A steel tank is completely filled with 2.80 m^3 of ethanol when both the tank and the ethanol ar

step 1 For this question, both the steel tank and the ethanol undergo volumetric expansion. So we'll ne

A steel tank is completely filled with 2.80 m^3 of ethanol...

A steel tank is completely filled with 2.80 $\mathrm{m}^{3}$ of ethanol when both the tank and the ethanol are at a temperature of $32.0^{\circ} \mathrm{C} .$ When the tank and its contents have cooled to $18.0^{\circ} \mathrm{C},$ what additional volume of ethanol can be put into the tank?

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon in which materials change their dimensions (length, area, or volume) in response to a change in temperature. It is a fundamental property of matter that explains how and why most materials expand when heated and contract when cooled, affecting both solids and liquids in different ways.

Coefficient of Thermal Expansion

The coefficient of thermal expansion quantifies how much a material will expand or contract per unit temperature change. This property is essential in calculating the change in dimensions or volume of a material due to thermal effects and varies between substances, being typically smaller for solids like steel compared to liquids like ethanol.

Linear vs. Volumetric Expansion

For solids, thermal expansion is often described as linear expansion, where changes in one dimension are considered and then extended to volume changes through geometric relations. In contrast, liquids are characterized by volumetric expansion, where the volume change is directly proportional to the temperature change using the volumetric expansion coefficient.

Differential Thermal Expansion

Differential thermal expansion occurs when different materials in a system expand or contract by different amounts in response to the same temperature change. This concept is crucial when evaluating systems like a liquid inside a tank, where both the container and the liquid change volume, affecting how much additional liquid can be accommodated following a temperature change.

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