Examkin

Question

Difficulty: MediumThermal Expansion of Liquids and Anomalous Expansion of Water

A glass vessel has a linear expansivity of 9.0×10−6 K−19.0 \times 10^{-6}\text{ K}^{-1}. When filled to the brim with 500 cm3500\text{ cm}^3 of a liquid at 20∘C20^\circ\text{C} and heated to 70∘C70^\circ\text{C}, 12 cm312\text{ cm}^3 of the liquid overflows. What is the real cubic expansivity of the liquid?

  1. A
    4.80×10−4 K−14.80 \times 10^{-4}\text{ K}^{-1}
  2. B
    4.89×10−4 K−14.89 \times 10^{-4}\text{ K}^{-1}
  3. 5.07×10−4 K−15.07 \times 10^{-4}\text{ K}^{-1}Answer
  4. D
    4.53×10−4 K−14.53 \times 10^{-4}\text{ K}^{-1}

Answer

The real cubic expansivity of the liquid is 5.07×10−4 K−15.07 \times 10^{-4}\text{ K}^{-1}.
The real cubic expansivity of a liquid accounts for both the liquid's apparent expansion (overflow volume) and the expansion of the containing vessel. Calculating the apparent expansivity gives γa=12/(500×50)=4.80×10−4 K−1\gamma_a = 12 / (500 \times 50) = 4.80 \times 10^{-4}\text{ K}^{-1}. Combining this with the vessel's cubical expansivity γv=3×9.0×10−6=2.70×10−5 K−1\gamma_v = 3 \times 9.0 \times 10^{-6} = 2.70 \times 10^{-5}\text{ K}^{-1} yields γr=4.80×10−4+0.27×10−4=5.07×10−4 K−1\gamma_r = 4.80 \times 10^{-4} + 0.27 \times 10^{-4} = 5.07 \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).
ΔT=70∘C−20∘C=50 K\Delta T = 70^\circ\text{C} - 20^\circ\text{C} = 50\text{ K}. γa=ΔVaV0ΔT=12 cm3500 cm3×50 K=4.80×10−4 K−1\gamma_a = \frac{\Delta V_a}{V_0 \Delta T} = \frac{12\text{ cm}^3}{500\text{ cm}^3 \times 50\text{ K}} = 4.80 \times 10^{-4}\text{ K}^{-1}.
Apparent expansion corresponds directly to the volume of liquid that overflows.
2
Calculate the cubic expansivity of the glass vessel (γv\gamma_v).
γv=3×αv=3×(9.0×10−6 K−1)=2.70×10−5 K−1=0.27×10−4 K−1\gamma_v = 3 \times \alpha_v = 3 \times (9.0 \times 10^{-6}\text{ K}^{-1}) = 2.70 \times 10^{-5}\text{ K}^{-1} = 0.27 \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).
γr=γa+γv=4.80×10−4 K−1+0.27×10−4 K−1=5.07×10−4 K−1\gamma_r = \gamma_a + \gamma_v = 4.80 \times 10^{-4}\text{ K}^{-1} + 0.27 \times 10^{-4}\text{ K}^{-1} = 5.07 \times 10^{-4}\text{ K}^{-1}.
Real cubic expansivity accounts for both the apparent expansion of the liquid and the volume expansion of the container.

Key Concept

Relationship between real cubic expansivity, apparent cubic expansivity, and container volume expansion: γr=γa+γv\gamma_r = \gamma_a + \gamma_v.
Rate this question

Topics

Thermal Expansion of Liquids and Anomalous Expansion of Water

More questions from this topic