Question

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 \(\beta=181 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\).

Short answer

Based on the calculated percentage difference of 2975%, the thermometer made of quartz will not work within the expected uncertainty of 5%. Thus, the manufacturer should discard this batch of thermometers made of the wrong material.

Step-by-step solution

1. Calculate the expansion of fused silica tube for the given temperature range

To calculate the expansion, we will use the linear expansion formula: \(\delta L = L_0 \cdot \alpha \cdot \delta T\), where \(\delta L\) is the change in length, \(L_0\) is the original length, \(\alpha\) is the linear expansion coefficient, and \(\delta T\) is the change in temperature. Here, we are given the temperature range as \(20^\circ C\) to \(70^\circ C\). So, \(\delta T = 70 - 20 = 50^\circ C\). We don't have the initial length of the expansion tube, \(L_0\). However, as we are interested in the percentage difference in the readings, we can skip this value because it will get cancelled when we calculate the percentage difference. Now, let's calculate the expansion of the fused silica tube: \(\delta L_{\text{fused silica}} = \alpha_{\text{fused silica}} \cdot \delta T = 0.4 \cdot 10^{-6} \cdot 50 = 20 \cdot 10^{-6} \ \text{m}\).

2. Calculate the expansion of quartz tube for the given temperature range

Similarly, the expansion of the quartz tube can be calculated as: \(\delta L_{\text{quartz}} = \alpha_{\text{quartz}} \cdot \delta T = 12.3 \cdot 10^{-6} \cdot 50 = 615 \cdot 10^{-6} \ \text{m}\).

3. Calculate the percentage difference in readings

Now, we need to find the percentage difference in the readings: \(\text{Percentage difference} = \frac{\delta L_{\text{quartz}} - \delta L_{\text{fused silica}}}{\delta L_{\text{fused silica}}} \cdot 100\%\) \(\text{Percentage difference} = \frac{615 \cdot 10^{-6} - 20 \cdot 10^{-6}}{20 \cdot 10^{-6}} \cdot 100\% = 2975\%\)

4. Check if the percentage difference is within the acceptable range

Since the percentage difference in the readings is \(2975\%\), which is significantly higher than the acceptable error of \(5\%\). Therefore, the manufacturer must scrap the batch of thermometers made of quartz.

Key concepts

Mercury Thermometer

A mercury thermometer is a common type of thermometer used for measuring temperature. It uses mercury, a liquid metal, which expands as it is heated. This expansion is what allows the thermometer to measure temperature changes. In a mercury thermometer:

• The lower part of the thermometer consists of a bulb that holds the liquid mercury.
• An attached narrow tube, called a capillary, allows the mercury to travel up or down, showing the temperature.
• Mercury is used because it has a uniform rate of expansion, which makes it highly accurate for temperature readings.

This type of thermometer is sensitive and suitable for precise temperature measurements over a defined range, typically from text{0°C to 100°C.} For our discussion, it's important to understand that mercury's behavior under heat directly impacts the accuracy of the thermometer. In experiments and technologies where precision is crucial, the correct material for the thermometer housing matters greatly. Even small miscalculations, like using the wrong material with a different expansion rate, can lead to inaccurate temperature readings.

Linear Expansion Coefficient

The linear expansion coefficient is a crucial parameter in thermal expansion, particularly important when precision in manufacturing thermometers is considered. It quantifies how much a material expands per degree change in temperature. Given by the symbol text{\(\alpha\)}, the linear expansion formula is: \(\delta L = L_0 \cdot \alpha \cdot \delta T\)where

• \(\delta L\) is the change in length,
• \(L_0\) is the original length,
• \(\alpha\) is the linear expansion coefficient,
• \(\delta T\) is the change in temperature.

