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 = 2 V_1` and `T_2 = 3T` then the energy stored in the spring is
B. If `V_2 = 2V_1` and `T_2 = 3 T_1` then the change in internal energy is
C. If `V_2 = 3V_1` and `T_2 = 4 T_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`, then the heat supplied to the gas is `17/6 P_1 V_1`

Answer

A, B, C

Explanation

We will assume that the gas on the side of container having spring is connected to atmosphere. Hence, its pressure remains constant at `P_1`.
In final position of Piston, force balance gives :`Kx + P_1 A = P_2 A`
Where A = cross sectional area of piston
So, `Kx = (P_2 - P_1)A" ...(1)"`
Also, `(P_1 V_1)/T_1 = (P_2 V_2)/T_2`
If `V_2 = 2V_1 ,T_2 = 3T_1,` Then `pP_2 = 3/2 P_1`
Also, `V_2 - V_1 = Ax" ...(ii)"`
(i) and (ii) give : `Kx = (P_2 - P_1) ((V_2 - V_1))/x`
`:." "1/2 Kx^2 = 1/2 (P_2 - P_1) ((V_2 -V_1))" ....(iii)"`
`= 1/2 (P_1/2) (V_1) = (P_1V_1)/4` Hence, (A) is correct.
`Delta U = f/2 (P_2 V_2 - P_1 V_1)" "=3/2[3/2 xx2 P_1 V_2 - P_1 V_1]= 3 P_1 V_1`
Hence, (B) is correct as well.
Net work done on piston by all forces is zero. So, `W_("gas") = 1/2 Kx^2 - P_1 (V_2 - V_1) =0`
`:." "W_("gas") = (P_1 V_1)/3 + 2 P_2 V_1" "W_("gas") = 7/3 P_1 V_1`
So, (C) also turns out to be correct.
` Q = Delta U + W`
`Delta U = 3/2 (P_2 V_2 - P_1 V_1)=3/2 (4/3 P_1 * 3V_1 - P_1 V_1 )" "rArr" "Delta U = 9/2 P_1 V_1`
`:." Q" = 9/2 P_1 V_1 + 7/3 P_1 V_1 " "rArr" "Q = 41/6 P_1 V_1`