Question

Jodi, a sculptor, has prepared a mold for casting a bronze figure. The figure has a volume of \(225 \mathrm{~mL}\). If bronze has a density of \(7.8 \mathrm{~g} / \mathrm{mL}\) and Jodi likes to use \(90.0 \%\) of the bronze melted (she sometimes spills), how many ounces of bronze does Jodi need to melt in the preparation of the bronze figure?

Short answer

Jodi needs approximately 68.79 ounces of bronze.

Step-by-step solution

- Calculate the total mass of the bronze

First, use the volume of the figure and the density of the bronze to calculate the total mass required. The formula to find mass is: \[ \text{Mass} = \text{Volume} \times \text{Density} \] Given: \( \text{Volume} = 225 \, \text{mL} \) and \( \text{Density} = 7.8 \, \text{g/mL} \), \[ \text{Mass} = 225 \, \text{mL} \times 7.8 \, \text{g/mL} = 1755 \, \text{g} \]

- Adjust for spillage

Since Jodi likes to use 90.0% of the bronze melted, calculate the total amount of bronze she needs to melt. This can be found using the formula: \[ \text{Total Mass Required} = \frac{\text{Mass}}{\text{Utilization Rate}} \] Given: \( \text{Utilization Rate} = 90.0\% = 0.90 \) \[ \text{Total Mass Required} = \frac{1755 \, \text{g}}{0.90} = 1950 \, \text{g} \]

- Convert grams to ounces

Convert the total mass of bronze needed from grams to ounces. The conversion factor is: \[ 1 \, \text{ounce} = 28.3495 \, \text{grams} \] Thus, \[ \text{Total Mass Required in Ounces} = \frac{1950 \, \text{g}}{28.3495 \, \text{g/ounce}} \approx 68.79 \, \text{ounces} \]

Key concepts

Volume to Mass Conversion

To solve the given exercise, the first step involves converting the volume of the bronze figure to its mass.
This type of calculation is crucial in chemistry when dealing with materials and their properties.
We start by using the given volume of the figure, which is 225 mL, and the density of bronze, listed as 7.8 g/mL.
The formula to convert volume to mass is: \[ \text{Mass} = \text{Volume} \times \text{Density} \] By plugging in the values, we get:
\[ \text{Mass} = 225 \text{ mL} \times 7.8 \text{ g/mL} = 1755 \text{ g} \] This gives us the total mass of bronze required to fill the mold assuming there is no spillage.

Density Formula

Understanding the density formula is essential for performing volume to mass conversions.
Density is defined as mass per unit volume and is typically expressed in units such as g/mL or g/cm³.
The density formula is given by:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] By rearranging this formula, we obtain the expression for mass:
\[ \text{Mass} = \text{Volume} \times \text{Density} \] This is the formula we used in the previous section to calculate the mass of bronze from its volume and known density.
Density values are often provided in problems or can be found in reference materials and they play a key role in relating volume to mass.

Unit Conversion

Unit conversion is a vital skill in chemistry, especially when dealing with practical problems.
In the given exercise, after computing the total mass of bronze required to account for spillage, we must convert the mass from grams to ounces.
The conversion factor between grams and ounces is: \[ 1 \text{ ounce} = 28.3495 \text{ grams} \] To find the mass in ounces, we use:
\[ \text{Total Mass Required in Ounces} = \frac{1950 \text{ g}}{28.3495 \text{ g/ounce}} \] \[ \text{Total Mass Required in Ounces} \thickapprox 68.79 \text{ ounces} \] Learning and remembering common conversion factors, and understanding how to apply them, is critical for solving real-world problems.
It ensures accurate measurement and proper material usage.
Estimating conversions can help spot potential calculation errors.