Problem 89 - Chemistry Principles and Practice

Question

Lead has a density of \(11.4 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the mass, in kilograms, of a lead brick measuring \(8.50 \times 5.10 \times 3.20 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

The mass of the lead brick is approximately 1.58 kg.

Step 1: Find the Volume of the Brick

Calculate the volume of the lead brick using the formula for volume of a rectangular prism, \( V = \text{length} \times \text{width} \times \text{height} \). For the given brick, this becomes \( V = 8.50 \text{ cm} \times 5.10 \text{ cm} \times 3.20 \text{ cm} \).

Step 2: Perform the Multiplication for the Volume

Multiply the given dimensions: \( 8.50 \times 5.10 \times 3.20 \). The calculation gives \( V = 138.72 \text{ cm}^3 \).

Step 3: Calculate the Mass Using the Density

Use the relationship between mass, volume, and density: \( m = \rho \times V \), where \( m \) is the mass, \( \rho \) is the density, and \( V \) is the volume. Given \( \rho = 11.4 \text{ g/cm}^3 \) and \( V = 138.72 \text{ cm}^3 \), calculate the mass: \( m = 11.4 \times 138.72 \).

Step 4: Perform the Calculation for Mass

Calculate \( m = 11.4 \times 138.72 \) to get \( m = 1582.408 \text{ g} \).

Step 5: Convert Mass from Grams to Kilograms

Convert the mass from grams to kilograms since 1 kilogram equals 1000 grams. Therefore, \( m = 1582.408 \text{ g} = 1.582408 \text{ kg} \).

Step 6: Round the Final Mass

Round the mass to a reasonable number of significant figures, given the precision of the measurements, typically here it's three significant figures: \( m \approx 1.58 \text{ kg} \).

Key Concepts

Mass and Volume

Understanding mass and volume is essential when dealing with problems involving density. Mass is the measure of the amount of matter in an object, typically measured in grams (g) or kilograms (kg). Volume, on the other hand, is the amount of space that an object occupies, usually measured in cubic centimeters (\( ext{cm}^3\),) or liters (L).
When calculating the mass of a solid object like a lead brick, knowing its volume is necessary. Volume can be easily determined if you know the object's dimensions. For a rectangular prism, like our brick, volume is calculated as length × width × height. In our case, the brick's volume is \(8.50 ext{ cm} imes 5.10 ext{ cm} imes 3.20 ext{ cm} = 138.72 ext{ cm}^3\), which is a straightforward multiplication.
Once the volume is known, mass can be calculated using the object's density. Density is defined as mass divided by volume and provides a measure of how much matter is packed into a given space. For lead, a typical density is \(11.4 ext{ g/cm}^3\).

Unit Conversion

Unit conversion is a vital skill in science and everyday life. It allows values to be changed from one unit of measure to another, ensuring that the numbers can be practically and correctly interpreted.
In our exercise, the mass of the lead brick was initially calculated in grams but was required in kilograms for the final solution. Since 1 kilogram is equal to 1000 grams, converting from grams to kilograms involves dividing the mass in grams by 1000.
Thus, the calculated mass \(1582.408 \text{ g}\) is converted to kilograms by calculating \(\frac{1582.408}{1000} = 1.582408 \text{ kg}\). This conversion simplifies the way we express the mass in a larger unit, suitable for contextual understanding.

Significant Figures

Significant figures are crucial in scientific calculations because they help communicate the precision of the fields measured or calculated. The number of significant figures in a measurement includes all the certain digits and the first uncertain digit.
In our brick problem, the given dimensions are provided to three significant figures. Therefore, after completing all calculations, the final mass value should also reflect this level of precision. Hence, the mass is rounded appropriately.
Initially, the calculated mass was \(1.582408 \text{ kg}\). For consistency and accuracy based on the given data, it makes sense to round this number to three significant figures, resulting in \(1.58 \text{ kg}\). This ensures that the answer is as precise as the least precise measurement, following standard significant figures rules.