question_answer

A sample of gas expands from volume \[{{\text{V}}_{\text{1}}}\]to \[{{\text{V}}_{\text{2}}}\text{.}\]The amount of work done by the gas is greatest when the expansion is :

A)  adiabatic                            

B)  isobaric

C)  isothermal                        

D)  equal in all above cases

Correct Answer: B

Solution :

                The P-V diagram for isobaric, isothermal and adiabatic processes of an ideal gas is shown in graph below: In thermodynamics, for same change in volume, the work done is maximum for the curve having area enclosed with the volume axis. Area enclosed by the curve \[\propto {{(Slope\,of\,curve)}^{-1}}\] As shown, \[{{(slope)}_{isobaric}}<{{(slope)}_{isothermal}}<{{(slope)}_{adiabatic}}\] \[\Rightarrow \]\[{{(Area)}_{isbaric}}>{{(Area)}_{isothermal}}>{{(Area)}_{adiabatic}}\] Hence, work done is maximum in isobaric process. NOTE:   \[{{(Slope)}_{adiabatic}}=-\gamma \left( \frac{P}{V} \right)\] and    \[{{(Slope)}_{isothemal}}=-\frac{P}{V}\] \[\therefore \]  \[{{(Slope)}_{adiabtic}}=\gamma \times {{(Slope)}_{isothemal}}\]