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Difficulty: HardThermal Expansion of Liquids and Anomalous Expansion of Water

A liquid has a real cubic expansivity of 7.5×10−4 K−17.5 \times 10^{-4}\text{ K}^{-1}. It is heated inside a metal vessel whose material has a linear expansivity of 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}. What is the ratio of the real cubic expansivity of the liquid to its apparent cubic expansivity?

  1. 2523\frac{25}{23}Answer
  2. B
    7573\frac{75}{73}
  3. C
    2527\frac{25}{27}
  4. D
    7577\frac{75}{77}

Answer

The ratio of the real cubic expansivity of the liquid to its apparent cubic expansivity is \(\frac{25}{23}\).
The real cubic expansivity of a liquid \(\gamma_r\) is related to its apparent cubic expansivity \(\gamma_a\) and the cubic expansivity of the container \(\gamma_v\) by the equation \(\gamma_r = \gamma_a + \gamma_v\). For a vessel made of material with linear expansivity \(\alpha\), \(\gamma_v = 3\alpha\). Substituting \(\alpha = 2.0 \times 10^{-5}\text{ K}^{-1}\) yields \(\gamma_v = 6.0 \times 10^{-5}\text{ K}^{-1} = 0.6 \times 10^{-4}\text{ K}^{-1}\). The apparent expansivity is \(\gamma_a = 7.5 \times 10^{-4} - 0.6 \times 10^{-4} = 6.9 \times 10^{-4}\text{ K}^{-1}\). The ratio \(\frac{\gamma_r}{\gamma_a}\) is therefore \(\frac{7.5 \times 10^{-4}}{6.9 \times 10^{-4}} = \frac{25}{23}\).

Step-by-Step Solution

1
Calculate the cubic expansivity of the metal vessel (\(\gamma_v\)) from its linear expansivity (\(\alpha\)).
\(\gamma_v = 3\alpha = 3 \times (2.0 \times 10^{-5}\text{ K}^{-1}) = 6.0 \times 10^{-5}\text{ K}^{-1} = 0.6 \times 10^{-4}\text{ K}^{-1}\)
The volume expansion coefficient of a solid vessel is three times its linear expansion coefficient.
2
Determine the apparent cubic expansivity (\(\gamma_a\)) using the relationship between real expansivity, apparent expansivity, and vessel expansivity.
\(\gamma_a = \gamma_r - \gamma_v = 7.5 \times 10^{-4}\text{ K}^{-1} - 0.6 \times 10^{-4}\text{ K}^{-1} = 6.9 \times 10^{-4}\text{ K}^{-1}\)
Real cubic expansivity of a liquid equals the sum of its apparent cubic expansivity and the cubic expansivity of the containing vessel.
3
Compute the ratio of real cubic expansivity to apparent cubic expansivity (\(\frac{\gamma_r}{\gamma_a}\)).
\(\frac{\gamma_r}{\gamma_a} = \frac{7.5 \times 10^{-4}}{6.9 \times 10^{-4}} = \frac{75}{69} = \frac{25}{23}\)
Dividing the given real cubic expansivity by the calculated apparent cubic expansivity gives the simplified fraction.

Key Concept

Thermal Expansion of Liquids: Relationship between Real and Apparent Expansivity
Estimated Time:1m 30s
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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