91Ó°ÊÓ

• Work done in isothermal expansion is$50\text{ J}$
• Work done in isothermal compression is$30\text{ J}$
• Work done in adiabatic expansion is$40\text{ J}$
• Work done in adiabatic compression is$25\text{ J}$

The expression for the heat from the first law of thermodynamics is given by,

$Q=W+\mathrm{Δ}U$

Here Q is the heat required, W is the work done, $\mathrm{Δ}U$is the change in internal energy.

So, the first law of thermodynamics gives the relation between heat, work and internal energy.

Since internal energy is a state function, change in internal energy along path 1 and 2 are equal.

$\mathrm{Δ}{U}_1=\mathrm{Δ}{U}_2$

For adiabatic expansion,

Work is done by the system, so work done is positive.

$W=40\text{J}$

First law of thermodynamics gives,

$Q=W+\mathrm{Δ}{U}_1$

For adiabatic process$\text{Q}=0$

So,

$\begin{array}{c}0=40+\mathrm{Δ}{U}_1\\ \mathrm{Δ}{U}_1=−40\text{ J}\end{array}$

For adiabatic compression,

Work is done on the system, so work done is negative.

$W=−25\text{ J}$

$Q=W+\mathrm{Δ}{U}_2$

For adiabatic process$Q=0$

So,

$\begin{array}{c}0=−25+\mathrm{Δ}{U}_2\\ \mathrm{Δ}{U}_2=25\text{ J}\end{array}$

So total change in internal energy is

$\begin{array}{c}\mathrm{Δ}U=\mathrm{Δ}{U}_1+\mathrm{Δ}{U}_2\\ =−40+25\\ ⇒\mathrm{Δ}U=−15\text{ J}\end{array}$

Therefore, the change in internal energy along path 2 is$−15\text{ J.}$