Prove that the work done by a thermodynamic system depend on the … - Vidyadip

Question

Prove that the work done by a thermodynamic system depend on the path on which the expansion take place.

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Answer

Three possible reactions leading to the following are shown in the figure:

(i) If the transformation is done through $i$ a path, then
$\begin{aligned}d \omega_{(i a f)} & =\text { Area of rectangle (iafed) } \\& =d \omega_{(i a)}+d \omega_{(a f)} \\d \omega_{(i a f)} & =P_1\left(V_2-V_1\right)+0 \quad \because d V=0 \\& =P_1\left(V_2-V_1\right)\end{aligned}$
(ii) If the transformation is done through $i b f$ path, then
$\begin{aligned}d \omega_{(i b f)} & =\text { Area of rectangle }(\text { ibfcdi }) \\& =d \omega_{(ib)}+d \omega_{(bf)} \\& =0+P_2\left(V_2-V_1\right) \quad \because d V=0 \\d \omega_{(i b f)} & =P_2\left(V_2-V_1\right)\end{aligned}$
Similar do d $\omega$ (ief) = area of curve (iefcdi) path hence it clear that the value of $d \omega$ is different for different paths. It also clear that :
$d \omega$ (iaf) is higher
$d \omega$ (ief) is less
And $d \omega( ibf )$ is the lowest
Therefore, the work done by a thermodynamic system depends on the path on which the expansion takes place.

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