Problem 32 - Chemistry: The Central Science

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 of the United States, which was \(\$ 13.5\) trillion in 2006 ? (The GDP is the total market value of the nation's goods and services.)

Step-by-Step Solution

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. In comparison, this amount is \(1.487\) times greater than the GDP of the United States in 2006, which was \(\$13.5 \times 10^{12}\).

Step 1: Find the number of pennies per person

Divide Avogadro's number by the population of the United States: \[ \frac{6.022\times 10^{23}}{3\times 10^{8}} = 2.007\times 10^{15} \] So, each person would receive \(2.007\times 10^{15}\) pennies.

Step 2: Convert pennies to dollars

Convert the number of pennies per person into dollars by dividing the number of pennies by 100: \[ \frac{2.007\times 10^{15}}{100}=2.007\times 10^{13} \] So, each person would receive \(2.007\times 10^{13}\) dollars.

Step 3: Compare the value with the GDP of the United States in 2006

The GDP of the United States in 2006 was \(\$13.5\times 10^{12}\). To compare it with the value we just found, let's first find out how many times larger the value is than the GDP. Divide the value of dollars each person would receive by the GDP of 2006: \[ \frac{2.007\times 10^{13}}{13.5\times 10^{12}} = 1.487 \] Now we can say that if Avogadro's number of pennies were divided equally among the population of the United States, each person would receive an amount equivalent to \(1.487\) times the GDP of the United States in 2006.

Key Concepts

Pennies to Dollars Conversion

When dealing with large numbers, like Avogadro's number, converting units can be tricky but also useful. Avogadro's number, which is approximately \(6.022 \times 10^{23}\), represents a vast quantity. It is famously used to express the number of atoms or molecules in a mole of a substance.
In this exercise, we start by contemplating what would happen if you had Avogadro's number of pennies. To make this comprehensible in terms of everyday currency, we convert pennies to dollars.

• There are 100 pennies in a dollar.
• To convert from pennies to dollars, you divide by 100.

This means that if you have \(2.007 \times 10^{15}\) pennies, to find out how much that's worth in dollars, you compute \(\frac{2.007 \times 10^{15}}{100}\), resulting in \(2.007 \times 10^{13}\) dollars.

Gross Domestic Product (GDP)

GDP stands for Gross Domestic Product, a key metric in determining the economic performance of a country. It accounts for the total market value of all the goods and services produced in a country over a specified period, usually annually.
In 2006, the GDP of the United States was \(13.5\) trillion dollars, or \(13.5 \times 10^{12}\) dollars. This figure serves as a comparator for the wealth from Avogadro's number of pennies.

• GDP provides insight into the economic health of a country.
• It is an indicator of the standard of living and a way to compare economic activity between different periods or countries.

In the exercise, dividing the value from pennies by the GDP gives an insight into the magnitude of such a vast number of pennies.

Population of the United States

Understanding the population figures is crucial when dividing large sums. The exercise assumes a population of 300 million people in the United States. These figures help in visualizing how resources or amounts of money would be distributed among individuals.

• The population of a country can impact economic indicators like GDP per capita, showing wealth distribution.
• Dividing total amounts by population can provide a more relatable per-person value.

In our scenario, sharing Avogadro’s number of pennies among 300 million people would each person end up with an impressive \(2.007 \times 10^{13}\) dollars, which highlights the concept of sharing and distributing vast resources.