Question

Derive an expression for work done during isothermal process

Answer 1

Suppose one mole of ideal gas undergoes isothermal expansion from V1 to V2.
Then the work done is given by

Answer 2

In isothermal process the work done is $\bold{W=R T \ln \frac{P_{1}}{P_{2}}}$.

Given:

As we know that work done during a process is the integral product of all the work done.  

Solution:

$\begin{array}{l}{W=\int d W} \\ {W=\int V_{1} V_{2} P d V}\end{array}$

For one mole of gas, PV = RT  

So, $\begin{aligned} P &=\frac{R T}{V} \\ W &=\int V_{1} V_{2} P d V \\ W &=\int V_{1} V_{2}\left(\frac{R T}{V}\right) d V \\ W &=R T \int V_{1} V_{2}\left(\frac{1}{V}\right) d V \\ W &=R T \ln \frac{V_{2}}{V_{1}} \end{aligned}$

For the isothermal process, we have the temperature during the whole process be constant,  

$\begin{array}{l}{P_{1} V_{1}=P_{2} V_{2}} \\ {\frac{P_{1}}{P_{2}}=\frac{V_{2}}{V_{1}}}\end{array}$

Thereby we have, $W = R T \ln \frac{V_{2}}{V_{1}}$

$W=R T \ln \frac{P_{1}}{P_{2}}$.