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

In a thermometer manufacturing plant, a type of mercury thermometer is built at room temperature $\left(20^{\circ} \mathrm{C}\right)$ to measure temperatures in the $20^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ range, with $\mathrm{a}$ $1-\mathrm{cm}^{3}$ spherical reservoir at the bottom and a $0.5-\mathrm{mm}$ inner diameter expansion tube. The wall thickness of the reservoir and tube is negligible, and the $20^{\circ} \mathrm{C}$ 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 $(\alpha=$ $\left.0.4 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .$ 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 Coefficient

This is a measure of how much a material changes in size (length, area, or volume) per degree change in temperature. In the context of the thermometer, both the container (made of glass or quartz) and the mercury inside expand when heated. The coefficients, whether linear for solids or volumetric for liquids, help predict how much each component will change with temperature, which is essential for ensuring accurate temperature measurements.

Linear versus Volumetric Expansion

Most solid materials expand in a lengthwise or linear fashion, described by their linear expansion coefficient, while liquids expand in all three dimensions, described by their volumetric expansion coefficient. In thermometers, the weak expansion of the container (linear expansion) compared to the much larger volumetric expansion of the mercury is critical in determining whether the resulting change in dimensions significantly alters the reading of the liquid column.

Error Analysis in Temperature Measurements

Error analysis involves quantifying how the combined expansion of the thermometer’s container and its mercury affects the measurement accuracy. In this scenario, assessing whether the increased expansion of quartz relative to fused silica is significant compared to the inherent uncertainty (5%) in the temperature reading is key. This analysis helps determine if the systematic error introduced by the container expansion remains within acceptable limits.

Material Properties and Measurement Uncertainty

The specific physical properties of the materials used, such as their thermal expansion coefficients, directly impact the performance of precision instruments like thermometers. Understanding these properties and their influence on measurement uncertainty is crucial when evaluating whether a manufacturing error, such as using a material with a higher expansion coefficient, will compromise the instrument's accuracy or remain within the allowed tolerance range.

Transcript

00:01 So we'll start with a couple of equations.

00:02 We can say that the change in the volume of the mercury would be equalling to the volumetric thermal expansion coefficient for mercury multiplied by the volume, the initial volume, multiplied by delta t.

00:23 So the circumference of the capillary, 2 pi multiplied by r, we can say that the change in the circumference would be equalling the linear expansion, coefficient multiplied by the original circumference multiplied by delta t and the volume of a cylinder is equaling to pi r squared h and so we can say that the change in the volume of the spherical reservoir would be equalling then to three multiplied by alpha v sub s delta t and when the volume of the mercury expands, we can say that the excess mercury goes into the excess volume of the sphere.

01:12 And the height reached by this excess mercury would then be equal to the volume of the cylinder divided by pi r squared.

01:22 We know that r is equaling the circumference divided by 2 pi, which we know to be equal to the original circumference plus the change in the circumference divided by 2 pi, we can say that then r would be equal to c not multiplied by 1 plus alpha delta t.

01:45 This would then be divided by 2 pi again, and this would then be equal to 2 pi r not multiplied by 1 plus alpha delta t divided by 2 pi.

02:01 Of course the 2 pi is cancel out, and we find that r is going to be equaling to the original radius multiplied by 1 plus alpha delta t.

02:14 We can say from the equation for h, h would then be the volume of the volume change of the mercury minus the volume change of the spherical reservoir.

02:29 This would be divided by pi r squared.

02:34 And we can substitute h would then be equal to beta, mercury v -s -s -d delta -t minus 3 alpha v -s -s -d delta -t divided by pi, and then here instead of r, we'll substitute and say r not squared, multiplied by 1 plus alpha -delta -ttt quantity squared.

03:06 And simplifying a little bit, we have h equaling volume sub s delta t, volumetric thermal expansion coefficient for mercury minus three times the linear expansion coefficient.

03:25 This would be divided by, again, pi r not squared, multiplied by one plus alpha delta t quantity squared.

03:37 So for quarts, we have the, we're going to be using the, linear expansion coefficient for quartz.

03:43 We know the fractional change f would be equaling to 1 minus h for quartz divided by h for silica and after substituting both of these in we have that the fractional change is going to be equal to 1 minus and we then have beta for mercury minus 3 alpha quarts pi r not squared 1 plus alpha for silica the lunar efficiency coefficient for silica multiplied by delta t quantity squared this second term will then be divided by again beta for mercury minus 3 alpha for silica pi r not squared...

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