A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The relation between temperature and volume of this process is ${\mathrm{TV}}^{\mathrm{x}}=$constant, then $\mathrm{x}$ is:

The correct option is B

$\frac{2}{5}$

Explanation for the correct option:

(B) Step 1:

${\mathrm{PV}}^{\gamma }=$constant for adiabatic process

we know, $PV=nRT$

so $V=\frac{nRT}{P}$

$\frac{nRT}{V}\left(V^{\gamma }\right)=$constant

so, $T{V}^{\gamma -1}=$constant$---------\left(1\right)$

Step 2:

for diatomic gas $\gamma =\frac{7}{5}$

Step 3:

from given question $T{V}^x=$constant$-------\left(2\right)$

comparing equations 1 and 2 we get

$x$=$\gamma -1$

so,$x$=$\frac{7}{5}-1$

=$\frac{2}{5}$

Explanation for incorrect option:

(A): The above detailed mathematical explanation clearly shows that option A is correct.

(C): The above detailed mathematical explanation clearly shows that option C is correct.

(D): The above detailed mathematical explanation clearly shows that option D is correct.

Hence, option (B) is correct.