Physics

Physics

Problem

For a solid whose mass remains constant during heating, the density changes as the volume changes with temperature. a. (i) Derive an expression showing that the fractional decrease in density of a solid on heating through ΔT\Delta T is equal to γ ΔT\gamma\,\Delta T, where γ\gamma is the volume coefficient of expansion. b. (ii) Using the above result, calculate the percentage decrease in density of brass when it is heated from 25 ∘C25\,^{\circ}\text{C} to 125 ∘C125\,^{\circ}\text{C}, given γbrass=5.7×10−5 K−1\gamma_{\text{brass}} = 5.7\times10^{-5}\,\text{K}^{-1}."
Intuitive explanation

When a solid heats up, its atoms vibrate more vigorously and push farther apart, so the volume expands—but the mass stays the same. Since density is just mass divided by volume, a larger volume means lower density. The key insight is that thermal expansion is predictable: if you know the volume expansion coefficient, you can calculate how density changes with temperature, and this relationship is often linear for small temperature changes.

Intuitive explanation

When a solid heats up, its atoms vibrate more vigorously and push farther apart, so the volume expands—but the mass stays the same. Since density is just mass divided by volume, a larger volume means lower density. The key insight is that thermal expansion is predictable: if you know the volume expansion coefficient, you can calculate how density changes with temperature, and this relationship is often linear for small temperature changes.

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