Heart leaks into a vessel, containing one mole of an ideal monoatomic gas at a constant rate of `(R )/(4)s^(-1)` where R is universal gas constatant. It is observed that the gas expands at a constant rate `(dV)/(dt) = (V_(0))/(400("second"))` where `V_(0)` is intial volume. The inital temperature is given by `T_(0)=40K`. What will be final temperature of gas at time `t = 400 s`.

Options

A. `40 K`
B. `80 K`
C. `200 K`
D. `260 K`

Answer

B

Explanation

`(dV)/(dt) = (V_(0))/(400)`
`V = V_(0)(1+(t)/(400))`
`(DeltaR)/(Deltat)=(dU)/(dt)+p(dV)/(dt)`
` (R )/(4) = (3)/(2)R (dT)/(dt) + (RT)/(V_(0)(1+(t)/(400)))V_(0)/(400)`
`(1)/(4) = (3)/(2)(dT)/(dt)+(T)/((t+400))`
`(dT)/(dt)+(2)/(3)(T)/((t+400)) = (1)/(6)`
`T=(t+400)/(10)+C(t+400)^(-2//3)`
at `t = 0 T = 40 K implies C=0`
at `t = 400 S`
`T=80 K`