Problem 85 - Chemistry

Question

The density of silver is \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\). What is the mass (in kilograms) of a cube of silver that measures \(0.62 \mathrm{~m}\) on each side?

Step-by-Step Solution

Answer

The mass of the silver cube is 2502.444 kg.

Step 1: Calculate the Volume of the Cube

The side length of the cube is given as \(0.62 \text{ m}\). To find the volume of the cube, use the formula for the volume of a cube, \( V = a^3 \), where \( a \) is the side length. Therefore, the volume \( V \) is: \[V = (0.62)^3 = 0.238328 \text{ m}^3\]

Step 2: Convert Volume to Cubic Centimeters

Since the density is given in grams per cubic centimeter, we need to convert the volume from cubic meters to cubic centimeters. Use the conversion factor: \(1 \text{ m}^3 = 1,000,000 \text{ cm}^3\). Thus, \[V = 0.238328 \text{ m}^3 \times 1,000,000 \text{ cm}^3/\text{m}^3 = 238328 \text{ cm}^3\]

Step 3: Calculate the Mass of the Silver

Using the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), rearrange it to find Mass: \( \text{Mass} = \text{Density} \times \text{Volume} \). Plug in the values: \[\text{Mass} = 10.5 \text{ g/cm}^3 \times 238328 \text{ cm}^3 = 2502444 \text{ g}\]

Step 4: Convert Mass to Kilograms

Since the problem asks for the mass in kilograms, convert grams to kilograms using the conversion factor \(1 \text{ kg} = 1000 \text{ g}\). Therefore, \[\text{Mass} = \frac{2502444 \text{ g}}{1000} = 2502.444 \text{ kg} \]

Key Concepts

Volume Conversion

Volume conversion is critical when dealing with measurements in science and math, especially if different units are used for inputs and outputs. This can often be seen when density is involved, as densities are commonly expressed in one set of units (like grams per cubic centimeter) while measurements might be recorded in another set. To convert from cubic meters to cubic centimeters, you need to know the basic conversion factor:

• 1 meter (m) equals 100 centimeters (cm), but when you're dealing with cubic dimensions, you must cube this factor.
• Thus, 1 cubic meter (m³) is equal to 1,000,000 cubic centimeters (cm³). This conversion is crucial when given measurements in meters but needed for calculations in centimeters.

For instance, if the volume of a cube is 0.238328 cubic meters and you need to convert it to cubic centimeters, you multiply by 1,000,000, resulting in 238328 cm³. Thus, always ensure your volume unit aligns with the unit used in the density formula or other relevant measurements.

Cube Volume Calculation

The volume of a cube calculation is one of the simplest, thanks to the cube's uniform structure. Here's how it works:The formula for finding the volume of a cube is straightforward:

• It is given by \[V = a^3\]where \(a\) is the length of one side of the cube.
• The result is the cube's volume in cubic units.

For instance, if each side of a cube is 0.62 meters long, you would calculate the volume by cubing the side length:\[ V = (0.62 \, \text{m})^3 = 0.238328 \, \text{m}^3 \]Always ensure that the units are consistent when performing calculations. This keeps the mathematical relationships accurate and helps in obtaining the correct volume.

Density Formula

Density is a key concept that links mass and volume together. This is essential in determining the mass of an object if you know its volume and material characteristics.The formula for density is:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]This can be rearranged to solve for mass when density and volume are known:

• \(\text{Mass} = \text{Density} \times \text{Volume}\)
• Make sure that the units for volume match the units required for density.
• Typically, density is given in grams per cubic centimeter (g/cm³) for small objects and deals with grams and cubic centimeters, but might involve other units like kilograms and cubic meters.

For instance, if a cube of silver has a volume of 238328 cm³ and the density of silver is 10.5 g/cm³, the mass of the silver is:\[ \text{Mass} = 10.5 \, \text{g/cm}^3 \times 238328 \, \text{cm}^3 = 2502444 \, \text{g} \]Always remember to convert units if needed—for example, from grams to kilograms when asked—using conversion factors, such as 1 kg = 1000 g. This helps ensure accurate and uniform results in your calculations.