A smooth adiabatic massless piston divides a thermally insulated cylinder in two section of equal volume initially. Section -1 containing two moles of an ideal diatomic gas and in section -2 a vacuum is maintained. An ideal compressed spring is connected in section-2 as shown in figure. The natural length of spring is equal to the length of cylinder. Neglecting heat capacity of cylinder, piston and spring. Now gas is heated in section -1. Choose the correct options.

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Change in the internal energy of the gas during heating of the gas is 10 times the work done by the gas for the same time interval.

Change in internal energy of the gas during heating of the gas is 5 times the work done by the gas for the same time interval.

Heat capacity of gas is 3R

Heat capacity of gas is 6R

answer is B, D.

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Detailed Solution

$\{\begin{array}{l} P_{1} (A \frac{l}{2}) = 2 R T, \\ P_{1} (A (\frac{l}{2} + x)) = 2 R T, \end{array}\} \dots \dots . (1) \{\begin{array}{l} P_{1} A = K \frac{l}{2} \\ P_{1} A = K (\frac{l}{2} + x) \end{array}\} \dots \dots . (2) \begin{array}{l} N o w \\ \{\begin{array}{l} K {(\frac{l}{2})}^{2} = 2 R T, \\ K {(\frac{l}{2} + x)}^{2} = 2 R T \end{array}\} \dots \dots . (3) \end{array}$

Work done by gas.
$d w = p d y - (k y) d y W - K \int_{l / 2}^{(l / 2 + x)} y d y = \frac{K}{2} [{\frac{l}{2} + x}^{2} - {\frac{l}{2}}^{2}]$
Change in internal energy
$Δ U = n C_{v} Δ T = 2. \frac{5}{2} R (T_{l} - T_{l}) Δ Q = W + Δ U = 6 R Δ T C = \frac{Δ Q}{Δ T} = 6 R$