A thermally insulated piece of metal is heated under atmospheric...

A thermally insulated piece of metal is heated under atmospheric pressure by an electric current so that it receives electric energy at constant power

P

. This leads to an increase in the absolute temperature

T

of the metal with time

t

T\left(t\right) = {T}_{0}\left[1 + a\left(t - {t}_{0}\right)\right]\dfrac{1}{4}

, where

a

{t}_{0}

and

{T}_{0}

are constants. The heat capacity

{C}_{P} \left(T\right) of the metal is

Options

A $\dfrac{4P}{a{T}_{0}}$
B $\dfrac{4PT^{3}}{a{T}_{0}^{4}}$
C $\dfrac{2PT^{3}}{a{T}_{0}^{4}}$
D $\dfrac{2P}{a{T}_{0}}$

Correct answer

B. $\dfrac{4PT^{3}}{a{T}_{0}^{4}}$

Step-by-step solution

Heat given to the metal

dQ = Pdt = C_P(t) dT ...(i)

At constant pressure in time interval at

Given, T = T₀ [1 + a(t - t₀)]^1/4

\dfrac{\text{dT}}{\text{dt}} = \dfrac{{T}_{0}}{4} {\left[1 + \text{a} \left(\text{t} - {t}_{0}\right)\right]}^{- 3 / 4} \times \text{a}

...(ii)

From Eqs. (i) and (ii)

{C}_{P} \left(\text{T}\right) = \dfrac{\text{P}}{\left(\dfrac{\text{dT}}{\text{dt}}\right)} = \dfrac{4 \text{P} {\left[1 + \text{a} \left(\text{t} - {t}_{0}\right)\right]}^{3 / 4}}{{T}_{0} \text{a}} = \dfrac{4 {PT}^{3}}{\text{a} {T}_{0}^{4}}

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