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**Hint:**The heat capacity at constant volume (${{C}_{V}}$) for diatomic gas is $\dfrac{5}{2}R$

For ideal gas, PV = nRT

Where P is pressure, V is volume is temperature is number of moles and R is a constant Rydberg constant.

**Complete Solution :**

So in the question it is given that, there is a thermally insulated container with volume of 1L which contains diatomic gas and there is a change in pressure due to the action of a small paddle installed inside the container. We have to find the internal energy, for that,

We know the equation relating internal energy, molal heat capacity of gas at constant volume ${{C}_{V}}$, temperature and number of moles of gas as-

\[\Delta U=n{{C}_{V}}\Delta T\]

- Here volume remains the same. There is only change in pressure parameter. As pressure changes which results in the change in temperature also.

The heat capacity at constant volume (${{C}_{V}}$) for diatomic gas is $\dfrac{5}{2}$R

So the equation for internal energy becomes,

-\[\Delta U=\dfrac{5}{2}nR\Delta T\]

By ideal gas equation, $PV = nRT$

But in this case only volume changes so the equation is written as,

-$\Delta PV=nR\Delta T$

Comparing the equation of internal energy and ideal gas equation we can rearrange and write the equation as,

$\Delta U = \dfrac{5}{2}\Delta PV$

Now substitute the values,

Pressure (P) = ${{10}^{5}}Pa$

Volume (V) = 1L = $\dfrac{1}{1000}{{m}^{3}}={{10}^{-3}}{{m}^{3}}$

Substituting the values we get,

\[\Delta U=\dfrac{5}{2}\times {{10}^{5}}\times \dfrac{1}{{{10}^{-3}}}\]

\[\Delta U=\dfrac{5}{2}\times 100 = 250J\]

**So, the correct answer is “Option D”.**

**Note:**If in the place of diatomic gas, monoatomic was given then the value of ${{C}_{V}}$ is $\dfrac{3R}{2}$

Values must be substituted in the final equation, after converting all the values to standard form.

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