Question

If Avogadro's number of pennies is divided equally among the 300 million men, women, and children in the United States, how many dollars would each receive? How does this compare with the gross domestic product (GDP) of the United States, which was $\mathrm{\$}14.4$ trillion in $2008?$ (The GDP is the total market value of the nation's goods and services.)

Short answer

If Avogadro's number of pennies were divided equally among the 300 million people in the United States, each person would receive $2.007\times {10}^{13}$ dollars. This is approximately equal to 0.0139 times the GDP of the United States in 2008, which was $14.4$ trillion.

Step-by-step solution

Calculate the total amount of pennies each person would receive

Avogadro's number is approximately equal to 6.022 × 10^23. Since there are 300 million people in the United States, we can calculate the total amount of pennies each person would receive by dividing Avogadro's number by 300 million. Let's divide Avogadro's number by 300 million: $\frac{6.022\times {10}^{23}}{3\times {10}^8}$.

Convert the pennies to dollars

Since there are 100 pennies in a dollar, we can convert the total amount of pennies each person would receive to dollars by dividing by 100. Let's divide the result from step 1 by 100: $\frac{6.022\times {10}^{23}}{3\times {10}^8\times {10}^2}$.

Compare the resulting number with the GDP of the United States in 2008

The GDP of the United States in 2008 was $14.4trillion,whichisequalto$14.4 × 10^12. To compare the dollars per person calculated in step 2 with the GDP, we can divide the dollars per person by the GDP of the United States in 2008. Let's divide the result from step 2 by the GDP of the United States in 2008: $\frac{6.022\times {10}^{23}}{3\times {10}^8\times {10}^2\times 14.4\times {10}^{12}}$. After calculating the values for all steps, we have: 1. Total amount of pennies each person would receive: $\frac{6.022\times {10}^{23}}{3\times {10}^8}=2.007\times {10}^{15}$ pennies. 2. Convert the pennies to dollars: $\frac{6.022\times {10}^{23}}{3\times {10}^8\times {10}^2}=\mathrm{\$}2.007\times {10}^{13}$ per person. 3. Compare the resulting number with the GDP of the United States in 2008: $\frac{6.022\times {10}^{23}}{3\times {10}^8\times {10}^2\times 14.4\times {10}^{12}}=1.39\times {10}^{-2}$, which means that each person would have received 0.0139 times the GDP of the United States in 2008. If Avogadro's number of pennies was divided equally among the 300 million men, women, and children in the United States, each person would have received $2.007\times {10}^{13}$ dollars. This is approximately equal to 0.0139 times the GDP of the United States in 2008, which was $14.4$ trillion.

Key concepts

Chemical Calculations

Understanding chemical calculations is critical when dealing with large quantities of substances, as in the field of chemistry. These calculations often use Avogadro's number, which is approximately equal to 6.022 x 10^23. This constant represents the number of particles, like atoms or molecules, in one mole of a substance. In our exercise, Avogadro's number represents a hypothetical situation where it equals the number of pennies.
To contextualize this vast quantity, imagine distributing these pennies evenly among the population of the United States. The sheer volume of calculations required to transition from the microscopic scale of atoms to the macroscopic scale of pennies showcases the versatility and necessity of chemical calculations in different scenarios. Chemical calculations are not just academic exercises but also form the basis for understanding the quantitative aspects of the physical world around us.

Mole Concept

The mole concept is a bridge between the atomic world and the world we experience every day. It allows chemists to count particles as groups or 'moles,' a fundamental skill that aids in various types of calculations. One mole is defined as Avogadro's number of particles, whether those are atoms, ions, molecules, or—as in the quirky textbook question—pennies.
When calculations are done using the mole concept, they typically involve converting between mass, number of particles, and volume in the case of gases. In our exercise, by assigning Avogadro's number to a quantity of pennies, we apply the mole concept at a scale that's massive and easier to picture. While pennies and atoms differ greatly in size, the principle remains the same and shows the immense power behind the concept of the mole in comparing entities of vastly different magnitudes.

Economics in Chemistry

Economics in chemistry might not be immediately obvious, but the principles of managing resources and analyzing costs versus benefits are certainly applicable. In the exercise, when comparing the hypothetical distribution of wealth (in the form of Avogadro's number of pennies) to an actual economic indicator like the GDP, we integrate economics into a chemistry context.
This comparison provides a vivid illustration of economic principles by contrasting individual wealth to the total economic output of a country. The result of such a comparison could spark a conversation about wealth distribution and economic policies, showing that even in the seemingly abstract world of chemistry, there can be correlations with economic considerations that have a significant impact on society.

Quantitative Problem Solving

Quantitative problem solving involves applying mathematical methods to solve problems with numerical data. In chemistry, this might involve conversions between units or scales, calculations involving chemical equations, or determining substance quantities. From our exercise, we employ quantitative problem-solving methods to translate a simple scientific constant, Avogadro's number, into a meaningful financial figure per person.
By systematically breaking down the problem into smaller, more manageable steps, we can reach a comprehensible solution that tells us not only about chemistry but also its implications in real-life scenarios. This step-by-step approach is at the heart of quantitative problem solving, which, when applied correctly, can provide insights into complex issues across various disciplines.