Question

There are about \(7.0\) billion \(\left(7.0 \times 10^{9}\right)\) people on Earth. If exactly \(1 \mathrm{~mol}\) of dollars were distributed equally among these people, how many dollars would each person receive?

Short answer

Each person would receive \(8.6 \times 10^{13}\) dollars.

Step-by-step solution

Understand the Problem

Determine how many dollars each person would receive if one mole of dollars were distributed among 7.0 billion people.

Recall Avogadro's Number

A mole is defined as exactly \(6.022 \times 10^{23}\) units. In this case, one mole of dollars means \(6.022 \times 10^{23}\) dollars.

Set Up the Division Problem

Divide the total number of dollars (one mole, \(6.022 \times 10^{23}\) dollars) by the total number of people (7.0 billion or \(7.0 \times 10^{9}\) people): \[ \frac{6.022 \times 10^{23}}{7.0 \times 10^{9}} \].

Perform the Division

Calculate the result of the division: \[ \frac{6.022 \times 10^{23}}{7.0 \times 10^{9}} = 8.6 \times 10^{13} \].

Interpret the Result

Each person would receive \(8.6 \times 10^{13}\) dollars if one mole of dollars were distributed equally among 7.0 billion people.

Key concepts

Avogadro's Number

Avogadro's Number is a key concept in chemistry. It is defined as the number of units (such as atoms, molecules, or dollars) in one mole of a substance. This number, denoted as \(6.022 \times 10^{23}\), is incredibly large. Understanding Avogadro's Number helps in grasping concepts involving very large quantities, which are common in chemistry and many real-world applications.
When dealing with such large numbers, it's useful to know how to handle and manipulate them effectively, especially when they are used in practical calculations like our exercise with dollars and people.

Division of Large Numbers

Dividing large numbers can seem challenging, but it becomes manageable with the right steps. In our exercise, we need to divide \(6.022 \times 10^{23}\) by \(7.0 \times 10^{9}\). Here’s the step-by-step method:

• First, write both numbers in scientific notation.
• Next, divide the coefficients (the numbers in front of the powers of 10). This means dividing 6.022 by 7.0.
• Then, subtract the exponents: 23 (from \(10^{23}\)) minus 9 (from \(10^{9}\)) which gives 14.
• Combine the result of the coefficient division with the new power of 10 exponent.

The final result is \(8.6 \times 10^{13}\). Each person receives \(8.6 \times 10^{13}\) dollars if one mole of dollars is divided among 7 billion people.

Scientific Notation

Scientific notation is a way to express very large or very small numbers compactly. It allows easy manipulation and understanding of these numbers in a manageable form. For example, in our exercise, instead of writing 7.0 billion as 7,000,000,000, we write it as \(7.0 \times 10^{9}\). Likewise, a mole of anything, which is 602,200,000,000,000,000,000,000, is more neatly written as \(6.022 \times 10^{23}\).
This technique simplifies calculations and helps avoid errors. When multiplying and dividing numbers in scientific notation, you simply handle the coefficients and exponents separately. This reduces the risk of mistakes and makes understanding the magnitude of these numbers much easier.
Remember, manipulating exponents can be summarized as:

• Multiply: Add the exponents.
• Divide: Subtract the exponents.

This skill is fundamental in chemistry and physics and is widely used in scientific calculations.