Question

What is the density of lithium metal (in \(\mathrm{g} / \mathrm{cm}^{3}\) ) if a cube measuring \(0.82 \mathrm{~cm} \times 1.45 \mathrm{~cm} \times 1.25 \mathrm{~cm}\) has a mass of \(0.794 \mathrm{~g}\) ?

Short answer

The density of lithium metal is approximately \(0.534\, \text{g/cm}^3\).

Step-by-step solution

Calculate the Volume

To find the density, we first need the volume of the lithium metal cube. Since it's a rectangular prism, use the formula for volume: \[ V = ext{length} \times ext{width} \times ext{height} \]Substitute the given dimensions: \[ V = 0.82\, \text{cm} \times 1.45\, \text{cm} \times 1.25\, \text{cm} \]Calculate the product for volume: \[ V = 1.48585 \, \text{cm}^3 \]

Use the Density Formula

The formula for density is: \[ \rho = \frac{\text{mass}}{\text{volume}} \]We have a mass of \(0.794\, \text{g}\) and a volume of \(1.48585\, \text{cm}^3\). Substitute these values into the density formula: \[ \rho = \frac{0.794}{1.48585} \]

Calculate the Density

Perform the division to find the density:\[ \rho = 0.534\, \text{g/cm}^3 \]

Round the Result

Round the calculated density to three significant figures, as the given measurements suggest a precision of three significant figures: \[ \rho = 0.534\, \text{g/cm}^3 \]

Key concepts

Volume of a Rectangular Prism

The volume of a rectangular prism is a simple concept in geometry. A rectangular prism has three dimensions: length, width, and height. To find the volume, you multiply these three dimensions together. The formula for the volume of a rectangular prism can be written as: \[ V = ext{length} \times ext{width} \times ext{height} \] This formula gives the measure of how much space the prism occupies, in cubic units. In our example, the lengths given were "0.82 cm, 1.45 cm, and 1.25 cm". By substituting these values into the formula, the volume is calculated as "1.48585" cubic centimeters. It's crucial to ensure that all units are the same when performing these calculations, otherwise, the final measurement will be incorrect.

Density Formula

Density is a fundamental concept in physics and chemistry, representing the mass of an object divided by its volume. The formula for density is expressed as: \[ \rho = \frac{\text{mass}}{\text{volume}} \] Where \(\rho\) (rho) stands for density. It tells us how much mass is contained within a specific volume.

• The mass should be measured in grams (g).
• The volume should be in cubic centimeters (cm³) for consistency with the metric system.

In our example, lithium metal has a mass of "0.794 g" and a previously determined volume of "1.48585 cm³". By dividing the mass by the volume, we find that the density is approximately "0.534 g/cm³".

Significant Figures

Significant figures are critical in scientific measurements and calculations, as they convey the precision of a number. For this exercise, identifying the number of significant figures impacts how we record our final answer. In our calculation, each measurement was provided to three significant figures: "0.82 cm", "1.45 cm", and "1.25 cm". We employ the rule stating that the result must not be more precise than the least precise measurement used in the calculation. Since all input measurements had three significant figures, the final density, "0.534 g/cm³", is also rounded to three significant figures. Properly using significant figures ensures the accuracy and reliability of scientific results.

Measurement Precision

Measurement precision refers to the degree of exactness associated with a measured value. When you conduct experiments or solve problems involving calculations, it is important to consider how precise your measurements are.

• Precision affects the clarity and dependability of results.
• Significant figures are a tool to express precision.

In scientific measurements, being precise allows for consistent and replicable results. Using our exercise as an example, the dimensions of the cube were measured to two decimal places, suggesting an awareness of precision. This awareness guided the rounding step in the solution, making sure that the calculated density provided a meaningful representation of the data without suggesting false accuracy. By maintaining consistency with measurement precision, we ensure results accurately reflect the data gathered in experiments or problem-solving scenarios.