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 tehn 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 o the energy stored in the spiring is `1/4P_1V_1`
B. If `V_2=2V_1 and T_2=3T_1`, then the change in internal energy is `3P_1V_1`
C. If `V_2=3V_1 and T_2=4T_1`, then the work done by the gas is `7/3P_1V_1`
D. If `V_2=3V_1 and T_2=4T_1`, then the heat supplied to the gas is `17/6P_1V_1`

Answer

B, C

Explanation

(b,c)
Applying combined gas law
`(P_1V_1)/(T_1)=(P_2V_2)/(T_2)`
If `V_2=2V_1 and T_2=3T_1` then
`(P_1V_1)/(T_1)=(P_2xx2V_1)/(3T_1) rArr P_1=2/3P_2`
Now change in internal energy
`DeltaU=f/2[nR(T_2-T_1)]=f/2[P_2V_2-P_1V_1]`
For monoatomic gas `f=3`
`DeltaU=3/2[3/2P_1xx2V_1-P_V_1]=3P_1V_1`
`:. (b) is the correct option.
Now assuming that the pressure on the piston on the right
hand side (not considering the affect of spiring ) remains the
same throughout the motion of the piston then,
`Pressure of gas=P_1+(kx)/A rArr P_2=P_1+(kx)/A`
Where k is spring constant and `A = area of piston`
`Energy stored=1/2kx^2`
`P_2=P_1+(kx)/A rArr 3/2P_1=P_1+(kx)/A`
`(P_1)/2=(kx)/2`
`:. kx=(P_1A)/2`
Also,
`V_2=V_1+Ax`
`V_1=Ax`
`:. x=(V_1)/A`
`:. Energy =1/2(P_1A)/2xx(V_1)/A=1/4P_1V_1`
`:. A is correct `
Now
`w=int pdV=int(P_1+(kx)/A)dV=intP_1dV+int(kx)/adV`
`:. W=intP_1dV+int(kx)/Axx(dx)A`
`:. W=P_1(V_2-V_1)+(kx^2)/2`
`[Here on applying `(P_1V_1)/(T_1)=(P_2V_2)/(T_2) we get P_2=(4P_1)/3
and `V_2=V_1+Ax rArr x=(2V_1)/A[:' V_2=3V_1]]`
`:. W=2P_1V_1+1/2xx(P_1A)/3xx(2V_1)/A=7/3P_1V_1`
C is correct option
Heat supplied
`Q=W+DeltaU`
`=(7P_1V_1)/3+3/2(P_2V_2-P_1V_1)`
`=(7P_1V_1)/3+3/2[4/3P_13V_1-P_1V_1]=41/4P_1V_1`