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Difficulty: HardThermal Expansion of Liquids and Anomalous Expansion of Water

A liquid has a density of 840 kg m−3840\text{ kg m}^{-3} at 10∘C10^\circ\text{C}. It is heated inside a container whose material has a linear expansivity of 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}. If the measured apparent cubic expansivity of the liquid in this container is 4.4×10−4 K−14.4 \times 10^{-4}\text{ K}^{-1}, what is the density of the liquid at 110∘C110^\circ\text{C} in kg m−3\text{kg m}^{-3}?

Answer: 800 kg m^-3

Answer

800 kg m^-3
To find the density of the liquid at the elevated temperature, we must use its real cubic expansivity γr\gamma_r. The vessel expands with a cubic expansivity of γv=3α=6.0×10−5 K−1\gamma_v = 3\alpha = 6.0 \times 10^{-5}\text{ K}^{-1}. The real cubic expansivity of the liquid is γr=γa+γv=4.4×10−4+0.6×10−4=5.0×10−4 K−1\gamma_r = \gamma_a + \gamma_v = 4.4 \times 10^{-4} + 0.6 \times 10^{-4} = 5.0 \times 10^{-4}\text{ K}^{-1}. Using the density variation formula ρ2=ρ11+γrΔT\rho_2 = \frac{\rho_1}{1 + \gamma_r \Delta T}, we evaluate 8401+(5.0×10−4×100)=8401.05=800 kg m−3\frac{840}{1 + (5.0 \times 10^{-4} \times 100)} = \frac{840}{1.05} = 800\text{ kg m}^{-3}.

Step-by-Step Solution

1
Calculate the cubic expansivity of the container material
γv=6.0×10−5 K−1\gamma_v = 6.0 \times 10^{-5}\text{ K}^{-1}
The volume (cubic) expansivity of a solid container is three times its linear expansivity (γv=3α\gamma_v = 3\alpha).
2
Determine the real cubic expansivity of the liquid
γr=5.0×10−4 K−1\gamma_r = 5.0 \times 10^{-4}\text{ K}^{-1}
The real cubic expansivity is the sum of the apparent cubic expansivity and the cubic expansivity of the container (γr=γa+γv\gamma_r = \gamma_a + \gamma_v).
3
Determine the change in temperature
ΔT=100 K\Delta T = 100\text{ K}
Subtract the initial temperature from the final temperature: 110∘C−10∘C=100 K110^\circ\text{C} - 10^\circ\text{C} = 100\text{ K}.
4
Calculate the final density of the liquid at 110∘C110^\circ\text{C}
ρ2=800 kg m−3\rho_2 = 800\text{ kg m}^{-3}
Density varies inversely with volumetric expansion according to ρ2=ρ11+γrΔT\rho_2 = \frac{\rho_1}{1 + \gamma_r \Delta T}.

Key Concept

Relationship between real expansivity, apparent expansivity, container expansion, and density change in fluids
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Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

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