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 $0.00400 \mathrm{cm}$, (FIGURE CANNOT COPY) 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 $0.00400 \mathrm{cm}$, (FIGURE CANNOT COPY)
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

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.

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.

Transcript

00:01 To solve this exercise, i have to recall a few things about thermal expansion.

00:05 Specifically, we have to remember that increasing volume of an object due to the increase in temperature is going to be the initial volume of the object times its initial temperature, times the coefficient of volumetric expansion.

00:22 And this quantity, beta, varies depending with the material that we are working with.

00:29 And if you check table 19 .1, you're going to have several values of beta for different materials.

00:36 Now, in this exercise, suppose that we have a thermometer that looks like this.

00:43 So this is a thermometer made of glass, and it's made of a long tube and a spherical part.

00:58 So suppose that a diameter of the tube, i'm going to call d1, is 0 .004 centimeters, and that the diameter of this spherical part that i'm going to call d2 is 0 .25 centimeters.

01:19 And the exercise asks us to suppose that these things here, these diameters are not going to change with temperature.

01:29 So these things we're going to treat as constant with temperature.

01:35 Okay? now let's suppose that at a temperature, t .i.

01:41 We put some mercury in the thermometer, and the mercury completely fills the spherical part of our thermometer.

01:53 The exercise asks us, what is going to be the height of the mercury? so what is going to be this height of the mercury after it expands due to an increase of temperature of 30 degrees? okay, so delta t is 30 degrees, celsius, the final temperature is going to be ti plus delta t.

02:32 Okay, now first things first, let's calculate what is going to be the volumetric.

02:38 Expansion of mercury.

02:41 So we have that the expansion of the mercury, the volumetric expansion, i'm going to call delta v, is going to be the initial volume of the mercury at a temperature ti, which is the volume of the spherical parts.

02:59 So i'm going to call this vs times a variation in temperature, times the constant for mercury.

03:11 The coefficients of expansion for mercury is 1 .82 times 10 to the minus 4 per degree celsius.

03:26 So let's calculate the volume of the sphere...

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