Solved Problem on Thermal Expansion

A container is filled with 125 cm³ of mercury at a temperature of 20 °C. The average coefficient of expansion of the mercury is 180 × 10⁻⁶ °C⁻¹, and the coefficient of linear expansion of the glass is 9 × 10⁻⁶ °C⁻¹. Find the volume of mercury spilled out when the temperature rises to 28 °C.

Problem data:

Volume of the container at 20°C: V₀ = 125 cm³;
Coefficient of expansion of mercury: γ_Hg = 180 × 10⁻⁶ °C⁻¹;
Coefficient of linear expansion of glass: α_v = 9 × 10⁻⁶ °C⁻¹;
Initial system temperature: t_i = 20 °C;
Final system temperature: t_f = 28 °C.

Problem diagram:

At the initial temperature, the container and the mercury have the same volume as 125 cm³, when the system is heated, it expands, but as the mercury expands more than the container a part of it overflows. This overflowing part is the apparent volume change ΔV_ap we want to find.

Figure 1

Solution

The coefficient of volumetric expansion of glass γ_v obtained from the coefficient of linear expansion of glass α_v

\[ \bbox[#99CCFF,10px] {\gamma_{v}=3\alpha _{v}} \]

\[ \begin{gather} \gamma_{v}=3 \times 9 \times 10^{-6}\\ \gamma_{v}=27 \times 10^{-6}\;°\text{C}^{-1} \end{gather} \]

The apparent coefficient of expansion of mercury, it is only of the part that overflowed, is given by

\[ \bbox[#99CCFF,10px] {\gamma_{ap}=\gamma_{Hg}-\gamma_{v}} \]

\[ \begin{gather} \gamma_{ap}=180 \times 10^{-6}-27 \times 10^{-6}\\ \gamma_{ap}=153 \times 10^{-6}\;°\text{C}^{-1} \end{gather} \]

The spilled volume will be

\[ \bbox[#99CCFF,10px] {\Delta V_{ap}=V_{0} \gamma_{ap} \Delta t} \]

\[ \begin{gather} \Delta V_{ap}=V_{0}\gamma_{ap}\;(t_{f}-t_{i})\\ \Delta V_{ap}=125 \times 153 \times 10^{-6} \times (28-20)\\ \Delta V_{ap}=125 \times 153 \times 8 \times 10^{-6} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\Delta V_{ap}=0.153\;\text{cm}^{3}} \]

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