Chemistry - Thermodynamics - 1 Question with Solution | TestHub

$\text{dE}=\text{dq}+\text{dw}$(F L T)
${\text{P}}_1={\text{K}}^{\prime }{\text{V}}_1$and${\text{P}}_2={\text{K}}^{\prime }{\text{V}}_2$Also${\text{V}}_1=\frac{{\text{P}}_1}{{\text{K}}^{\prime }}\text{, }{\text{V}}_2=\frac{{\text{P}}_2}{{\text{K}}^{\prime }}$
${\text{T}}_2={\text{T}}_1\left(\frac{{\text{P}}_2{\text{V}}_2}{{\text{P}}_1{\text{V}}_1}\right)={\text{T}}_1\left(\frac{{\text{P}}_2{\text{P}}_2/{\text{K}}^{\prime }}{{\text{P}}_1{\text{P}}_1/{\text{K}}^{\prime }}\right)={\text{T}}_1\frac{{\text{P}}_2^2}{{\text{P}}_1^2}$
${\text{T}}_2={\text{T}}_1\frac{4^2}{2^2}=4{\text{ T}}_1$
$\Delta \text{E}={\text{C}}_{\text{v}}\left(\Delta \text{T}\right)={\text{C}}_{\text{v}}\left(4{\text{T}}_1-{\text{T}}_1\right)=3{\text{C}}_{\text{v}}{\text{T}}_1$
$=3\frac{3}{2}\text{R }{\text{T}}_1=9\frac{\text{R }{\text{T}}_1}{2}$
$\text{w}=-\underset{{\text{v}}_1}{\overset{{\text{v}}_2}{\int }}\text{pdv}=-\underset{{\text{v}}_1}{\overset{{\text{v}}_2}{\int }}{\text{K}}^{\prime }\text{v}\text{dv}=-{\left|\frac{{\text{K}}^{\prime }{\text{v}}^2}{2}\right|}_{{\text{v}}_1}^{{\text{v}}_2}$
$\left({\text{K}}^{\prime }{\text{v}}^2={\text{K}}^{\prime }\text{v}\text{v}=\text{P}\text{v}=\text{RT}\right)$
$\text{w}=-{\left|\frac{\text{RT}}{2}\right|}_{{\text{T}}_1}^{{\text{T}}_2}=-\frac{\text{R}}{2}\left({\text{T}}_2-{\text{T}}_1\right)=-\frac{\text{R}}{2}3{\text{T}}_1$
$\frac{\Delta \text{E}}{\text{w}}=\frac{9\text{R}{\text{T}}_1/2}{-3\text{R}{\text{T}}_1/2}=-3$