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Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A brass container with an initial capacity of 500 cm3500\text{ cm}^3 at 20∘C20^\circ\text{C} is filled completely with ethanol. When the container and its contents are uniformly heated to 70∘C70^\circ\text{C}, a volume of 12 cm312\text{ cm}^3 of ethanol overflows from the container. Given that the linear expansivity of brass is 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}, calculate the real cubic expansivity of ethanol in units of 10−4 K−110^{-4}\text{ K}^{-1}.

Answer: 5.4 10^-4 K^-1

Answer

The real cubic expansivity of ethanol is 5.4×10−4 K−15.4 \times 10^{-4}\text{ K}^{-1}, which corresponds to a value of 5.45.4 in units of 10−4 K−110^{-4}\text{ K}^{-1}.
The real volume increase of a liquid consists of the apparent expansion (observed overflow volume) plus the volume expansion of the container itself. By finding the apparent cubic expansivity γa=12500×50=4.8×10−4 K−1\gamma_a = \frac{12}{500 \times 50} = 4.8 \times 10^{-4}\text{ K}^{-1} and adding the vessel's cubic expansivity γv=3×2.0×10−5=0.6×10−4 K−1\gamma_v = 3 \times 2.0 \times 10^{-5} = 0.6 \times 10^{-4}\text{ K}^{-1}, we obtain the real cubic expansivity γr=5.4×10−4 K−1\gamma_r = 5.4 \times 10^{-4}\text{ K}^{-1}, which yields 5.45.4 in the required units.

Step-by-Step Solution

1
Determine the temperature increase of the system
\Delta T = 70^\circ\text{C} - 20^\circ\text{C} = 50\text{ K}
The thermal expansion is driven by the change in temperature.
2
Calculate the apparent cubic expansivity of ethanol (\gamma_a)
\gamma_a = \frac{\Delta V_{overflow}}{V_0 \cdot \Delta T} = \frac{12\text{ cm}^3}{500\text{ cm}^3 \times 50\text{ K}} = 4.8 \times 10^{-4}\text{ K}^{-1}
The overflow volume represents the apparent volume increase of the liquid relative to the expanding vessel.
3
Calculate the cubic expansivity of the brass vessel (\gamma_v)
\gamma_v = 3 \times \alpha_{brass} = 3 \times 2.0 \times 10^{-5}\text{ K}^{-1} = 6.0 \times 10^{-5}\text{ K}^{-1} = 0.6 \times 10^{-4}\text{ K}^{-1}
Cubic expansivity of a solid vessel is three times its linear expansivity.
4
Compute the real cubic expansivity of ethanol (\gamma_r)
\gamma_r = \gamma_a + \gamma_v = 4.8 \times 10^{-4}\text{ K}^{-1} + 0.6 \times 10^{-4}\text{ K}^{-1} = 5.4 \times 10^{-4}\text{ K}^{-1}
The real expansion of a liquid is the sum of its apparent expansion and the expansion of the containing vessel.

Key Concept

Relationship between Real and Apparent Cubic Expansivity of Liquids
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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