Materials with higher linear expansion coefficients expand more with temperature changes. In the context of the original problem, fused silica and quartz were compared. Fused silica has a very low linear expansion coefficient, \(0.4 \cdot 10^{-6}{}^{\circ}C^{-1}\), indicating minimal expansion. Quartz, on the other hand, has a coefficient of \(12.3 \cdot 10^{-6}{}^{\circ}C^{-1}\), which means it expands significantly more for the same temperature increase.Understanding this difference helps in choosing suitable materials for applications where thermal precision is essential, such as manufacturing thermometers.

Volume Expansion Coefficient

The volume expansion coefficient is a measure of how much a given volume of material expands as the temperature changes. Just like the linear expansion coefficient, it provides insight into the thermal properties of materials, but it applies to volume changes rather than just one dimension. The formula to express this is:\(\Delta V = V_0 \cdot \beta \cdot \Delta T\)where

• \(\Delta V\) is the change in volume,
• \(V_0\) is the original volume,
• \(\beta\) is the volume expansion coefficient,
• \(\Delta T\) is the change in temperature.

In the context of mercury thermometers, the volume expansion of mercury is significant. With a \(\beta = 181 \cdot 10^{-6}{}^{\circ}C^{-1}\), mercury's volume significantly increases with temperature, making it a suitable liquid for thermometers. The interplay between the expansion of mercury and the material of the thermometer (like quartz or fused silica) determines the overall accuracy and performance of the thermometer. Hence, understanding the volume expansion helps in designing accurate thermometric devices.

Material Properties

Material properties are fundamental in determining how a material responds to changes in temperature among other environmental conditions. This involves understanding the coefficients of thermal expansion (both linear and volume), but it also includes considering other factors like:

• Thermal conductivity - how well a material conducts heat.
• Elasticity - the ability of a material to return to its original shape after deformation.
• Specific heat capacity - the amount of heat per unit mass required to raise the temperature by one degree Celsius.

In thermometer manufacturing, selecting the right material ensures the device functions correctly over its operational range. Different materials change dimensions differently when heated or cooled. This is why precision instruments, such as thermometers, must utilize materials with well-understood and appropriate material properties. Choosing a material like quartz or fused silica will affect the thermometer's sensitivity and accuracy. For instance, using quartz, which has a higher expansion rate, might lead to significant error in temperature readings if not correctly calibrated. Thus, careful material selection is vital to achieve reliable and accurate measurements.

Fused Silica

Fused silica is a type of glass made from pure silicon dioxide (SiO₂) with exceptional thermal and physical properties. It is particularly notable for

• Its very low linear expansion coefficient \(0.4 \cdot 10^{-6}{ }^{\circ}C^{-1}\),
• High thermal resistance,
• Excellent clarity and transparency.

These qualities make it an ideal material for precision instruments such as thermometers.The low thermal expansion means that fused silica expands very little in response to temperature changes, providing a stable, consistent measurement which is crucial in thermometer manufacturing. This minimizes any distortion or measurement inaccuracies due to the material expanding or contracting. Additionally, its high thermal resistance ensures that it can withstand different temperature ranges without deforming or losing its structural integrity.Overall, when making accurate thermometric devices, using fused silica can significantly enhance the reliability and precision of the instrument.

Quartz

Quartz is a crystalline form of silicon dioxide, distinguishable by its properties compared to its amorphous counterpart, fused silica. It is frequently used in applications requiring thermal and mechanical stability, but its thermal expansion characteristics differ noticeably:

• Quartz has a relatively high linear expansion coefficient of \(12.3 \cdot 10^{-6}{}^{\circ}C^{-1}\),
• It is crystal-clear and has good hardness and chemical resistance.

The higher thermal expansion coefficient of quartz means it expands significantly more upon heating than fused silica. This can be problematic in situations where precision is required, such as in thermometer manufacturing, as it can lead to greater measurement errors if not adequately accounted for.In the context of the original exercise, using quartz meant there was an unacceptable level of expansion (text{2975%} difference compared to the ideal), leading to inaccuracies. Therefore, while quartz has many valuable properties, its use where minimal expansion is critical should be carefully considered and often, materials like fused silica are preferred for accurate measurement devices. This is an excellent learning point for the application of materials science in practical situations.