Question

Your physics teacher assigned a thermometer-building project. He gave you a glass tube with an inside diameter of \(1.00 \mathrm{~mm}\) and a receptacle at one end. He also gave you \(8.63 \mathrm{~cm}^{3}\) of mercury to pour into the tube, which filled the receptacle and some of the tube. You are to add marks indicating degrees Celsius on the glass tube. At what increments should the marks be put on? The volume expansion coefficient of mercury is \(1.81 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\).

Short answer

Answer: The marks should be placed at increments of approximately 1.989 mm.

Step-by-step solution

Find the cross-sectional area of the glass tube

To find the cross-sectional area of the glass tube, we can use the formula for the area of a circle (since the tube has a circular cross-section): \(A = \pi r^2\). Given the diameter is 1.00 mm, the radius is half of that, \(r = 0.50 \mathrm{~mm}\). Convert the radius to meters: \(r=0.50 \times 10^{-3} \mathrm{~m}\). Now, calculate the area: \(A = \pi r^2 = \pi (0.50 \times 10^{-3})^2 \approx 7.853 \times 10^{-7} \mathrm{~m}^2\).

Determine the volume expansion per degree Celsius

To find the volume expansion of mercury for each degree Celsius, we can use the volume expansion coefficient \(\beta\) and the initial volume \(V_0\) of mercury. The formula for volume expansion is \(V-T = V_0(1+\beta T)\), where \(V-T\) is the volume of mercury at temperature \(T\), and \(V_0\) is the initial volume. For a 1-degree Celsius change, we have \(\Delta T = 1^\circ \mathrm{C}\). The change in volume per degree Celsius \(V_1\) will be: \(V_1 = V_0(1+\beta \Delta T) - V_0 = V_0\beta \Delta T = 8.63 \times 10^{-6} \mathrm{~m}^3\times(1.81 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1})(1^\circ \mathrm{C}) \approx 1.562 \times 10^{-9} \mathrm{~m}^3\).

Calculate the increments for the marks

Now that we have the cross-sectional area of the glass tube and the volume expansion per degree Celsius, we can find the increments at which the marks should be put on by dividing the change in volume by the cross-sectional area: \(\text{Increments} = \frac{V_1}{A} = \frac{1.562 \times 10^{-9} \mathrm{~m}^3}{7.853 \times 10^{-7} \mathrm{~m}^2} \approx 1.989 \times 10^{-3} \mathrm{~m}\) or \(1.989 \mathrm{~mm}\). Marks should be placed on the glass tube at increments of approximately 1.989 mm to indicate 1-degree Celsius changes.

Key concepts

Volume Expansion Coefficient

Understanding how materials expand with temperature is crucial in applications such as thermometer design. The volume expansion coefficient is a measure of this property, defining how much a unit volume of a substance expands per degree of temperature increase. Specifically, in your thermometer-building project, the coefficient for mercury is given as \(1.81 \cdot 10^{-4}{ }^\circ \mathrm{C}^{-1}\), which is a relatively small number, indicating that mercury expands only slightly with each degree Celsius.

However, over a range of temperatures, this slight expansion can accumulate, significantly affecting the volume. The accurate measurement of this coefficient ensures your thermometer can reliably indicate temperature changes. The volume expansion coefficient is central to calculating the required increments on the tube where each mark represents an equal temperature interval.

Circular Cross-Section Area

When dealing with tubes or columns, the internal area available for a liquid to occupy is critical. The circular cross-section area of your glass tube directly influences how the volume of mercury within it will translate into a height change as the temperature varies. This area is calculated using the formula \(A = \pi r^2\), where \(\pi\) is a constant (approximately 3.14159), and \(r\) is the radius of the tube's interior.

In your project, converting the diameter to radius and then to the cross-sectional area allows for finding the height increment that corresponds to a 1-degree Celsius change by dividing the volume expansion by the cross-sectional area. This step is crucial for placing temperature marks accurately along the length of your thermometer.

Mercury Volume Expansion

Mercury is often used in thermometers due to its predictable expansion properties when heated. The mercury volume expansion becomes the focal point for determining the incremental marks on the thermometer. The change in volume per degree Celsius \(V_1\) is derived from the initial volume of mercury and the volume expansion coefficient.

In simple terms, as mercury's temperature increases, it needs more space. This volume increase can be visualized as the mercury column rising within the tube. By calculating \(V_1\), you can determine how much the mercury will rise with each degree Celsius, which directly corresponds to the spacing of the marks on the thermometer for accurate temperature readings.