Question

If you had a mol of U.S. dollar bills and equally distributed the money to all of the people of the world, how rich would every person be? Assume a world population of 6 billion.

Short answer

Each person would be approximately \(1.0037 \times 10^{14}\) U.S. dollars rich.

Step-by-step solution

Calculate the total number of U.S. dollar bills in a mol

A mol of U.S. dollar bills is equal to Avogadro's number multiplied by the number of dollar bills. So, the total number of U.S. dollar bills can be calculated by: Total number of dollar bills = 1 mol × (6.022 x 10^23 dollar bills/mol)

Find the amount of money in a mol of U.S. dollar bills

Since each U.S. dollar bill is worth one U.S. dollar, the total amount of money in a mol is equal to the total number of dollar bills we calculated in step 1. Total amount of money = Total number of dollar bills = 6.022 x 10^23 U.S. dollars

Calculate the population share for each person

Now, we need to distribute the total amount of money equally among 6 billion (6 x 10^9) people. To do this, we will divide the total amount of money by the world population: Share of each person = (Total amount of money) / (World population) Share of each person = (6.022 x 10^23 U.S. dollars) / (6 x 10^9 people)

Simplify and evaluate

Simplify the expression by dividing the two numbers: \(Share \, of \, each \, person = \frac {6.022 \times 10^{23} \, U.S. \, dollars}{6 \times 10^9 \, people}\) Evaluating this gives: \(Share \, of \, each \, person ≈1.0037 \times 10^{14} \, U.S. \, dollars \, per \, person\) Thus, if a mol of U.S. dollar bills were distributed equally among the 6 billion people in the world, every person would be approximately 1.0037 x 10^14 U.S. dollars rich.

Key concepts

Understanding Avogadro's Number

Imagine having enough pieces of anything to make a very specific, huge number - that's where Avogadro's number comes into play in chemistry. Avogadro's number, approximately equal to \(6.022 \times 10^{23}\), represents the number of atoms, molecules, ions, or any other chemical entities in one mole of a substance. It's named after the scientist Amedeo Avogadro, and it's a fundamental unit in the mole concept.

Knowing Avogadro's number is like knowing the exact number of jellybeans that would fill a large jar to the brim - it's a very precise count. Just as you would expect the jar to contain the same number of jellybeans each time you fill it (assuming it's always the same size and shape), a mole of any substance always contains Avogadro's number of particles, regardless of what it is. This constancy makes stoichiometry and chemical calculations much simpler. If you visualize a mole of U.S. dollar bills as in the original exercise, we use Avogadro's number to count them, even though they're not atoms or molecules, but for educational purposes, the concept remains the same.

Furthermore, Avogadro's number is essential when converting between atomic-scale measurements and macro-scale quantities, which is a fundamental aspect of chemistry.

Demystifying Stoichiometry

Stoichiometry may sound complicated, but it is essentially the chemistry equivalent of a recipe. In cooking, a recipe outlines the quantities of each ingredient you need. Similarly, stoichiometry provides the proportions of reactants and products involved in chemical reactions.

In the demonstration using U.S. dollar bills, stoichiometry helped to relate the abstract idea of moles to a tangible quantity of money. By knowing that one mole equals Avogadro's number, we can determine the number of dollar bills in a mole and then, using stoichiometric calculations, divide this total amount evenly across the world's population.

Steps in Stoichiometric Calculations

• Identify the 'recipe', or the balanced chemical equation.
• Determine the mole ratio between the reactants and products.
• Use mole-to-mole conversion to relate the quantities.

With the example provided, we took the concept of a mole and applied it to an everyday concept, dividing a very large number by the world population to find out how much money each individual would receive. It's these kinds of stoichiometric calculations that reinforce the practical applications of chemistry in the world around us.

Making Sense of Unit Conversion

Unit conversion is akin to translating one language to another. It's converting a given value from one system of measurement to another, and it's a critical skill not only in chemistry but in everyday life. For example, when you convert kilometers to miles, you're using unit conversion.

In chemistry, especially when dealing with the mole concept, we often need to convert between units such as grams, moles, particles, and sometimes more creative units like dollar bills in the given exercise. This requires a clear understanding of the 'conversion factors'—the rates at which two different units equate. Here, the conversion factor is Avogadro's number, which relates moles to individual entities.

Basic Steps in Unit Conversion:

• Identify the unit you are starting with and the unit you are converting to.
• Find a conversion factor that relates the two units.
• Use the conversion factor to 'cancel out' units until you are left with the desired unit.

In the problem solution, the process took us from moles to number of dollar bills (using Avogadro's number as the conversion factor), and eventually to dollars per person by dividing by the world population. The ability to switch between units seamlessly is a powerful tool in understanding and solving chemistry problems and it also helps to explain scientific concepts in ways that are more relatable to our daily experiences, like understanding just how much money might be in a mole of dollar bills.