Two moles of an ideal monatomic gas are taken through cycle ABCA as shown in the P-T diagram. During the process A B, the pressure and temperature of the gas vary such that $mathrm{PT}=$ constant. If $mathrm{T}_1=300 mathrm{~K}$,the work done on the gas in the process ABis

Two moles of an ideal monatomic gas is taken through cycle ABCA as shown in the P-T diagram. During the process A B, pressure and temperatue of the gas vary such that <img alt="\mathrm{PT}=\

Two moles of an ideal monatomic gas is taken through cycle ABCA as shown in the P-T diagram.
During the process A B, pressure and temperatue of the gas vary such that $\mathrm{PT}=\text{constant}$ . If $\mathrm{\mathrm{T}_{1}=300 \mathrm{~K}}$ , the work done on the gas in the process $\mathrm{\mathrm{AB}}$ is

Option: 1
$600 \mathrm{R}$

Option: 2
$800 \mathrm{R}$

Option: 3
$1000 \mathrm{R}$

Option: 4
$1200 \mathrm{R}$

Answers (1)

Number of moles, $\mathrm{n}=2, \mathrm{~T}_{1}=300 \mathrm{~K}$
$\mathrm{\text{During the process} \, \mathrm{A} \rightarrow \mathrm{B}}$

$\mathrm{PT}=\text { constant or } \mathrm{P}^{2} \mathrm{~V}=\text { constant }=\mathrm{K} \text { (say) }$
$\mathrm{\therefore \quad \mathrm{P} =\frac{\sqrt{K}}{\sqrt{V}} }$

$\mathrm{\mathrm{~W}_{\mathrm{A} \rightarrow \mathrm{B}} =\int_{V_{A}}^{v_{B}} P d v=\int_{V_{A}}^{V_{B}} \frac{\sqrt{K}}{\sqrt{V}} d v }$
                $\mathrm{ =2 \sqrt{K}\left[\sqrt{V_{B}}-\sqrt{V_{A}}\right] }$
                $\mathrm{=2\left[\sqrt{K V_{B}}-\sqrt{K V_{A}}\right]}$
               $\mathrm{ =2\left[\sqrt{\left(P_{B}^{2} B_{B}\right) V_{B}}-\sqrt{\left(P_{A}^{2} V_{A}\right) V_{A}}\right]\left(\mathrm{K}=\mathrm{P}^{2} \mathrm{~V}\right)}$
              $\mathrm{ =2\left[\mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}}-\mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}\right] }$
            $\mathrm{ =2\left[\mathrm{nRT} \mathrm{T}_{\mathrm{B}}-\mathrm{nRT}_{\mathrm{A}}\right] }$
          $\mathrm{ =2 \mathrm{nR}\left[\mathrm{T}_{1}-2 \mathrm{~T}_{1}\right] }$
            $\mathrm{ =(2)(2)(\mathrm{R})[300-600]}$
          $\mathrm{ =-1200 \mathrm{R}}$

         $\mathrm{ \therefore}$ Work done on the gas in the process AB is $\mathrm{ 1200 \mathrm{R}}$ .