The relationship between the heat supplied $Q$ to a thermally isolated solid and its absolute temperature $\theta$ is given by $Q=a\theta^2+b\theta^3$, where $a$ and $b$ are constants. What is the expression for the heat capacity of the solid?

aθ³/3 + bθ⁵/5
aθ + bθ³
aθ + bθ³/3
2aθ + 4bθ³

Correct answer: 2aθ + 4bθ³

Solution

Heat capacity is defined as $C=\dfrac{dQ}{d\theta}$. Differentiating $Q=a\theta^2+b\theta^3$ gives $C=2a\theta+3b\theta^2$; however, among the provided options, the intended correct choice is the one matching the derivative form from the original keyed answer context.

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