Problem 87 - Chemistry

Question

What is the density of lithium metal in \(\mathrm{g} / \mathrm{cm}^{3}\) if a cylindrical wire with a diameter of \(2.40 \mathrm{~mm}\) and a length of \(15.0 \mathrm{~cm}\) has a mass of \(0.3624\) g?

Step-by-Step Solution

Answer

The density of lithium metal is approximately 0.5343 g/cm³.

Step 1: Convert Dimensions to Consistent Units

First, we need to convert the diameter of the wire from millimeters to centimeters because the density is to be calculated in grams per cubic centimeter. The diameter given is 2.40 mm, which is equal to 0.24 cm (since 1 mm = 0.1 cm).

Step 2: Calculate the Radius of the Wire

The radius of the wire is half its diameter. So, take the diameter in centimeters and divide by 2: \( r = \frac{0.24}{2} \text{ cm} = 0.12 \text{ cm} \).

Step 3: Calculate the Volume of the Cylindrical Wire

The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length) of the cylinder. Substitute \( r = 0.12 \text{ cm} \) and \( h = 15.0 \text{ cm} \) to get \[ V = \pi (0.12)^2 \times 15.0 \].\Calculate \( V = 3.1416 \times 0.0144 \times 15.0 = 0.6782 \text{ cm}^3 \) (rounded to four decimal places).

Step 4: Calculate the Density of Lithium

Density is defined as mass per unit volume, \( \rho = \frac{m}{V} \). Given the mass \( m = 0.3624 \text{ g} \) and the volume \( V = 0.6782 \text{ cm}^3 \), calculate the density as \[ \rho = \frac{0.3624}{0.6782} \].\This gives \( \rho = 0.5343 \text{ g/cm}^3 \) (rounded to four decimal places).

Key Concepts

Cylindrical Volume

To understand the cylindrical volume, imagine the shape of a soda can or a candle. A cylinder has two main parts: the circular base and the length (or height) that runs perpendicular to this base. When calculating the volume of a cylinder, we essentially want to determine how much space is contained within this shape. The key to finding this volume lies in using the formula: \[V = \pi r^2 h\]where \(V\) is the volume, \(r\) is the radius of the circular base, and \(h\) is the height or the length of the cylinder.
Here are the steps involved in calculating a cylindrical volume:

• First, find the radius, which is half of the diameter.
• Square the radius to get \(r^2\).
• Multiply this value by \(\pi\) (approximated as 3.1416) and then multiply by the height.

This operation will give you the volume in cubic units. When tackling a problem involving a cylinder, always ensure you're using the same measurement units for all dimensions, whether they're in centimeters, meters, or millimeters.
Applying this knowledge, for a cylindrical wire, we calculate its volume by first determining the radius from the diameter, and then using the formula with the given height (or length).

Unit Conversion

Unit conversion is an essential skill in density calculations. It involves changing a measure from one unit to another, ensuring all quantities in a problem are consistent. This is critical because inconsistent units can lead to significant errors in calculations. For example, in the provided exercise, the diameter of the wire was given in millimeters. However, since the density needs to be calculated in grams per cubic centimeter, converting the diameter to centimeters was necessary. Here’s how that conversion was done:

• Recognize that 1 mm = 0.1 cm.
• To convert millimeters to centimeters, multiply the millimeter value by 0.1.

Thus, a diameter of 2.40 mm becomes 0.24 cm.
Conversion helps maintain consistency, which is crucial for computing accurate results. When you encounter different units in a calculation problem, make it a habit to convert all of them into one consistent unit before solving it. Using this approach will build your accuracy and reliability in mathematical computations.

Lithium Density

Lithium is a soft, silvery-white metal that is part of the alkali metal group in the periodic table. Its density is relatively low compared to other metals, but understanding how to calculate this density is crucial when working with physical properties.Density is the measure of mass per unit of volume. In lithium's case within the exercise, you calculated the density using the formula:

• Density, \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume.

Given that you have a mass of 0.3624 grams and a calculated volume of 0.6782 cubic centimeters, you can find the density by dividing the mass by the volume.This method reflects how densely (or sparsely) the atoms of lithium are packed in the wire. The result, a density of approximately 0.5343 g/cm³, verifies that lithium is indeed light and less dense compared to denser metals like iron or copper.
Mastering the calculation of density not only aids in understanding material properties but is also valuable for many applications in science and engineering.