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Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A container made of a metal with a linear expansivity of 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1} has a volume of 1000 cm31000\text{ cm}^3 at 20∘C20^\circ\text{C}. It is completely filled with a liquid at this temperature. When the system is heated to 120∘C120^\circ\text{C}, 40 cm340\text{ cm}^3 of the liquid overflows. If the real cubic expansivity of the liquid is expressed as X×10−4 K−1X \times 10^{-4}\text{ K}^{-1}, what is the value of XX?

Answer: 4.6

Answer

The value of XX is 4.6.
The real cubic expansivity of a liquid is given by γr=γa+γv\gamma_r = \gamma_a + \gamma_v. Calculating apparent expansivity gives γa=401000×100=4.0×10−4 K−1\gamma_a = \frac{40}{1000 \times 100} = 4.0 \times 10^{-4}\text{ K}^{-1}. The vessel's volume expansivity is 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}. Adding them together yields γr=4.6×10−4 K−1\gamma_r = 4.6 \times 10^{-4}\text{ K}^{-1}, so X=4.6X = 4.6.

Step-by-Step Solution

1
Calculate the temperature change ΔT\Delta T
\Delta T = 120^\circ\text{C} - 20^\circ\text{C} = 100\text{ K}
The thermal expansion is driven by the change in temperature.
2
Calculate the apparent cubic expansivity γa\gamma_a
\gamma_a = \frac{40\text{ cm}^3}{1000\text{ cm}^3 \times 100\text{ K}} = 4.0 \times 10^{-4}\text{ K}^{-1}
Apparent expansivity is defined as the fraction of initial volume overflowed per degree rise in temperature.
3
Calculate the volume expansivity of the container γv\gamma_v
\gamma_v = 3\alpha = 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 an isotropic solid container is three times its linear expansivity.
4
Sum apparent expansivity and vessel cubic expansivity to find real cubic expansivity γr\gamma_r
\gamma_r = \gamma_a + \gamma_v = 4.0 \times 10^{-4} + 0.6 \times 10^{-4} = 4.6 \times 10^{-4}\text{ K}^{-1}
Real expansivity accounts for both the apparent expansion of the liquid and the expansion of the container.

Key Concept

Relationship between real cubic expansivity, apparent cubic expansivity, and vessel expansivity
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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