Examkin

Question

Difficulty: HardThermal Expansion of Liquids and Anomalous Expansion of Water

A metal container with an initial volume of 500 cm3500\text{ cm}^3 is completely filled with an organic liquid at 15∘C15^\circ\text{C}. The real cubic expansivity of the liquid is 4.0×10−4 K−14.0 \times 10^{-4}\text{ K}^{-1}. When the container and liquid are heated together to a final temperature of 65∘C65^\circ\text{C}, exactly 8.5 cm38.5\text{ cm}^3 of the liquid overflows. What is the linear expansivity of the metal container, in units of 10−5 K−110^{-5}\text{ K}^{-1}?

Answer: 2 10^-5 K^-1

Answer

The linear expansivity of the metal container is 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}, which corresponds to a numerical value of 2.02.0 in units of 10−5 K−110^{-5}\text{ K}^{-1}.
The apparent expansivity of the liquid is derived from the overflow volume (8.5 cm^3 / (500 cm^3 * 50 K) = 3.4 x 10^-4 K^-1). Subtracting this from the liquid's real expansivity (4.0 x 10^-4 K^-1) yields the container's cubic expansivity of 0.6 x 10^-4 K^-1 (or 6.0 x 10^-5 K^-1). Dividing by 3 gives the linear expansivity of the container material as 2.0 x 10^-5 K^-1.

Step-by-Step Solution

1
Determine the temperature increase
\Delta T = 50\text{ K}
Temperature change drives the thermal expansion process.
2
Compute the apparent cubic expansivity of the liquid
\gamma_a = 3.4 \times 10^{-4}\text{ K}^{-1}
Apparent expansion is measured directly from the liquid overflow relative to initial volume and temperature rise.
3
Calculate the cubic expansivity of the metal container
\gamma_v = 6.0 \times 10^{-5}\text{ K}^{-1}
The difference between real cubic expansivity of the liquid and its apparent cubic expansivity equals the cubic expansivity of the container.
4
Calculate the linear expansivity of the container material
\alpha = 2.0 \times 10^{-5}\text{ K}^{-1}
For isotropic solids, volume expansivity is three times linear expansivity (\gamma_v = 3\alpha).

Key Concept

Thermal expansion of liquids in expanding vessels: Real expansivity equals apparent expansivity plus vessel cubic expansivity (\gamma_r = \gamma_a + 3\alpha).
Estimated Time:2m 0s
Rate this question

Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

More questions from this topic