Correct Answer - Option 1 : 2/5

**Concept:**

Given

Relation between temperature and volume for the given adiabatic process is

TV^{x} = constant

For an ideal gas undergoing an adiabatic process at room temperature,

pV^{γ} = constant or TV^{(γ-1)} = constant

**Calculation:**

For a diatomic gas, degree of freedom, f = 5

\(\therefore \gamma = 1 + \frac{2}{f}\)

\(\gamma = 1 + \frac{2}{5} = \frac{7}{5}\)

As for adiabatic process, TV^{(γ-1)} = constant ----(i)

And it is given that, here TV^{x} = constant ----(ii)

Comparing equations (i) and (ii), we get

γ – 1 = x

\(\frac{7}{5} - 1 = x\)

\(x = \frac{{7 - 5}}{5} = \frac{2}{5}\;\)

Then, the value of x in the given relation is equal to

\(\frac{2}{5}.\)