Question

U.S. quarters are composed of \(91.67 \%\) copper and \(8.33 \%\) nickel. The mass of a quarter is \(5.670 \mathrm{~g}\). (a) Write out the dimensional analysis factors that can be derived from the two percentages given above. Be sure to include units. (b) If the U.S. mint buys \(3.0\) tons of copper for making quarters, how many quarters can it make?

Short answer

523883 quarters can be made from 3.0 tons of copper.

Step-by-step solution

Determine Mass of Copper in Each Quarter

The percentage of copper in a quarter is given as 91.67%. To find the mass of copper per quarter, multiply the total mass of a quarter by this percentage. Mass of copper in each quarter (in grams) is calculated as follows: Material mass of copper per quarter = 0.9167 * 5.670g = 5.1977g.

Convert Tons of Copper to Grams

Convert the given amount of copper (3.0 tons) to grams. Use the conversion factor between tons and grams where 1 ton = 907184.74 grams. Amount of copper in grams = 3.0 tons * 907184.74 g/ton = 2721554.22g.

Calculate the Number of Quarters that can be Made

Divide the total amount of copper available by the mass of copper per quarter to find out how many quarters can be made. Number of quarters = 2721554.22g / 5.1977g/quarter = 523883.07 quarters.

Key concepts

Dimensional Analysis

Dimensional analysis is a method in chemistry that helps ensure units are consistent and calculations are correct. It uses conversion factors to switch between different units.
For instance, when dealing with the mass of copper in quarters, a percentage of 91.67% is given. This can be used as a conversion factor to find out the mass of copper in a single quarter.
Here, the factor 0.9167 is derived from 91.67/100. To find the mass of copper in each quarter, you multiply:
Mass of copper in each quarter = mass of quarter x 0.9167 = 5.670g x 0.9167 = 5.1977g.
Dimensional analysis ensures each step has the correct units and the calculations flow logically.

Percent Composition

Percent composition tells you how much of each element is present in a compound or material by percentage. In this case, the quarter is composed of 91.67% copper and 8.33% nickel.
To find the mass of each component, multiply the total mass by the percentage of the component as a decimal.
For copper in quarters: 91.67% translates to 91.67/100 = 0.9167. Multiply this with the mass of a quarter:
mass of copper per quarter = 0.9167 x 5.670g = 5.1977g.
Similarly, for nickel: 8.33% translates to 8.33/100 = 0.0833. Multiply this with the mass of a quarter:
mass of nickel per quarter = 0.0833 x 5.670g = 0.4723g.

Mass Conversion

Mass conversion is essential when switching between different units like tons, kilograms, and grams.
Often, we deal with large quantities. For instance, converting tons to grams requires a specific conversion factor.
In the example above, the U.S. mint buys 3.0 tons of copper. Knowing that 1 ton equals 907184.74 grams helps convert this large unit into a more manageable one:
Amount of copper in grams = 3.0 tons x 907184.74 g/ton = 2721554.22g.
Converting between units ensures calculations are accurate and consistent throughout the problem.

Unit Factor Method

The unit factor method, also known as dimensional analysis or factor-label method, uses conversion factors to move from one unit to another.
Conversion factors are ratios that equal 1, making it easy to switch units without changing the value.
For our example, to calculate how many quarters can be made from 3.0 tons of copper:
First, determine the mass of copper in grams: 3.0 tons x 907184.74 g/ton.
Then, divide by the mass of copper in each quarter:
Number of quarters = total grams of copper/mass of copper per quarter.
Number of quarters = 2721554.22g / 5.1977g/quarter = 523883.07 quarters.
This method ensures that you're always keeping track of the units, leading to accurate results.