Problem 118 - Introductory Chemistry Atoms First

Question

When you bring your piggy bank of pennies into the bank, instead of counting them individually, the teller measures the mass of a single penny on a scale and then measures the mass of the entire contents of your piggy bank. If a single penny has a mass of \(2.49 \mathrm{~g}\) and your entire penny collection has a mass of \(5789.25 \mathrm{~g}\), how much money in dollars do you receive?

Step-by-Step Solution

Answer

You receive $23.24 in cash for your entire penny collection.

Step 1: Calculate the number of pennies in the piggy bank

To find the number of pennies in the piggy bank, we need to divide the total mass of the penny collection (\(5789.25 \mathrm{~g}\)) by the mass of one penny (\(2.49 \mathrm{~g}\)): Number of pennies = \(\frac{Total \; mass \; of \; the \; collection}{Mass \; of \; one \; penny}\)

Step 2: Plug the values into the equation

Now, let's plug the given values into the equation: Number of pennies = \(\frac{5789.25 \mathrm{~g}}{2.49 \mathrm{~g}}\)

Step 3: Divide the masses

Now, we just need to divide these masses: Number of pennies = \(\frac{5789.25}{2.49}\) = 2324 So, there are 2324 pennies in the piggy bank.

Step 4: Convert the number of pennies into dollars

To convert the number of pennies into dollars, we need to divide the number of pennies by 100 (since there are 100 pennies in a dollar): Amount in dollars = \(\frac{Number \; of \; pennies}{100}\) Amount in dollars = \(\frac{2324}{100}\) = $23.24 Thus, you receive $23.24 in cash for your entire penny collection.

Key Concepts

Understanding the Mole Concept

The mole concept is a fundamental pillar of chemistry that provides a bridge between the microscopic world of atoms and molecules and the macroscopic world of grams and kilograms that we can observe. In simple terms, a mole is a unit of measurement for amount of substance.

The number of entities in a mole, known as Avogadro's number, is approximately \(6.022 \times 10^{23}\), and this huge number provides a means to count particles by weighing them. When we say we have a mole of a substance, we are talking about a fixed number of particles, whether those particles are atoms, molecules, ions, or electrons, which is analogous to a dozen meaning twelve of something.

Converting Mass to Moles

The process of converting mass to moles is an essential skill in chemical calculations. It starts by understanding the molar mass of a substance, which is the mass (in grams) of one mole of particles of the substance. To convert the mass of a substance to moles, the formula used is:

\[\text{Number of moles (mol)} = \frac{\text{Mass of the substance (g)}}{\text{Molar mass of the substance (g/mol)}}\]

Where the molar mass is a substance-specific value usually found on the periodic table for elements and calculated by summing the molar masses of each element in a compound for molecules.

Mass-to-Mole Conversion in Action

Performing a mass-to-mole conversion is straightforward when you've grasped the mole concept and molar mass. Suppose we want to convert 18 grams of water to moles. The molar mass of water (\(H_2O\)) is approximately 18 g/mol, so:

\[\text{Number of moles of water} = \frac{18 \text{ g}}{18 \text{ g/mol}} = 1 \text{ mole}\]

This tells us that 18 grams of water is equivalent to 1 mole of water molecules. Similar steps are taken whenever converting mass to moles for any substance, making it a universal approach in chemical calculations.

Chemical Calculations Simplified

Chemical calculations often involve conversions and the use of balanced chemical equations. Understanding the principles of stoichiometry, which is the quantitative relationship between reactants and products in a chemical reaction, allows us to predict the amounts of products formed or reactants required. Whether it's finding the mass of a product given the mass of a reactant or determining how much of a reactant is needed to completely react with another, these conversions are at the heart of stoichiometry.

Once you know how many moles are involved, using stoichiometric coefficients from the balanced equations, it is possible to perform calculations that tell you, for example, how much of a reactant you need to make a certain amount of product—or in our practical example, how many dollars' worth of pennies you have in your piggy bank.