Examkin

Question

Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A glass flask of volume 1000 cm31000\text{ cm}^3 is filled completely with mercury at a temperature of 10∘C10^\circ\text{C}. The linear expansivity of the glass is 9.0×10−6 K−19.0 \times 10^{-6}\text{ K}^{-1} and the real cubic expansivity of mercury is 1.8×10−4 K−11.8 \times 10^{-4}\text{ K}^{-1}. What volume of mercury (in cm3\text{cm}^3) will overflow when the system is heated to 110∘C110^\circ\text{C}?

Answer: 15.3 cm^3

Answer

The volume of mercury that overflows is 15.3 cm315.3\text{ cm}^3.
The apparent expansion of the liquid equals its real expansion minus the expansion of the container. Since γv=3α=2.7×10−5 K−1=0.27×10−4 K−1\gamma_v = 3\alpha = 2.7 \times 10^{-5}\text{ K}^{-1} = 0.27 \times 10^{-4}\text{ K}^{-1}, the apparent cubic expansivity is γa=1.8×10−4−0.27×10−4=1.53×10−4 K−1\gamma_a = 1.8 \times 10^{-4} - 0.27 \times 10^{-4} = 1.53 \times 10^{-4}\text{ K}^{-1}. Multiplying by initial volume (1000 cm31000\text{ cm}^3) and temperature change (100 K100\text{ K}) yields an overflow volume of 15.3 cm315.3\text{ cm}^3.

Step-by-Step Solution

1
Calculate the volume expansivity of the glass vessel (γv\gamma_v)
γv=3×9.0×10−6 K−1=2.7×10−5 K−1=0.27×10−4 K−1\gamma_v = 3 \times 9.0 \times 10^{-6}\text{ K}^{-1} = 2.7 \times 10^{-5}\text{ K}^{-1} = 0.27 \times 10^{-4}\text{ K}^{-1}
The volumetric (cubic) expansivity of a solid container is three times its linear expansivity.
2
Determine the apparent cubic expansivity of mercury (γa\gamma_a)
γa=γr−γv=1.8×10−4−0.27×10−4=1.53×10−4 K−1\gamma_a = \gamma_r - \gamma_v = 1.8 \times 10^{-4} - 0.27 \times 10^{-4} = 1.53 \times 10^{-4}\text{ K}^{-1}
The apparent expansion of a liquid accounts for both the expansion of the liquid itself and the expansion of the containing vessel.
3
Calculate the overflow volume (apparent expansion ΔVa\Delta V_a)
\Delta V_a = V_0 \times \gamma_a \times \Delta T = 1000 \times 1.53 \times 10^{-4} \times 100 = 15.3\text{ cm}^3
The volume of liquid that overflows corresponds directly to its apparent volume increase.

Key Concept

Real and Apparent Cubical Expansivity of Liquids
Estimated Time:1m 30s
Rate this question

Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

More questions from this topic