Physics - Heat & Thermo Question with Solution | TestHub

As ${\mathrm{TV}}^{\gamma -1}=$ constant $\Rightarrow \mathrm{T}\propto {\mathrm{V}}^{1-\gamma }$

$$\begin{array}{rlr}& \Rightarrow \frac{\mathrm{dT}}{\mathrm{T}}=(1-\gamma )\frac{\mathrm{dV}}{\mathrm{V}}\Rightarrow \frac{\mathrm{dV}}{\mathrm{dT}}=\frac{-\mathrm{V}}{(\gamma -1)\mathrm{T}}=\tan 45^{\circ }=1& \Rightarrow \gamma =1+\frac{{\mathrm{V}}_0}{{\mathrm{T}}_0}\end{array}$$

Bulk modulus

$$\begin{array}{rlr}\mathrm{K}& =\frac{\mathrm{dP}}{-\mathrm{dV}/\mathrm{V}}=\gamma \mathrm{P}=\left(1+\frac{{\mathrm{V}}_0}{{\mathrm{T}}_0}\right){\mathrm{P}}_0=\left(1+\frac{{\mathrm{V}}_0}{{\mathrm{T}}_0}\right)\left(\frac{2{\mathrm{RT}}_0}{{\mathrm{V}}_0}\right)& =2\mathrm{R}\left(1+\frac{{\mathrm{T}}_0}{{\mathrm{V}}_0}\right)\end{array}$$