In a thermometer manufacturing plant, a type of mercury...

In a thermometer manufacturing plant, a type of mercury thermometer is built at room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ to measure temperatures in the $20.0^{\circ} \mathrm{C}$ to $70.0^{\circ} \mathrm{C}$ range, with a $1.00-\mathrm{cm}^{3}$ spherical reservoir at the bottom and a 0.500 -mm inner diameter expansion tube. The wall thickness of the reservoir and tube is negligible, and the $20.0^{\circ} \mathrm{C}$ mark is at the junction between the spherical reservoir and the tube. The tubes and reservoirs are made of fused silica, a transparent glass form of $\mathrm{SiO}_{2}$ that has a very low linear expansion coefficient $\left(\alpha=0.400 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$ By mistake, the material used for one batch of thermometers was quartz, a transparent crystalline form of $\mathrm{SiO}_{2}$ with a much higher linear expansion coefficient $\left(\alpha=12.3 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .$ Will the manufacturer have to scrap the batch, or will the thermometers work fine, within the expected uncertainty of $5 \%$ in reading the temperature? The volume expansion coefficient of mercury is $\beta=181 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. This phenomenon is described by coefficients that quantify how much a given material expands per degree change in temperature. It is a fundamental concept in materials science and engineering, particularly important in the design of precise instruments like thermometers, where even small changes can impact accuracy.

Linear vs. Volumetric Expansion

Materials expand in different ways depending on their structure. Linear expansion refers to the change in length of a material, while volumetric expansion concerns the change in volume. For example, the container of a thermometer may expand linearly, whereas the fluid inside (such as mercury) typically expands volumetrically. The differing expansion behaviors must be carefully matched to ensure accurate calibration and functioning of a device.

Differential Thermal Expansion Effects

When two materials in a device expand at different rates due to different thermal expansion coefficients, the resulting differential expansion can lead to measurement errors. In precision instruments like thermometers, where one component is a container and the other a working fluid, any mismatch in expansion characteristics can alter the relationship between the observed scale and the actual temperature, potentially introducing systematic errors.

Measurement Uncertainty

Measurement uncertainty quantifies the range within which the true value of a measurement lies, acknowledging that instruments have inherent limitations in accuracy and precision. In engineering applications, understanding and accounting for the uncertainty is crucial when assessing whether deviations due to factors like thermal expansion are acceptable or if they necessitate recalibration or redesign of the device.

Transcript

00:01 So we can say that first defining the change in volume of the mercury as the volume expansion or volume thermal expansion coefficient for mercury multiplied by the volume of the reservoir, the spherical reservoir, multiplied by the change in temperature.

00:18 C would simply be equal to 2 pi r, the circumference of the capillary.

00:23 We can say that then delta c would be equalling the thermal expansion coefficient c -not delta t and then the volume of a cylinder would be equaling to pi r squared times h and we can then say that the change in the volume of the circular reservoir would be equaling to three times the linear expansion coefficient multiplied by the volume of the reservoir multiplied by delta t and so when the volume of the mercury expands by that change in volume the excess mercury goes into the excess volume of the sphere and the remainder goes up into the capillary with a height defined as a volume of the cylinder so this would be the height reached in the capillary divided by pi r squared volume of the cylinder would then be equalling to the change in the volume of the mercury minus the change in the volume of the spherical reservoir.

01:37 So r is equaling then c over 2 pi, equaling then c not plus delta c divided by 2 pi, equaling then c not multiplied by 1 plus alpha delta t divided by 2 pi.

02:00 So a lot of this is simple algebraic manipulation essentially.

02:05 This is r is then equaling 2 pi, r0, 1 plus alpha delta t divided by 2 pi.

02:24 Of course the 2 pi is cancel out, and we're left with r0 plus 1 plus alpha delta t.

02:37 And so h would be given as the change in the volume of the mercury, delta v.

02:45 Hg minus delta v...

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