Question

You are designing a precision mercury thermometer based on the thermal expansion of mercury \(\left(\beta=1.8 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\right)\) which causes the mercury to expand up a thin capillary as the temperature increases. The equation for the change in volume of the mercury as a function of temperature is \(\Delta V=\beta V_{0} \Delta T\) where \(V_{0}\) is the initial volume of the mercury and \(\Delta V\) is the change in volume due to a change in temperature, \(\Delta T .\) In response to a temperature change of \(1.0^{\circ} \mathrm{C}\), the column of mercury in your precision thermometer should move a distance \(D=1.0 \mathrm{~cm}\) up a cylindrical capillary of radius \(r=0.10 \mathrm{~mm} .\) Determine the initial volume of mercury that allows this change. Then find the radius of a spherical bulb that contains this volume of mercury.

Short answer

Answer: The initial volume of mercury in the thermometer is approximately \(30.83\mathrm{~cm}^3\), and the radius of the spherical bulb containing this volume is approximately \(1.98\mathrm{~cm}\).

Step-by-step solution

Calculate total change in volume

We are given that the mercury column rises 1 cm (\(D = 1.0\) cm) up a cylindrical capillary of radius \(r = 0.1\) mm. We can use this to find the total change in volume due to the temperature change. The formula for the volume of a cylinder is \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. For our situation, the height is the distance the mercury rises, which is \(D\). So, \(\Delta V = \pi r^2 D\). \(\Delta V = \pi (0.1 \cdot 10^{-1}\mathrm{~cm})^2 (1.0\mathrm{~cm}) = \pi \cdot 10^{-3}\mathrm{~cm}^3\)

Calculate initial volume of mercury, \(V_0\)

Now we want to find the initial volume of mercury, \(V_0\). We can use the equation provided for change in volume: \(\Delta V = \beta V_0 \Delta T\). We are given that \(\Delta T = 1.0 ^\circ\mathrm{C}\) Solving for \(V_0\), we get \(V_0 =\frac{\Delta V}{\beta \Delta T} =\frac{\pi \cdot 10^{-3}\mathrm{~cm}^3}{(1.8 \cdot 10^{-4}\ ^{\circ}\mathrm{C}^{-1})(1.0 ^\circ\mathrm{C})} = \frac{5.55\ldots}{1.8 \cdot 10^{-4} }\mathrm{~cm}^3\) \(V_0 \approx 30.83\mathrm{~cm}^3\) So, the initial volume of mercury is approximately \(30.83\mathrm{~cm}^3\).

Calculate the radius of spherical bulb

Finally, we want to find the radius of the spherical bulb that contains this volume of mercury. The volumne of a sphere is given by the formula, \(V = \frac{4}{3}\pi r^3\). We can solve for \(r\), the radius of the sphere as follows: \(r = \sqrt[3]{\frac{3V}{4\pi}} = \sqrt[3]{\frac{3 \cdot 30.83\mathrm{~cm}^3}{4\pi}} \approx \sqrt[3]{7.74\ldots} \approx 1.98\mathrm{~cm}\) So, the radius of the spherical bulb that contains the initial volume of mercury is approximately \(1.98\mathrm{~cm}\).

Key concepts

Mercury Thermometer

A mercury thermometer is a classic device used to measure temperature. It consists of mercury encapsulated within a glass tube. As the temperature changes, the mercury expands or contracts, moving up or down the tube. This movement corresponds to different temperature readings marked on the side.

• Mercury's Properties: Mercury is an ideal choice for thermometers because it remains liquid over a wide range of temperatures and expands uniformly with temperature changes.
• Design: It consists of a bulb filled with mercury connected to a thin capillary tube. The precise measurement of temperature is possible due to the predictable expansion of the mercury.
• Safety Considerations: Modern alternatives sometimes replace mercury due to its toxicity if the thermometer breaks. However, mercurial devices are still praised for their accuracy in controlled environments.

Volume Change

The concept of volume change describes how the volume of a substance adjusts when subjected to temperature variation. For mercury in a thermometer, this phenomenon is predictable and quantifiable using the formula: \(\Delta V = \beta V_0 \Delta T\)

• Key Terms:
  • \(\Delta V\) is the change in volume.
  • \(\beta\) is the coefficient of volumetric expansion, specific to the material.
  • \(V_0\) is the initial volume.
  • \(\Delta T\) is the change in temperature.
• Application: In the exercise, knowing the mercury's coefficient of expansion allows calculation of how much mercury will increase in volume per degree increase in temperature, thus causing it to rise in the thermometer tube.
• Predictability: The uniform expansion property makes calculations straightforward and reliable, utilizing the formula to understand volume reactions over temperature shifts.

Temperature Change

Understanding temperature change is crucial in many scientific and daily life applications, including using a thermometer. Temperature change affects how substances expand or contract.

• Temperature Effect: In a mercury thermometer, a temperature increase leads to an expansion of the liquid mercury. This is due to the added heat energy causing mercury atoms to move more vigorously, pushing apart molecules and increasing volume.
• Thermal Expansion Coefficient: For mercury, this coefficient is given as \(1.8 \times 10^{-4} \, ^{\circ}C^{-1}\), indicating how sensitive mercury is to temperature shifts.
• Practical Example: Knowing that a \(1^{\circ}C\) change causes measurable displacement in mercury level allows engineers to design thermometers with known precision, crucial in laboratory settings.

Precision Measurement

Precision measurement is key to scientific discovery and everyday technology applications, such as those involving a mercury thermometer. These measurements depend on accurate and reliable tools.

• Reasons for Precision: Accurate temperature readings are essential in processes like chemical reactions, regulated climate systems, and medical diagnostics.
• Mercury Properties: Mercury's consistent expansion coefficient enables the design of high-precision thermometers, important in experiments where even a small temperature difference is significant.
• Focus Areas for Improvement: In thermometer design, reducing external factors like pressure or ambient temperature fluctuations can enhance the precision and reliability of the readings.

The exercise demonstrates how an accurate understanding and application of physics can directly improve technological solutions for measuring temperature changes accurately in diverse settings.