A thermally isulated piece of metal is heated under atmospheric pressure by an electric current so that it receives electric energy at a constant power P. This leads to an increase of absolute temperature T of the metal with time t as follows:
`T(t)=T_0[1+a(t-t_0)]^(1//4)`. Here, a, `t_0` and `T_0` are constants. The heat capacity `C_p(T)` of the metal is

Options

A. `(4P)/(aT_0)`
B. `(4PT)/(aT_0^4)`
C. `(2PT)/(aT_0^4)`
D. `(2P)/(aT_0)`

Answer

B

Explanation

Heat given to the metal
`dQ=Pdt=C_P(t)dT` ..(i)
At constant pressure in time interval at
Given
`T=T_0[1+a(t-t_0)]^(1//4)`
`(dT)/(dt)=(T_0)/(4)[1+a(t-t_0)^(-3//4)xxa` ..(ii)
From Eqs. (i) and (ii)
`C_P(T)=(P)/(((dT)/(dt)))=(4P[1+a(t-t_0)]^(3//4))/(T_0a)=(4PT^3)/(aT_0^4)`