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A solid of mass m has temperature-dependent specific heat as C(T) = C0 + αT, where C0 and α are constants. The solid is heated from T1 to T2. Which one of the following is the correct expression for quantity of heat (Q) of the solid mass?
Explanation
The quantity of heat (Q) required to change the temperature of a mass (m) is calculated by integrating the temperature-dependent specific heat capacity C(T) over the temperature range from T1 to T2:
Q = ∫T1T2 m C(T) dT
Given the expression C(T) = C0 + αT, the integral becomes:
Q = m ∫T1T2 (C0 + αT) dT
Performing the integration:
Q = m [C0T + (αT2)/2]T1T2
Q = m [C0(T2 - T1) + (α/2)(T22 - T12)]
Using the identity (T22 - T12) = (T2 - T1)(T2 + T1), we can factor out (T2 - T1):
Q = m(T2 - T1) [C0 + α/2 (T1 + T2)]
This matches the expression provided in option D.