Problem 34 - Chemistry The Central Science

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 would receive approximately \(1.875 \times 10^{15}\) dollars if Avogadro's number of pennies were divided equally among the 321 million people in the United States. This is approximately 107.69 times the GDP of the United States in 2015, which was \(\$17.419\) trillion.

Step 1: Use Avogadro's Number

Avogadro's number is a scientific constant representing the number of units (atoms, ions, molecules, etc.) in one mole of a substance. Avogadro's number is approximately \(6.022 \times 10^{23}\). In this exercise, we'll use this number to represent the total number of pennies.

Step 2: Divide Pennies Among the Population

We need to divide the Avogadro's number of pennies equally among the 321 million people in the United States. To accomplish this, we will divide Avogadro's number (total number of pennies) by the population size. \[ \frac{6.022 \times 10^{23} \mathrm{\,pennies}}{321 \times 10^6 \mathrm{\,people}} \]

Step 3: Calculate Number of Pennies per Person

Perform the division to find out how many pennies each person would receive: \[ \frac{6.022 \times 10^{23}}{321 \times 10^6} = 1.875 \times 10^{17} \mathrm{\,pennies\,per\,person} \]

Step 4: Convert Pennies to Dollars

Now, we will convert the number of pennies each person receives to dollars, using the conversion factor that 100 pennies is equal to 1 dollar. \[ 1.875 \times 10^{17} \mathrm{\,pennies\,per\,person} \times \frac{1\mathrm{\,dollar}}{100 \mathrm{\,pennies}} = 1.875 \times 10^{15} \mathrm{\,dollars\,per\,person} \]

Step 5: Compare to Gross Domestic Product (GDP) of the United States

The GDP of the United States in 2015 was \(\$17.419\) trillion which is equal to \(\$17.419 \times 10^{12}\). To compare this with the amount each person would receive, we can divide the dollars per person by the US GDP: \[ \frac{1.875 \times 10^{15}}{17.419 \times 10^{12}} \approx 107.69 \] This means that each person would receive approximately 107.69 times the GDP of the United States in 2015 if Avogadro's number of pennies were divided equally among the population.

Key Concepts

Mole Concept

The mole concept is a fundamental part of chemistry that helps us bridge the microscopic world of atoms and molecules with the macroscopic world we observe. It introduces the term "mole" as a count of units, like atoms or molecules, similar to a "dozen" representing twelve items. This is useful because atoms and molecules are so small that we need an immense number to have a significant amount we can use. One mole is defined as containing exactly Avogadro's number of units, which is approximately \(6.022 \times 10^{23}\).

• A mole helps chemists calculate how much of a substance will react or is produced in a chemical reaction.
• Avogadro's number is a large number because it relates to particles, which are incredibly tiny.
• By using the mole concept, we can convert measurable amounts of substances from grams to moles and vice versa, which simplifies many chemical calculations.

Understanding the mole and Avogadro's number is pivotal for scientific calculations, especially when talking about reactions in chemistry where actual molecular counts are required.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It's especially useful in science where such numbers frequently occur. Instead of writing out all the zeros, we express the number as a product of two parts: a coefficient and a power of ten. For instance, Avogadro's number is written as \(6.022 \times 10^{23}\) to avoid the cumbersome approach of writing out 602 followed by 21 zeros. This notation makes calculations more manageable within scientific and mathematical work.

• Scientific notation simplifies operations with large or tiny numbers, such as addition, subtraction, multiplication, and division.
• It helps compare numbers by focusing on their scale (large or small) via exponents.
• Understanding scientific notation is crucial for disciplines like physics, chemistry, and computer science.

Learning to use scientific notation can also enhance day-to-day tasks like finance or engineering, where precision and accuracy are paramount.

Gross Domestic Product (GDP)

The gross domestic product (GDP) represents the total monetary value of all finished goods and services produced within a country's borders in a specific time period, typically calculated annually. It's a comprehensive measure of a nation's economic activity and trends, widely used by economists to assess the health of an economy and make comparisons over time or between different countries. By illustrating the economic output, GDP helps indicate whether a country is economically strong or weak.

• GDP can be measured in several ways, including production, income, and expenditure approaches.
• It includes consumption, government spending, investments, and net exports.
• A rising GDP signifies economic growth, while a falling GDP indicates economic slowdown.

Understanding GDP is essential for interpreting economic data and making informed decisions, whether on a policy level, for businesses, or individuals seeking insights into economic patterns.