Question

An ideal $C{O}_2$ gas obeys the law $P{V}^x=$ constant. The range of $x$ for which the gas has non-positive molar specific heat at normal temperature is

• A. $x\ge -0.5$
• B. $x\le 1.7$
• C. $1<x\le 1.4$
• D. $x\le -1.4$

Answer 1

The correct option is C $1<x\le 1.4$

Molar Specific heat of a gas during a polytropic process is given by $\left(C\right)=C_V+\frac{R}{1-x}..........\left(1\right)$
Where $x$ is the exponent of polytropic process which can take values in the range of $(-\infty <x<+\infty )$

Since, $C{O}_2$ is a linear triatomic molecule, at normal temperatures we can say that $C_V=\frac{5R}{2}$
Thus, from $\left(1\right)$ we get,
$\frac{5R}{2}\le \frac{R}{x-1}\Rightarrow \frac{5}{2}\le \frac{1}{x-1}\text{For}x>1\Rightarrow x-1\le \frac{2}{5}\Rightarrow x\le 1.4$
$\therefore$ For values of $x$ between $1<x\le 1.4$ heat capacity of gas becomes non-positive.
Thus, option (c) is the correct answer.

Alternate solution:
we know that,
Heat capacity of a gas can be negative if $1<x<\gamma$
we can say that, Heat capacity of a gas can be non-positive iff $1<x\le \gamma$
For $C{O}_2$ we know that, $\gamma =\frac{7}{5}=1.4$
$\therefore$ we can conclude that $1<x\le 1.4$
Hence, option (c) is the correct answer.