Work — Question 9

Question 9

A measured work value identifies an unknown pressure peak

A quasistatic expansion starts at $(1.00\,\mathrm L,100\,\mathrm{kPa})$, follows a straight line to $(3.00\,\mathrm L,p_m)$, then another straight line to $(5.00\,\mathrm L,100\,\mathrm{kPa})$. The measured work done by the gas is 800 J.

1. Infer $p_m$ using signed areas.

2. Find the work if the gas returns directly at 100 kPa and hence the net work for the closed loop. What would the same measured 800 J mean for a constant-pressure expansion over 4 L?

Question 9 - Solution

A measured work value identifies an unknown pressure peak

(a) Method and calculation

$$W=\frac{100+p_m}{2}(3-1)+\frac{p_m+100}{2}(5-3).$$$$800=200+2p_m\ \Rightarrow\ p_m=300\,\mathrm{kPa}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{return}}=100(1-5)=-400\,\mathrm J,\quad W_{\mathrm{loop}}=400\,\mathrm J.$$$$p_{\mathrm{constant}}=800/4=200\,\mathrm{kPa}.$$Work constrains an area; without a specified path shape it cannot uniquely determine a peak pressure.

Physical check

The extra work above the return pressure equals the area of the triangular pressure excess.