Problem 33 - Physics

Question

At the bottom of an old mercury-in-glass thermometer is a \(45-\mathrm{mm}^{3}\) reservoir filled with mercury. When the thermometer was placed under your tongue, the warmed mercury would expand into a very narrow cylindrical channel, called a capillary, whose radius was \(1.7 \times 10^{-2} \mathrm{~mm}\). Marks were placed along the capillary that indicated the temperature. Ignore the thermal expansion of the glass and determine how far (in \(\mathrm{mm}\) ) the mercury would expand into the capillary when the temperature changed by \(1.0 \mathrm{C}^{\circ}\).

Step-by-Step Solution

Answer

The mercury expands about 9.03 mm into the capillary.

Step 1: Identify Known Information

We are given the initial reservoir volume of mercury to be \(45 \, \text{mm}^3\), the capillary radius \(r = 1.7 \times 10^{-2} \, \text{mm}\), and a temperature change \(\Delta T = 1.0 \, \text{C}^{\circ}\). Mercury's volume expansion coefficient \(\beta\) is typically \(1.82 \times 10^{-4} \, \text{C}^{\circ}\).

Step 2: Calculate Volume Expansion of Mercury

The change in volume of mercury due to thermal expansion can be calculated using the formula: \(\Delta V = \beta V_0 \Delta T\), where \(V_0\) is the initial volume. Here, \(\Delta V = 1.82 \times 10^{-4} \times 45 \, \text{mm}^3 \times 1.0 \).

Step 3: Simplify the Volume Expansion Calculation

Simplify to find \(\Delta V\):\[\Delta V = 1.82 \times 10^{-4} \times 45 = 0.00819 \, \text{mm}^3\].

Step 4: Relate Volume to Capillary Length

The expanded volume \(\Delta V\) fills a cylindrical section of the capillary with volume \(\pi r^2 h\), where \(h\) is the height/length in the capillary. Set \(\Delta V = \pi r^2 h\) to solve for \(h\).

Step 5: Solve for Height of Mercury in Capillary

Rearrange and solve: \[h = \frac{\Delta V}{\pi r^2} = \frac{0.00819}{\pi \times (1.7 \times 10^{-2})^2}\].

Step 6: Compute and Conclude the Height

Substitute and compute the value: \[h \approx \frac{0.00819}{\pi \times 0.000289} \approx \frac{0.00819}{0.000907} \approx 9.03 \, \text{mm}\].

Key Concepts

Mercury-in-Glass Thermometer

A classic tool for measuring temperature, the mercury-in-glass thermometer operates through the visible expansion of mercury, a liquid metal with a high degree of thermal sensitivity. The thermometer consists of a glass bulb which holds a reservoir of mercury. Above this reservoir, there is a long, narrow path called a capillary, where the mercury rises as it expands. The glass itself is marked with temperature calibrations to indicate readings.

This method is known for its precision because mercury expands uniformly with temperature. When the mercury absorbs heat, it expands and moves up the capillary tube. This type of thermometer has been used for many years for tasks ranging from medical to meteorological purposes. Despite modern advancements such as digital thermometers, understanding how a mercury-in-glass thermometer works is fundamental to grasping basic concepts of temperature measurement.

Volume Expansion Coefficient

The volume expansion coefficient, often denoted by the Greek letter \( \beta \), is a constant that quantifies how much a material's volume changes in response to temperature changes. It is intrinsic to the material itself, with each material having a unique coefficient.

For mercury, this coefficient is typically \( 1.82 \times 10^{-4} \, \text{C}^{\circ} \). This means that for every degree Celsius of temperature increase, a given volume of mercury will increase by a fraction of this coefficient.

• The formula for change in volume due to thermal expansion is \( \Delta V = \beta V_0 \Delta T \), where \( V_0 \) is the initial volume and \( \Delta T \) is the temperature change.
• This is an important concept in understanding how materials behave under varying temperatures, whether in scientific experiments or everyday applications, like thermometers.

By applying this coefficient, one can predict how a material will behave and ensure accurate measurements are taken.

Cylindrical Capillary

In a mercury-in-glass thermometer, the cylindrical capillary is the part of the thermometer where the mercury expands. It's typically very narrow and cylindrical, designed to magnify the visual effect of even small expansions in the mercury volume, making it easier to read.

• The capillary's radius plays a crucial role in this design. In our example, it's given as \( 1.7 \times 10^{-2} \, \text{mm} \). The tight radius means that any small change in the volume of mercury results in a noticeable change in the height of mercury inside the capillary.
• The formula for finding the height \( h \) that mercury reaches in the capillary is derived from the volume of a cylinder: \( \pi r^2 h \). Here, \( r \) is the radius, and \( h \) is the height or length of the mercury column.

By analyzing the capillary design, one can see how it magnifies the expansion of mercury, thus making small temperature changes noticeable.

Temperature Change Effect

Temperature change has a direct effect on the volume and height of the mercury in the thermometer. When the temperature increases, the heat is absorbed by the mercury, causing it to expand. This expansion results in its movement along the narrow capillary, as we calculated.

• The relationship between temperature change and mercury expansion is quantified by the volume expansion coefficient. Therefore, even a small change in temperature, like the \( 1.0 \, \text{C}^{\circ} \) example, can cause the mercury to travel a significant distance in the capillary.
• Understanding this phenomenon is crucial not just for operating thermometers but for a host of scientific and industrial applications where temperature-induced volume changes need to be managed or measured.

This fundamental concept illustrates the dynamic relationship between heat and material response, a cornerstone of thermodynamics.