An aluminum can is filled to the brim with a liquid. The can...

An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. The can's initial volume at $5^{\circ} \mathrm{C}$ is $3.5 \times 10^{-4} \mathrm{m}^{3} .$ The coefficient of volume expansion for aluminum is $69 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1} .$ When the can and the liquid are heated to $78^{\circ} \mathrm{C}, 3.6 \times 10^{-6} \mathrm{m}^{3}$ of liquid spills over. What is the coefficient of volume expansion of the liquid?

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Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. As materials are heated, the increased kinetic energy causes their particles to move apart, resulting in an increase in volume. This concept applies to both solids and liquids, though the rates of expansion can vary significantly between different materials.

Coefficient of Volume Expansion

The coefficient of volume expansion is a property of a material that quantifies how much its volume changes per degree change in temperature. It is defined as the fractional change in volume per unit temperature change. When applied to both the container (a solid) and the liquid inside, differences in their coefficients determine whether excess liquid will be forced out of the container upon heating.

Differential Expansion of Materials

Differential expansion occurs when different materials in a system expand by different amounts when subjected to the same temperature change. In a scenario where a liquid is contained within a solid container, if the liquid expands more than the container due to a higher coefficient of volume expansion, the excess liquid can spill over. This concept is critical in understanding the outcome when both the container and its contents are heated simultaneously.

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