Question

A thermally insulated 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: $T\left(t\right)=T_0{[1+a(t-t_0\left)\right]}^{0.25}$. Here, $a$, $t_0$ and $T_0$ are constants. The heat capacity $C_p\left(T\right)$ of the metal is

• A. $\frac{4P}{a{T}_0}$
• B. $\frac{4P{T}^3}{a{T_0}^4}$
• C. $\frac{2P{T}^3}{a{T_0}^4}$
• D. $\frac{2P}{a{T}_0}$

Answer 1

Answer: (B)

$\frac{4P{T}^3}{a{T_0}^4}$

By definition, heat capacity, $C_p=\frac{dQ}{dT}=\frac{Pdt}{dT}$

Given, $T^4=T_0^4[1+a(t-t_0\left)\right]\Rightarrow \frac{dt}{dT}=\frac{4T^3}{a{T}_0^4}$
Thus, $C_p=\frac{4P{T}^3}{a{T}_0^4}$