Question

A sunscreen preparation contains \(2.50 \%\) by mass benzyl salicylate. If a tube contains \(4.0\) ounces of sunscreen, how many kilograms of benzyl salicylate are needed to manufacture 325 tubes of sunscreen?

Short answer

0.921859 kilograms

Step-by-step solution

- Convert ounces to grams

There are 28.3495 grams in one ounce. Convert the total mass of one tube of sunscreen from ounces to grams by multiplying the number of ounces by 28.3495.\[4.0 \text{ ounces} \times 28.3495 \text{ grams/ounce} = 113.398 \text{ grams}\]

- Calculate mass of benzyl salicylate in one tube

The sunscreen contains 2.50% by mass benzyl salicylate. Multiply the total mass of one tube of sunscreen by 0.025 (2.50%) to find the mass of benzyl salicylate in one tube.\[113.398 \text{ grams} \times 0.025 = 2.83495 \text{ grams}\]

- Calculate total mass of benzyl salicylate for 325 tubes

Multiply the mass of benzyl salicylate in one tube by the total number of tubes (325).\[2.83495 \text{ grams/tube} \times 325 \text{ tubes} = 921.85875 \text{ grams}\]

- Convert grams to kilograms

There are 1000 grams in one kilogram. Convert the total mass of benzyl salicylate from grams to kilograms by dividing by 1000.\[921.85875 \text{ grams} \times \frac{1 \text{ kilogram}}{1000 \text{ grams}} = 0.921859 \text{ kilograms} \]

Key concepts

percent by mass

Percent by mass is a way of expressing the concentration of a substance in a mixture. It is calculated by taking the mass of the substance and dividing it by the total mass of the mixture, then multiplying by 100 to get the percentage.
For example, in the sunscreen exercise, the sunscreen contains 2.50% benzyl salicylate by mass. That means there are 2.50 grams of benzyl salicylate for every 100 grams of sunscreen.
To find the mass of benzyl salicylate in any quantity of sunscreen, multiply the total mass of the sunscreen by the percentage (in decimal form) of benzyl salicylate. For instance, if we have 113.398 grams of sunscreen, the mass of benzyl salicylate is calculated as follows: \[113.398\text{ g} \times 0.025 = 2.83495\text{ g}\]

unit conversion

Unit conversion is a crucial skill in chemistry to switch from one unit of measurement to another. This often involves using conversion factors.
In our exercise, we need to convert ounces to grams and then grams to kilograms.
Here’s how we do it:

• First, know the conversion factor: There are 28.3495 grams in one ounce.
• Multiply the number of ounces by this factor: \[4.0 \text{ ounces} \times 28.3495 \text{ grams/ounce} = 113.398 \text{ grams}\]
  For converting grams to kilograms, the conversion factor is 1000 grams in one kilogram. So, we divide the number of grams by 1000:
  \[921.85875 \text{ grams} \times \frac{1 \text{ kilogram}}{1000 \text{ grams}} = 0.921859 \text{ kilograms}\]
  These conversions ensure our final answer is in the correct units.

mass calculation

Calculating the mass involves multiplying or dividing quantities to find the desired amount. In our sunscreen exercise, we perform several mass calculations:

• First, we convert the mass of one tube from ounces to grams as shown above.
• Next, we calculate the mass of benzyl salicylate in one tube:
  \[113.398 \text{ grams} \times 0.025 = 2.83495 \text{ grams}\]
• Then, we find the total mass of benzyl salicylate needed for 325 tubes:
  \[2.83495 \text{ grams/tube} \times 325 \text{ tubes} = 921.85875 \text{ grams}\]

Finally, we convert that mass from grams to kilograms to get our final answer:
\[921.85875 \text{ grams} \times \frac{1 \text{ kilogram}}{1000 \text{ grams}} = 0.921859 \text{ kilograms}\]
This step-by-step approach helps ensure we correctly calculate the required mass for manufacturing many tubes of sunscreen.