Question

A inert gas obeys the law PV^x = C for what value of x, It has negative molar specific heat​

Answer 1

A inert gas obeys the law of PV^x = C.

then, $C=\frac{R}{\gamma-1}+\frac{R}{1-x}$.

question says " it has negative molar specific heat"

so, C < 0

or, $\frac{R}{\gamma-1}+\frac{R}{1-x}$ < 0

or, $\frac{1}{\gamma-1}+\frac{1}{1-x}$ < 0

inert gas is monoatomic gas.

so, $\gamma=1.67$

now, 1/(1.67 - 1) + 1/(1 - x ) < 0

or, 1/0.67 + 1/(1 - x) < 0

or, (1 - x + 0.67)/0.67(1 - x) < 0

or, (1.67 - x)/0.67(1 - x) < 0

or, (1.67 - x)/(x - 1) > 0

so, 1 < x < 1.67

hence, value of x belongs to (1, 1.67)