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?A.Q = mC0(T2 − T1)B.Q = m(T2 − T1)[C0 + α(T1 + T2)]C.Q = m(T2 − T1)[C0(T1 + T2) + α]D.Q = m(T2 − T1)[C0 + (α/2)(T1 + T2)]

Question

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?

Accepted Answer

Q = m(T2 − T1)[C0 + (α/2)(T1 + T2)]. When specific heat changes with temperature, we cannot just multiply C by the temperature change. We must integrate instead. Heat is Q = m ∫ C(T) dT, taken from T1 to T2. Here C(T) = C0 + αT. Integrating gives Q = m[C0T + αT²/2], evaluated between T1 and T2. This works out to Q = m[C0(T2−T1) + (α/2)(T2²−T1²)]. Factor (T2²−T1²) as (T2−T1)(T2+T1). Then pull out the common (T2−T1) term. This gives Q = m(T2−T1)[C0 + (α/2)(T1+T2)], which matches option (d).

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?

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