A mercury thermometer is constructed as shown in Figure...

A mercury thermometer is constructed as shown in Figure P19.47. The capillary tube has a diameter of 0.00400 $\mathrm{cm}$ , and the bulb has a diameter of $0.250 \mathrm{cm} .$ Neglecting the expansion of the glass, find the change in height of the mercury column that occurs with a temperature change of $30.0^{\circ} \mathrm{C} .$

A mercury thermometer is constructed as shown in Figure P19.47. The capillary tube has a diameter of 0.00400 $\mathrm{cm}$ , and the bulb has a diameter of $0.250 \mathrm{cm} .$ Neglecting the
expansion of the glass, find the change in height of the mercury column that occurs with a temperature change of $30.0^{\circ} \mathrm{C} .$

Key Concepts

Volumetric Thermal Expansion

This concept refers to the tendency of a material’s volume to change in response to temperature changes. For liquids like mercury, the change in volume is typically calculated using the relation ?V = ?V??T, where ? is the coefficient of volumetric expansion, V? is the initial volume, and ?T is the temperature change. Understanding this relationship is crucial when predicting how much a liquid will expand or contract with temperature variations.

Geometrical Relationship in Cylindrical Containers

In many physical applications, including thermometers, a change in volume within a cylindrical container corresponds to a change in height of the liquid column. The volume of a cylinder is given by V = A·h, where A is the cross?sectional area (?r²) and h is the height. Thus, when a liquid expands, the increase in volume divided by the fixed cross-sectional area of the tube directly gives the change in height.

Application in Capillary Tube Thermometers

Capillary tube thermometers are based on confining a liquid, such as mercury, in a narrow tube so that its volumetric expansion with temperature results in a measurable change in height. The design utilizes the precise geometric relationship between volume and column height, allowing small volume changes due to temperature variations to translate into significant and readable changes in the level of mercury along the narrow capillary.

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