A glass flask is filled 'to the mark" with 50.00 cm^3 of...

A glass flask is filled 'to the mark" with $50.00 \mathrm{~cm}^{3}$ of mercury at $18^{\circ} \mathrm{C}$. If the flask and its contents are heated to $38^{\circ} \mathrm{C}$, how much mercury will be above the mark? $\alpha_{\text {glass }}=9.0 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$ and $\beta_{\text {mercury }}=182 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$ We shall take $\beta_{\text {elass }}=3 \alpha_{\text {plass }}$ as a good approximation. The flask interior will expand just as though it were a solid piece of glass. Thus, Volume of mercury above mark $=(\Delta V$ for mercury) $-(\Delta V$ for glass) $$ \begin{array}{l} =\beta_{m} V_{0} \Delta T-\beta_{g} V_{0} \Delta T=\left(\beta_{m}-\beta_{g}\right) V_{0} \Delta T \\ =\left[(182-27) \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right]\left(50.00 \mathrm{~cm}^{3}\right)\left[(38-18){ }^{\circ} \mathrm{C}\right] \\ =0.15 \mathrm{~cm}^{3} \end{array} $$

A glass flask is filled 'to the mark" with $50.00 \mathrm{~cm}^{3}$ of mercury at $18^{\circ} \mathrm{C}$. If the flask and its contents are heated to $38^{\circ} \mathrm{C}$, how much mercury will be above the mark? $\alpha_{\text {glass }}=9.0 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$ and $\beta_{\text {mercury }}=182 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$
We shall take $\beta_{\text {elass }}=3 \alpha_{\text {plass }}$ as a good approximation. The flask interior will expand just as though it were a solid piece of glass. Thus,
Volume of mercury above mark $=(\Delta V$ for mercury) $-(\Delta V$ for glass)
$$
\begin{array}{l}
=\beta_{m} V_{0} \Delta T-\beta_{g} V_{0} \Delta T=\left(\beta_{m}-\beta_{g}\right) V_{0} \Delta T \\
=\left[(182-27) \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right]\left(50.00 \mathrm{~cm}^{3}\right)\left[(38-18){ }^{\circ} \mathrm{C}\right] \\
=0.15 \mathrm{~cm}^{3}
\end{array}
$$

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature. It is a fundamental physical property observed in solids, liquids, and gases, and it plays a crucial role in understanding how materials behave under varying thermal conditions.

Coefficient of Thermal Expansion

The coefficient of thermal expansion quantifies the extent to which a material expands per unit change in temperature. It can be expressed in linear or volumetric terms, depending on whether the dimensional change discussed is in one dimension (such as length) or three dimensions (such as volume), and it serves as a key parameter in design and analysis when temperature variations are involved.

Volumetric Expansion

Volumetric expansion refers to the increase in volume of a substance as its temperature rises. In liquids and solids, this change in volume is directly proportional to the initial volume, the temperature change, and the material's volumetric expansion coefficient, making it essential for calculations involving thermal stresses and container design.

Differential Expansion

Differential expansion describes the scenario where two materials with different thermal expansion coefficients change their dimensions at different rates when subjected to the same temperature change. This concept is important when considering the interaction between a container and its contents, as the net effect is determined by the difference in their respective expansion behaviors.

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