Problem 34 - Chemistry The Central Science In Si Units

Question

If Avogadro's number of pennies is divided equally among the 321 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 \(\$ 17.419\) trillion in \(2015 ?\) (The GDP is the total market value of the nation's goods and services.)

Step-by-Step Solution

Answer

Each person receives about \(1.876 \times 10^{13}\) dollars, which is larger than the GDP of the U.S. in 2015.

Step 1: Understand Avogadro's Number

Avogadro's number is commonly known as \(6.022 \times 10^{23}\), which represents the number of units (in this case, pennies) per mole.

Step 2: Calculate Total Pennies

Avogadro's number of pennies is \(6.022 \times 10^{23}\). Each penny is worth \(\$0.01\). Total value in dollars is calculated by multiplying \(6.022 \times 10^{23}\) by \(0.01\).

Step 3: Total Dollars from Pennies

Calculate the total value: \(6.022 \times 10^{23} \times 0.01 = 6.022 \times 10^{21}\) dollars.

Step 4: Divide Among U.S. Population

Divide the total amount of dollars by the population: \(\frac{6.022 \times 10^{21}}{321,000,000}\).

Step 5: Simplify the Division

This simplifies to approximately \(1.876 \times 10^{13}\) dollars per person.

Step 6: Compare with U.S. GDP

Compare each person's share (\(1.876 \times 10^{13}\) dollars) with the U.S. GDP (\(\$17.419\) trillion, or \(1.7419 \times 10^{13}\) dollars).

Key Concepts

Mole Concept

The mole concept is a fundamental principle in chemistry that helps us count and calculate particles such as atoms and molecules in a substance. It is similar to other counting terms we use in everyday life, like a dozen, which represents 12 of anything.
The mole, however, is used for counting very large numbers of tiny particles. One mole is defined as

• exactly 6.022 x 10²³ particles,
• which can be atoms, molecules, ions, or even pennies.

This fixed number is called Avogadro's number, named after the scientist who helped clarify the concept. In the given problem, we're applying the mole concept to estimate how much money it would represent if each of the 321 million people in the U.S. got an equal share of all the pennies in a mole. It’s a great way to grasp how vast Avogadro's number is, as it's large enough to astonishingly surpass the scope of everyday numerical quantities, even in financial contexts.

Unit Conversion

Unit conversion is a process of transforming a quantity expressed in one unit to an equivalent quantity in another unit, which is essential when solving various math, physics, and chemistry problems. In the context of our problem, we're converting pennies to dollars to make understanding easier and relate it to the GDP for a means of comparison.
This involves:

• Knowing the conversion factor: 1 penny is equal to 0.01 dollars.
• Using multiplication to find the total value in dollars. By multiplying Avogadro's number of pennies \[ 6.022 \times 10^{23} \times 0.01 = 6.022 \times 10^{21} \text{ dollars} \]

The calculation above illustrates a successful unit conversion by demonstrating how enormous quantities are managed using simple multiplication. This reinforces the importance of understanding and correctly applying unit conversions consistently to ensure accuracy in larger calculations.

Gross Domestic Product

The Gross Domestic Product (GDP) is a comprehensive measure of a nation's economic performance. It reflects the total market value of all goods and services produced within a country's borders in a given year. GDP is particularly important because it

• indicates economic health,
• aids in policy making decisions, and
• allows for comparisons between different economies.

In 2015, the United States' GDP was $17.419 trillion, which equates to 1.7419 x 10¹³ dollars.
When we divide the theoretical total dollar value from Avogadro's number of pennies (\[6.022 \times 10^{21}\] dollars) among the population, each person would get a substantial amount of roughly 1.876 x 10¹³ dollars.
This exercise highlights how mind-boggling Avogadro's number truly is when juxtaposed with significant real-world values like GDP, providing a unique perspective on understanding the scale of such abstract figures.