In the arrangement shown in the figure the cylinder is insulated from the surrounding. Both sides same diatomic gas is trapped by two insulting massless pistons with the help of an ideal spring. The natural length of the spring is equal to the length of the cylinder. Initial state of the gases are as shown in the figure.

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The value of energy stored in the spring is P₀V₀

The value of energy stored in the spring is 2P₀V₀

Now the gases are heated slowly, such that their temperature becomes three times to their initial temperature. The total heat given to the system is 12P₀V₀

Now the gases are heated slowly, such that their temperature becomes three times to their initial temperature. The total heat given to the system is 6P₀V₀

answer is A, C.

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Detailed Solution

If x be the initial length of the gas chambers and A be the area of cross section of the cylinder then,

$\begin{array}{l} x = \frac{V_{0}}{A} Also P_{0} A = 2 kx \\ \frac{P_{0}}{x} = 2 kx \Rightarrow P_{0} V_{0} = 2 {kx}^{2} \\ ∴ \frac{1}{2} k (2 x)^{2} = \frac{1}{2} K 4 x^{2} = 2 {kx}^{2} = P_{0} V_{0} \\ Similarly, P_{r} A = 2 k \frac{V_{r}}{A} \end{array}$

$\begin{array}{l} P_{0} A = 2 k \frac{V_{0}}{A} \\ \Rightarrow \frac{P_{r}}{P_{0}} = \frac{V_{r}}{V_{0}} \Rightarrow P = \frac{P_{0}}{V_{0}} v \\ Now ∴ Q = dU + W \\ = 2 \frac{5}{2} (P_{1} V_{1} - P_{0} V_{0}) + w \\ = 12 P_{0} V_{0} \end{array}$