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A thermally insulated rigid container of the one-litre volume contains a diatomic ideal gas at room temperature. A small paddle installed inside the container is rotated from the outside such that the pressure rises by ${{10}^{5}}$ Pa. The change in internal energy is to:
A) 0 J
B) 67 J
C) 150 J
D) 250 J

Last updated date: 13th Jun 2024
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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.