NDA 2026 · PYQ · Thermodynamics / Specific Heat · medium
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)]✓ Correct
Explanation
When specific heat depends on temperature, the heat required is calculated by integration: Q = m∫[T₁ to T₂] C(T) dT = m∫[T₁ to T₂] (C₀ + αT) dT. Evaluating the integral: Q = m[C₀T + αT²/2] from T₁ to T₂ = m[C₀(T₂ − T₁) + (α/2)(T₂² − T₁²)]. Using T₂² − T₁² = (T₂ − T₁)(T₂ + T₁), we get Q = m(T₂ − T₁)[C₀ + (α/2)(T₁ + T₂)]. Therefore, option (d) is correct.
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