Problem 109 - Chemistry and Chemical Reactivity

Question

Suppose you have a cylindrical glass tube with a thin capillary opening, and you wish to determine the diameter of the capillary. You can do this experimentally by weighing a piece of the tubing before and after filling a portion of the capillary with mercury. Using the following information, calculate the diameter of the capillary. Mass of tube before adding mercury \(=3.263 \mathrm{g}\) Mass of tube after adding mercury \(=3.416 \mathrm{g}\) Length of capillary filled with mercury \(=16.75 \mathrm{mm}\) Density of mercury \(=13.546 \mathrm{g} / \mathrm{cm}^{3}\) Volume of cylindrical capillary filled with mercury \(=(\pi)(\text { radius })^{2}(\text { length })\)

Step-by-Step Solution

Answer

The diameter of the capillary is approximately 0.0926 cm.

Step 1: Determine the Mass of Mercury

Calculate the difference in mass before and after filling the capillary with mercury. This will give the mass of the mercury in the capillary.\[ m_{\text{mercury}} = 3.416\,\text{g} - 3.263\,\text{g} = 0.153\,\text{g} \]

Step 2: Convert Length of Capillary to Centimeters

Convert the length of the capillary filled with mercury from millimeters to centimeters for use in the volume formula.\[ \text{Length in cm} = 16.75\,\text{mm} \times \frac{1\,\text{cm}}{10\,\text{mm}} = 1.675\,\text{cm} \]

Step 3: Calculate the Volume of Mercury

Use the formula for density to find the volume of the mercury. Density \( \rho \) is mass divided by volume.\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \implies \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]\[ V = \frac{0.153\,\text{g}}{13.546\,\text{g/cm}^3} = 0.01129\,\text{cm}^3 \]

Step 4: Solve for the Radius of the Capillary

Use the volume of a cylinder formula and solve for the radius of the capillary where the volume \( V = \pi r^2 h \).\[ V = \pi r^2 \cdot \text{Length} \]\[ 0.01129\,\text{cm}^3 = \pi r^2 \cdot 1.675\,\text{cm} \]\[ r^2 = \frac{0.01129\,\text{cm}^3}{\pi \cdot 1.675\,\text{cm}} \]\[ r^2 = \frac{0.01129}{5.26775} \approx 0.002143\,\text{cm}^2 \]\[ r = \sqrt{0.002143\,\text{cm}^2} \approx 0.0463\,\text{cm} \]

Step 5: Calculate the Diameter of the Capillary

The diameter is twice the radius.\[ \text{Diameter} = 2 \times 0.0463\,\text{cm} \approx 0.0926\,\text{cm} \]

Key Concepts

Density

Density is a fundamental concept in physics and chemistry that relates mass to the volume of a substance. It tells us how much matter is packed into a given space. For any material, density is defined as the mass per unit volume. The formula to calculate density, \( \rho \), is given by:
\[ \rho = \frac{m}{V} \] where:

• \( m \) is the mass of the substance.
• \( V \) is the volume occupied by the substance.

In the context of the exercise, you have mercury, a liquid metal, which is quite dense. It's important to note the units used for density in the calculation: grams per cubic centimeter (g/cm³). When calculating volumes and masses, make sure unit conversions are correct to avoid errors.

Volume of a Cylinder

The volume of a cylinder is a critical formula that helps calculate the amount of space inside a cylindrical shape. A cylinder, like a soda can or a tube, has two major components:

• The base, which is a circle.
• The height, which extends the shape into the third dimension.

The formula to calculate the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \] where:

• \( \pi \) is approximately 3.14159.
• \( r \) is the radius of the circular base.
• \( h \) is the height or length of the cylinder.

This formula is useful when determining volumes in situations involving cylindrical shapes, such as our mercury-filled capillary. It's essential to ensure that all measurements, especially the length, are in consistent units (e.g., centimeters).

Mercury

Mercury is a unique element primarily due to its liquid state at room temperature. Known as quicksilver, it's dense and heavy compared to other common liquids.

• In the context of our experiment, mercury is used to fill the capillary tube.
• Its density, given as 13.546 g/cm³, is essential for calculating its volume through the mass-volume relationship.

Handling mercury requires care due to its toxicity, especially in vapor form. In laboratory exercises, it helps to accurately measure small spaces or evaluate surface tensions. With its dense nature, mercury makes for an interesting substance in calculations, often yielding smaller volumes for a given mass than many other liquids.

Radius

The radius is a fundamental measure used to describe circles and cylindrical shapes. It extends from the center of a circle to any point on its boundary.

• When dealing with cylinders, knowing the radius is key to calculating surface areas and volumes.
• The radius is involved in the formula \( \pi r^2 h \) for calculating the volume of a cylinder, where \( r \) emerges squared and multiplied by height \( h \).

In our capillary exercise, we used the inverse approach, where we deduced the radius from known values of volume and length. The calculated radius allowed us to find the diameter, which is simply twice the radius. Ensure accuracy in squaring sections and solving through algebra to avoid inaccuracies in measurement outputs.