Figure shows a cyclic process performed on one mole of an ideal gas. A total of 1000 J of heat is withdrawn from the gas in a complete cycle. Find the work done on/by the gas, in joule, during the process B$\to$C. Given R = 8.3 J mol^-1 K^-1 

For a cyclic process, $\mathrm{\Delta U}=0$

$\Rightarrow {\mathrm{Q}}_{\text{cycle }}={\mathrm{W}}_{\text{cycle }}$   ……..(1)

where, ${\mathrm{W}}_{\text{cycle }}={\mathrm{W}}_{\mathrm{A}\to \mathrm{B}}+{\mathrm{W}}_{\mathrm{B}\to \mathrm{C}}+{\mathrm{W}}_{\mathrm{C}\to \mathrm{A}}$ …….(2)

Process C$\to$A is isochoric, hence ${\mathrm{W}}_{\mathrm{C}\to \mathrm{A}}=0$

Process A $\to$B is isobaric, as its T-V graph is a straight line passing through origin.

In an isobaric process, work done is

$\begin{array}{l} {\mathrm{W}}_{\mathrm{A}\to \mathrm{B}}=\mathrm{P}\left({\mathrm{V}}_{\mathrm{B}}-{\mathrm{V}}_{\mathrm{A}}\right)=\mathrm{nR}\left({\mathrm{T}}_{\mathrm{B}}-{\mathrm{T}}_{\mathrm{A}}\right)\\ \Rightarrow {\mathrm{W}}_{\mathrm{A}\to \mathrm{B}}=1\times 8.3(400-300)=830\mathrm{J} ........\left(3\right)\end{array}$

${\mathrm{W}}_{\mathrm{A}\to \mathrm{B}}$ is positive, so expansion of gas takes place.

From equations (1), (2) and (3), we get

$\begin{array}{l}-1000=830+{\mathrm{W}}_{\mathrm{B}\to \mathrm{C}}+0\\ \Rightarrow {\mathrm{W}}_{\mathrm{B}\to \mathrm{C}}=-1830 \mathrm{J}\end{array}$

Negative work done implies that compression of the gas takes place or work is done on the system/gas.