Question

For anideal gas, temperature varies as:
$T=K+\beta V^2$
Where $K$ and $\beta$ are positive constants and $‘V^{\prime }$ is the molar volume of the gas, if the minimum pressure $\left(P_{\text{min}}\right)$ of one mole of an ideal gas is expressed as:
$P_{\text{Min}}=n{R}^a{\beta }^b{K}^c$
Then, find the value of $\frac{n+a+b}{n-a+b-c}$

Answer 1

We know,
PV = RT (for T mole ideal gas)
Given, $T=K+\beta V^2$
$\therefore PV=R(K+\beta V^2)\cdot \cdot \cdot \left(1\right)$
$\Rightarrow P=\frac{RK}{V}+\beta RV$
For minimum pressure, the mathematical conditions are
$\frac{dP}{dV}=0$ and $\frac{d^{2}P}{d{V}^2}+ve$
$\frac{dP}{dV}=-\frac{RK}{V^2}+\beta R=0$
$\Rightarrow V=\sqrt{\frac{K}{\beta }}$
Again, $\frac{d^{2}P}{d{V}^2}=\frac{RK}{V^3},\text{which is positive}$
putting the value of $V$ on equation (1), we get
$P_{\text{min}}=2R\sqrt{\beta K}$
$\therefore n=2,a=1,b=0.5,c=0.5$
$\Rightarrow \frac{n+a+b}{n-a+b-c}=\frac{3.5}{1}=3.5$