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Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A sample of liquid has an initial volume of 500 cm3500\text{ cm}^3 inside a container at 20∘C20^\circ\text{C}. Upon heating the system to 70∘C70^\circ\text{C}, the liquid's apparent volume expansion is measured to be 10 cm310\text{ cm}^3. If the linear expansivity of the container material is 1.0×10−5 K−11.0 \times 10^{-5}\text{ K}^{-1}, what is the real cubic expansivity of the liquid?

  1. 4.3×10−4 K−14.3 \times 10^{-4}\text{ K}^{-1}Answer
  2. B
    4.1×10−4 K−14.1 \times 10^{-4}\text{ K}^{-1}
  3. C
    4.0×10−4 K−14.0 \times 10^{-4}\text{ K}^{-1}
  4. D
    3.7×10−4 K−13.7 \times 10^{-4}\text{ K}^{-1}

Answer

The real cubic expansivity of the liquid is 4.3×10−4 K−14.3 \times 10^{-4}\text{ K}^{-1}.
The real cubic expansivity of a liquid equals the sum of its apparent cubic expansivity and the cubic expansivity of its vessel (γr=γa+γv\gamma_r = \gamma_a + \gamma_v). Calculating apparent cubic expansivity gives 10500×50=4.0×10−4 K−1\frac{10}{500 \times 50} = 4.0 \times 10^{-4}\text{ K}^{-1}, and the container's cubic expansivity is 3×(1.0×10−5)=0.3×10−4 K−13 \times (1.0 \times 10^{-5}) = 0.3 \times 10^{-4}\text{ K}^{-1}. Adding these yields 4.3×10−4 K−14.3 \times 10^{-4}\text{ K}^{-1}.

Step-by-Step Solution

1
Calculate the temperature change (ΔT\Delta T) and the apparent cubic expansivity (γa\gamma_a) of the liquid.
ΔT=70∘C−20∘C=50 K\Delta T = 70^\circ\text{C} - 20^\circ\text{C} = 50\text{ K}, so γa=ΔVaV0ΔT=10500×50=4.0×10−4 K−1\gamma_a = \frac{\Delta V_a}{V_0 \Delta T} = \frac{10}{500 \times 50} = 4.0 \times 10^{-4}\text{ K}^{-1}.
Apparent expansivity relates apparent volume increase to initial volume and temperature change.
2
Convert the container's linear expansivity (αv\alpha_v) to its cubic expansivity (γv\gamma_v).
γv=3αv=3×(1.0×10−5 K−1)=3.0×10−5 K−1=0.3×10−4 K−1\gamma_v = 3\alpha_v = 3 \times (1.0 \times 10^{-5}\text{ K}^{-1}) = 3.0 \times 10^{-5}\text{ K}^{-1} = 0.3 \times 10^{-4}\text{ K}^{-1}.
Cubic expansivity of an isotropic solid container is three times its linear expansivity.
3
Calculate the real cubic expansivity of the liquid (γr\gamma_r) using the relation γr=γa+γv\gamma_r = \gamma_a + \gamma_v.
γr=4.0×10−4 K−1+0.3×10−4 K−1=4.3×10−4 K−1\gamma_r = 4.0 \times 10^{-4}\text{ K}^{-1} + 0.3 \times 10^{-4}\text{ K}^{-1} = 4.3 \times 10^{-4}\text{ K}^{-1}.
Real expansion of a liquid accounts for both its observed (apparent) expansion and the expansion of the containing vessel.

Key Concept

Relationship between real expansivity, apparent expansivity of liquids, and container expansivity
Estimated Time:1m 30s
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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