A solid of mass m has temperature-dependent specific heat as C(T) = C_0 + \alpha T, where C_0 and \alpha are constants. The solid is heated from T_1 to T_2. Which one of the following is the correct expression for quantity of heat (Q) of the solid mass?

  1. A. Q = m C_0 (T_2 - T_1)
  2. B. Q = m(T_2 - T_1)[C_0 + \alpha(T_1 + T_2)]
  3. C. Q = m(T_2 - T_1)[C_0(T_1 + T_2) + \alpha]
  4. D. Q = m(T_2 - T_1)\left[ C_0 + \frac{\alpha}{2} (T_1 + T_2) \right] ✓

Correct Answer: D. Q = m(T_2 - T_1)\left[ C_0 + \frac{\alpha}{2} (T_1 + T_2) \right]

Explanation

Heat Q = m \int_{T_1}^{T_2} C(T) dT = m \int_{T_1}^{T_2} (C_0 + \alpha T) dT = m \left[ C_0(T_2 - T_1) + \frac{\alpha}{2}(T_2^2 - T_1^2) \right]. Factoring out (T_2 - T_1) gives m(T_2 - T_1)\left[ C_0 + \frac{\alpha}{2}(T_1 + T_2) \right].

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