A Closer Look at Heat and Work — Question 1

Question 1

Two rectangular routes connect identical pressure-volume endpoints

A quasistatic gas process connects A $(V=2.00\,\mathrm L,p=100\,\mathrm{kPa})$ to B $(5.00\,\mathrm L,300\,\mathrm{kPa})$. Route H raises pressure first at fixed volume, then expands at 300 kPa. Route L expands first at 100 kPa, then raises pressure at fixed volume. Work done by the gas is $W=\int p\,dV$; $1\,\mathrm{kPa\,L}=1\,\mathrm J$.

1. Calculate work on each route and explain the contribution of the vertical legs.

2. Form a cycle by following H forward and L backward. Find net work and relate it to the enclosed area.

Question 1 - Solution

Two rectangular routes connect identical pressure-volume endpoints

(a) Method and calculation

$$W_H=0+300(5-2)=900\,\mathrm J.$$$$W_L=100(5-2)+0=300\,\mathrm J.$$A fixed-volume leg has $dV=0$, regardless of its pressure change.

(b) Interpretation and further calculation

$$W_{\mathrm{cycle}}=W_H-W_L=600\,\mathrm J.$$$$A_{pV}=(300-100)(5-2)=600\,\mathrm J.$$Expansion at the upper pressure and compression at the lower pressure produce positive clockwise-cycle work.

Physical check

Equal endpoints do not fix work because the pressure history during displacement matters.