Question
A solid of mass m has temperature-dependent specific heat as C(T) = C_0 + T, where C_0 and 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?
NDA
General Ability
General Science
2026
NDA 2026 (Phase 1)
NDA General Ability Question (2026) — Solution
Options
- A. Q = mC_0(T_2 - T_1)
- B. Q = m(T_2 - T_1)[C_0 + (T_1 + T_2)]
- C. Q = m(T_2 - T_1)[C_0(T_1 + T_2) + ]
- D. Q = m(T_2 - T_1) [C_0 + 2 (T_1 + T_2) ]
Answer
D. Q = m(T_2 - T_1) [C_0 + 2 (T_1 + T_2) ]
Step-by-step solution
The quantity of heat dQ required to raise the temperature of a mass m by dT is given by: dQ = m C(T) dT Substituting the given specific heat C(T) = C_0 + T: dQ = m (C_0 + T) dT Integrating both sides from T_1 to T_2: Q = _ T_1 ^ T_2 m (C_0 + T) dT Q = m [ C_0 T + T^2 2 ]_ T_1 ^ T_2 Q = m [ C_0 (T_2 - T_1) + 2 (T_2^2 - T_1^2) ] Using the identity T_2^2 - T_1^2 = (T_2 - T_1)(T_2 + T_1): Q = m (T_2 - T_1) [ C_0 + 2 (T_1 + T_2) ] Answer: Q = m(T_2 - T_1) [C_0 + 2 (T_1 + T_2) ]
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