A Closer Look at Heat and Work — Question 4

Question 4

A piston works against atmosphere and an added load

A frictionless piston of area $.0100\,\mathrm{m^2}$ rises quasistatically by .200 m against atmospheric pressure 100 kPa and an added mass 50.0 kg. Neglect piston mass. Take $g=9.80\,\mathrm{m/s^2}$.

1. Find the gas pressure and the total work done by the gas.

2. Separate work against the atmosphere from the load's potential-energy increase. What fraction of gas work raises the load?

Question 4 - Solution

A piston works against atmosphere and an added load

(a) Method and calculation

$$p=p_{\mathrm{atm}}+\frac{Mg}{A}=100000+49000=149000\,\mathrm{Pa}.$$$$\Delta V=A\Delta x=.00200\,\mathrm{m^3},\quad W=p\Delta V=298\,\mathrm J.$$

(b) Interpretation and further calculation

$$W_{\mathrm{atm}}=p_{\mathrm{atm}}A\Delta x=200\,\mathrm J,\quad\Delta E_g=Mg\Delta x=98.0\,\mathrm J.$$$$W=200+98=298\,\mathrm J,\quad\frac{\Delta E_g}{W}=\frac{98}{298}\approx.32886.$$Only about 32.9% raises the added load; the rest displaces the atmosphere.

Physical check

The pressure in the work integral is absolute pressure when atmospheric displacement is included.