Question

A fixed mass of a gas is first heated isobarically to double the volume and then cooled isochorically to decrease the temperature back to the initial value. By what factor would the work done by the gas decreased, had the process been isothermal?

• A. $2$
• B. $\frac{1}{2}$
• C. $\ln 2$
• D. $\ln 3$

Answer 1

Answer: (C)

$\ln 2$

Let initial pressure and volume be P and V respectively.
Then after isobaric expansion they are P and 2V. Here work done=$P\Delta V=P(2V-V)=PV$
To bring to initial temperature new pressure=$\frac{PV}{2V}=P/2$.

So after isochoric process they are $P/2and2V$ . Here, $\Delta V=0\Rightarrow$ work done=$P\Delta V=0$
Hence, total work done in the 2 successive processes=PV +0=PV
For isothermal expansion, work done=$PVln\left(\frac{V_f}{V_i}\right)=PV\ln 2$
Hence required factor=$\frac{PV\ln 2}{PV}=ln2$