Problem 78 - Chemistry The Molecular Science

Question

You have a U. S. penny that weighs \(2.458 \mathrm{~g}\) and contains \(2.40 \%\) copper by mass. (a) Calculate the number of pennies needed to contain \(2.458 \mathrm{~g} \mathrm{Cu}\) (b) Calculate the amount of copper in the penny. (c) Calculate the number of copper atoms in the penny.

Step-by-Step Solution

Answer

42 pennies contain 2.458 g Cu; each penny has 0.059 g Cu; 5.59 x 10^20 Cu atoms per penny.

Step 1: Mass of Copper per Penny

To find the mass of copper in one penny, we multiply the total mass of the penny by the percentage of copper it contains. Since the penny weighs \(2.458\, \text{g}\) and is \(2.40\%\) copper, calculate the mass of copper as follows:\[\text{Mass of Cu per penny} = \frac{2.40}{100} \times 2.458 = 0.059\, \text{g}\]

Step 2: Number of Pennies for Given Copper Mass

Now, we need to calculate how many pennies are required to make up \(2.458\, \text{g}\) of copper. Using the mass of copper per penny found in Step 1, calculate:\[\text{Number of pennies} = \frac{2.458}{0.059} \approx 41.66 \] Since you can't have a fraction of a penny when physically counting them, you need \(42\) pennies in reality.

Step 3: Calculate the Copper from One Penny

We already determined in Step 1 that each penny contains \(0.059\, \text{g}\) of copper. This value directly answers the question posed in part (b).

Step 4: Convert Copper Mass to Moles

To find the number of copper atoms, convert the mass of copper in one penny to moles using the atomic mass of copper \(63.55\, \text{g/mol}\):\[\text{Moles of Cu} = \frac{0.059}{63.55} \approx 0.000928otag\left( ext{mol of Cu} ight)\]

Step 5: Convert Moles to Number of Atoms

Using Avogadro's number \(6.022 \times 10^{23}\) atoms/mol, convert the moles of copper to the number of atoms:\[\text{Number of Cu atoms} = 0.000928 \times 6.022 \times 10^{23} \approx 5.59 \times 10^{20}\]

Key Concepts

Copper Mass Calculation

To calculate the mass of copper in an item, such as a penny, first identify the total mass of the item and the percentage of copper it contains. For example, if a penny weighs 2.458 grams and is composed of 2.40% copper, you calculate the copper mass using the formula:

• Mass of Copper = (Percentage of Copper / 100) × Total Mass of the Item

In this case, it equates to 0.059 grams of copper. Knowing how to calculate mass is crucial for various chemistry calculations, helping you understand the composition of an object.

Moles and Molar Mass

In chemistry, the mole is a fundamental unit used to measure the amount of substance. To convert the mass of a substance to moles, you use the molar mass, which is the weight of one mole of that substance in grams. The relationship is given by:

• Moles = Mass / Molar Mass

For copper with a molar mass of 63.55 g/mol, if you have 0.059 grams, you determine the moles of copper present. This step is crucial when dealing with reactions or when needing to find the number of atoms in a sample, showing the link between macroscopic mass and microscopic molecular amounts.

Avogadro's Number

Avogadro's number is a key constant in chemistry, valued at approximately \(6.022 \times 10^{23}\). It represents the number of atoms, ions, or molecules in one mole of substance, providing a bridge between the understandable macroscopic world and the invisible microscopic universe. Once you know the number of moles of any element, you can multiply by Avogadro's number to find the total number of atoms. For example, from 0.000928 moles of copper, using Avogadro’s number, you find approximately \(5.59 \times 10^{20}\) copper atoms. This conversion is useful for visualizing quantities that are otherwise hard to conceive.

Percentage Composition

Percentage composition is used to determine what fraction of a compound's mass is contributed by each component. It's calculated by dividing the mass of the individual ingredient by the total mass and multiplying by 100. In our case of the penny, the percentage of copper was stated to be 2.40%. This means:

• Percentage Composition of Copper = (Mass of Copper / Total Mass of Penny) × 100

It helps chemists understand both the structure of a compound and the ratio of elements, which is crucial for calculating other parameters or planning chemical reactions.

Unit Conversion in Chemistry

Unit conversion is an essential skill in chemistry, allowing scientists to transition between different types of measurements. This skill comes into play when converting between mass, moles, atoms, or even larger unit-based calculations. Always know which measurements you start with and need to convert to by using conversion factors or dimensional analysis. For example, converting grams to moles using molar mass, and then to number of atoms using Avogadro’s number, demonstrates converting mass ultimately to a countable quantity of atoms. Mastering unit conversions ensures accuracy and consistency in scientific calculations, preparing you to tackle more complex chemistry problem-solving.