Problem 88 - General Chemistry: Principles and Modern Applications

Question

The filament in an incandescent light bulb is made from tungsten metal \(\left(d=19.3 \mathrm{g} / \mathrm{cm}^{3}\right)\) that has been drawn into a very thin wire. The diameter of the wire is difficult to measure directly, so it is sometimes estimated by measuring the mass of a fixed length of wire. If a \(0.200 \mathrm{m}\) length of tungsten wire weighs \(42.9 \mathrm{mg}\), then what is the diameter of the wire? Express your answer in millimeters.

Step-by-Step Solution

Answer

The diameter of the tungsten wire is approximately 0.048 millimeters.

Step 1: Calculate the volume of the wire.

First determine the volume of the tungsten wire. As we know, density = mass/volume. Therefore, we can rearrange this formula to find the volume, which gives us volume = mass/density. Given mass = 42.9 mg and density = 19.3 g/cm³, it's important to convert milligrams to grams, so mass = 42.9*10^-3 g. Substitute these values into the volume formula to result in a volume of: volume = (42.9*10^-3) / 19.3 cm³.

Step 2: Convert length of wire to cm and calculate radius.

The length given (0.200 m) needs to be converted to centimeters since our volume measurement is in cm³, which results in L = 0.200*10^2 cm = 20 cm. Now, since wire is assumed as a cylinder, the volume of the cylinder is given by \(V=\pi r^{2}L\), where r is the radius. Rearranging this equation to find r gives us \(r=\sqrt{V/\pi L}\). Substituting the values into the formula, we get the radius of the wire.

Step 3: Calculate the diameter of the wire.

The diameter of a circle (or in this case, the wire) is simply twice the radius. So, by multiplying the calculated radius by 2, we obtain the diameter of the wire in centimeters.

Step 4: Convert Diameter from cm to mm.

The last step is to convert the found diameter from centimeters to millimeters. To do this, we multiply our diameter by 10 (since 1cm = 10mm). That gives us the final diameter in millimeters.

Key Concepts

Understanding Tungsten Wire

Tungsten wire serves as a key component in various applications, including incandescent light bulbs, because of its high melting point and durability. Tungsten itself is a rare metal, distinguished by its high density of 19.3 g/cm³. This dense property ensures that it remains stable under extreme conditions.
The wire is produced through a process called drawing, where the tungsten is stretched into a thin filament. Because the diameter is so small, it is often impractical to measure directly, making indirect methods like measuring a piece of wire's mass and length very useful.
These properties of tungsten make it ideal for use in environments where temperatures are high and durability is crucial. With tungsten wire, the method of detecting its diameter through weight and length can yield very accurate results, relying on fundamental principles of physics.

Cylindrical Volume and Its Calculation

To find the diameter of a tungsten wire, one starts by calculating the wire's volume, treating it as a cylinder. The formula for the volume of a cylinder is:

• Volume = \(\pi r^2 L\)

where \(r\) is the radius and \(L\) is the length of the wire.
In our problem, the density formula \(\text{density} = \frac{\text{mass}}{\text{volume}}\) is rearranged to solve for volume: \(\text{volume} = \frac{\text{mass}}{\text{density}}\).
This step is crucial to determine how much space the tungsten wire occupies. Given the mass in milligrams (42.9 mg), it is converted to grams to match the density units (g/cm³).
Converting units correctly is vital in ensuring that calculations are accurate, thereby providing reliable results.

The Diameter Calculation Process

Once the volume is known, it is necessary to calculate the radius of the wire to eventually find the diameter. The formula \(V = \pi r^2 L\) is pivotal in this step. We rearrange it to solve for the radius:

• \(r = \sqrt{\frac{V}{\pi L}}\)

Using the length in centimeters ensures consistency with the volume's units. The radius or half of the wire's width provides insight into how thick the wire is at any point in its length.
Once the radius is calculated, finding the diameter is straightforward: simply multiply the radius by 2. This step highlights the nature of cylindrical calculations, where precision in one step translates directly to the accuracy of the final diameter measurement.

Making the Right Unit Conversion

In physics and mathematics, unit conversions often occur to ensure consistency and coherence in calculations. Given the problem, this conversion plays a role both at the start (when converting mass units) and at the end (for the diameter measurement).
To find the diameter in millimeters, one must convert the value obtained in centimeters. Since 1 cm equals 10 mm, multiplying the diameter in centimeters by 10 provides the accurate measurement in millimeters.
This conversion not only standardizes units to be more understandable but also helps in presenting the diameter in a manner that aligns with common measurement systems, making the findings more practical and easily applicable in real-world scenarios.