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Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A copper container with an initial volume of 800 cm3800 \text{ cm}^3 at 25∘C25^\circ\text{C} is completely filled with oil. The real cubic expansivity of the oil is 6.5×10−4 K−16.5 \times 10^{-4} \text{ K}^{-1} and the linear expansivity of copper is 1.5×10−5 K−11.5 \times 10^{-5} \text{ K}^{-1}. What volume of oil (in cm3\text{cm}^3) will overflow when the temperature of the system is raised to 75∘C75^\circ\text{C}?

Answer: 24.2 cm^3

Answer

The volume of oil that overflows is 24.2 cm324.2 \text{ cm}^3.
When a container completely filled with liquid is heated, both liquid and container expand. The overflow volume equals the apparent volume expansion of the liquid ΔVa=V0γaΔT\Delta V_a = V_0 \gamma_a \Delta T. The apparent cubic expansivity γa\gamma_a is obtained by subtracting the container's volume expansivity (γv=3α=4.5×10−5 K−1\gamma_v = 3\alpha = 4.5 \times 10^{-5} \text{ K}^{-1}) from the liquid's real cubic expansivity (γr=6.5×10−4 K−1\gamma_r = 6.5 \times 10^{-4} \text{ K}^{-1}), giving γa=6.05×10−4 K−1\gamma_a = 6.05 \times 10^{-4} \text{ K}^{-1}. Multiplying by V0=800 cm3V_0 = 800 \text{ cm}^3 and ΔT=50 K\Delta T = 50 \text{ K} gives 24.2 cm324.2 \text{ cm}^3.

Step-by-Step Solution

1
Calculate the cubic expansivity of the copper container
γv=3α=3×(1.5×10−5 K−1)=4.5×10−5 K−1=0.45×10−4 K−1\gamma_v = 3 \alpha = 3 \times (1.5 \times 10^{-5} \text{ K}^{-1}) = 4.5 \times 10^{-5} \text{ K}^{-1} = 0.45 \times 10^{-4} \text{ K}^{-1}
The volumetric expansion coefficient of a solid container is three times its linear expansivity.
2
Determine the apparent cubic expansivity of the oil
γa=γr−γv=6.5×10−4 K−1−0.45×10−4 K−1=6.05×10−4 K−1\gamma_a = \gamma_r - \gamma_v = 6.5 \times 10^{-4} \text{ K}^{-1} - 0.45 \times 10^{-4} \text{ K}^{-1} = 6.05 \times 10^{-4} \text{ K}^{-1}
The apparent expansion of a liquid accounts for the concurrent thermal expansion of the containing vessel.
3
Compute the temperature increase
ΔT=75∘C−25∘C=50 K\Delta T = 75^\circ\text{C} - 25^\circ\text{C} = 50 \text{ K}
Temperature change is the final temperature minus the initial temperature.
4
Calculate the volume of oil that overflows
ΔVa=V0γaΔT=800 cm3×(6.05×10−4 K−1)×50 K=24.2 cm3\Delta V_a = V_0 \gamma_a \Delta T = 800 \text{ cm}^3 \times (6.05 \times 10^{-4} \text{ K}^{-1}) \times 50 \text{ K} = 24.2 \text{ cm}^3
The overflow volume is equal to the apparent increase in volume of the liquid.

Key Concept

Apparent and Real Expansion of Liquids
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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