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Difficulty: EasyThermal Expansion of Liquids and Anomalous Expansion of Water

A glass vessel has a linear expansivity of 1.0×10−5 K−11.0 \times 10^{-5} \text{ K}^{-1} and is filled with a liquid. If the apparent cubic expansivity of the liquid in this vessel is 1.5×10−4 K−11.5 \times 10^{-4} \text{ K}^{-1}, calculate the real cubic expansivity of the liquid in units of 10−4 K−110^{-4} \text{ K}^{-1}.

Answer: 1.8 10^-4 K^-1

Answer

The real cubic expansivity of the liquid is 1.8×10−4 K−11.8 \times 10^{-4} \text{ K}^{-1} (giving 1.81.8 in units of 10−4 K−110^{-4} \text{ K}^{-1}).
The real cubic expansivity of a liquid equals the sum of its apparent cubic expansivity and the volume expansivity of the vessel (\gamma_r = \gamma_a + \gamma_v). First, convert the linear expansivity of the glass vessel to volume expansivity: γv=3α=3×1.0×10−5 K−1=0.3×10−4 K−1\gamma_v = 3\alpha = 3 \times 1.0 \times 10^{-5} \text{ K}^{-1} = 0.3 \times 10^{-4} \text{ K}^{-1}. Adding this to the apparent cubic expansivity (1.5×10−4 K−11.5 \times 10^{-4} \text{ K}^{-1}) yields a real cubic expansivity of 1.8×10−4 K−11.8 \times 10^{-4} \text{ K}^{-1}.

Step-by-Step Solution

1
Determine the cubic expansivity of the vessel (\gamma_v)
\gamma_v = 3.0 \times 10^{-5} \text{ K}^{-1} = 0.3 \times 10^{-4} \text{ K}^{-1}
The volume (cubic) expansivity of a solid vessel is three times its linear expansivity (\gamma_v = 3\alpha).
2
Calculate the real cubic expansivity of the liquid (\gamma_r)
\gamma_r = 1.8 \times 10^{-4} \text{ K}^{-1}
Real cubic expansivity is the sum of apparent cubic expansivity and vessel cubic expansivity (\gamma_r = \gamma_a + \gamma_v).

Key Concept

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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