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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 t = T 0 1 + a t - t 0 1 4 , where a , t 0 and T 0 are constants. The heat capacity C P T of the metal is

Options

  1. A4 P a T 0
  2. B4 P T 3 a T 0 4
  3. C2 P T 3 a T 0 4
  4. D2 P a T 0

Correct answer

B. 4 P T 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 = T 0 [1 + a(t - t 0 )] 1/4 dT dt = T 0 4 1 + a t - t 0 - 3 / 4 × a ...(ii) From Eqs. (i) and (ii) C P T = P dT dt = 4 P 1 + a t - t 0 3 / 4 T 0 a = 4 PT 3 a T 0 4

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