An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature `T_1`, pressure `P_1` and volume `V_1`and the spring is in its relaxed state. The gas is then heated very slowly to temperature `T_2`,pressure `P_2`and volume `V_2`. During this process the piston moves out by a distance x. Ignoring the friction between the piston and the cylinder, the correct statement (s) is (are)

Options

A. If `V_(2) = 2V_(1) and T_(2)=3T_(1)`, then the energy stored in the string is `1/4 P_(1)V_(1)`
B. If `V_(2) and 2V_(1)` and `T_(2) = 3T_(1)`, then the charge in iternal energy is `3P_(1)V_(1)`.
C. If `V_(2)=3V_(1)` and `T_(2) = 4T_(1)`, then the work done by the gas is `7/3 P_(1)V_(1)`
D. If `V_(2) = 3V_(1)` and `T_(2) = 4T_(1)`, then the heat supplied to the gas is `17/6 P_(1)V_(1)`.

Answer

B

Explanation

From gas equation , `(P_(1)V_(1))/(T_1) = (P_(2)V_(2))/(T_2) = nR or P_(2) = (P_(1)V_(1)T_(2))/(V_(2)T_(1)) ` ...(i)
(a) when `V_(2) = 2 V_(1), T_(2) = 3T_(1)`, then
`P_(2) = (P_(1)V_(1) xx 3T_(1))/(2 V_(1) xx T_(1)) = 3/2 P_(1)`
If A is the area of cross- section of the piston, then
`P_(2)A = kx or P_(2)Ax = kx^(2)`
Energy in spring , `U = 1/2 kx^(2) = 1/2 P_(2) Ax`
`=1/2 P_(2)(V_(2)-V_(1))`
`=1/2 xx (3P_1)/(2) (2V_(1)-V_(1)) = 3/4 P_(1)V_(1)`
Thus, option (a) is wrong.
(b) When `V_(2) = 2V_(1) and T_(2) = 3T_(1) , P_(2) = 3/2P_(1)` [from (i) ]
Change in internal energy ,
`dU = nC_(v) dT = m((3R)/(2)) (T_(2)-T_(1))`
`= n xx (3R)/(2) [3T_(1)-T_(1)] = 3nRT_(1) = 3P_(1)V_(1)`
Thus option (b) is true.
(c) when `V_(2) = 3V_(1) and T_(2) = 4T_(1)`
`P_(2) = (P_(1)V_(1)xx4T_(1))/(3V_(1)xxT_(1)) = 4/2 P_(1)`
Workdone , dW = energy in spring
`1/2kx^(2) = 1/2 P_(2)(V_(2)-V_(1))`
`1/2 xx 4/3 P_(1)[3V_(1)-C_(1)] = 4/3 P_(1)V_(1)`
Thus option (c) is wrong.
(d) when `V_(2) = 3V_(1) and T_(2) = 4T_(1), P_(2) = 4/3 P_(1)`
Heat supplied to the gas, `dQ = dU + dW`
`dQ = n xx (3R)/(2)[T_(2)-T_(1)]+1/2 P_(2)(V_(2)-V_(1))`
`=3/2 nR [4T_(1)-T_(1)] + 1/2 4/3 P_(1)[3V_(1)-V_(1)]`
`= 9/2 nRT_(1)+4/2 P_(1)V_(1) = 9/2 P_(1)V_(1)+ 4/2 P_(1)V_(1) = 35/6 P_(1)V_(1)`.
Thus, option (d) is wrong.