Question

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 $t$ as $T\left(t\right)= \, T_{0}\left(\left[1 \, + a \left(t \, - \, t_{0}\right)\right]\right)^{1/4}$ , where $a$ , $t_{0}$ and $T_{0}$ are constants. The heat capacity $C_{P}\left(T\right)$ of the metal is

• A. $\frac{4 P}{a T_{0}}$
• B. $\frac{4 P T^{3}}{a T_{0}^{4}}$
• C. $\frac{2 P T^{3}}{a T_{0}^{4}}$
• D. $\frac{2 P}{a T_{0}}$

Answer 1

Answer: (B)

Heat given to the metal $ d Q=P d t=C_{P}(t) d T $
At constant pressure in time interval at
Given, $T=T_{0}\left[1+ a \left( t - t _{0}\right)\right]^{1 / 4}$
$ \frac{ dT }{ dt }=\frac{ T _{0}}{4}\left[1+ a \left( t - t _{0}\right)\right]^{-3 / 4} \times a $
From Eqs. (i) and (ii) $ C _{ P }( T )=\frac{ P }{\left(\frac{ dT }{ dt }\right)}=\frac{4 P \left[1+ a \left( t - t _{0}\right)\right]^{3 / 4}}{ T _{0} a }=\frac{4 PT ^{3}}{ aT _{0}^{4}} $