Problem 119 - Chemistry Matter and Change

Question

A mixture contains 0.250 mol of Fe and 1.20 g of C. What is the total number of atoms in the mixture?

Step-by-Step Solution

Answer

The total number of atoms in the mixture is approximately \(2.11 \times 10^{23}\) atoms.

Step 1: Calculate moles of Carbon

To find the number of atoms, we need to first calculate the moles of carbon in the mixture. Since the mass of carbon is given as 1.20 g, we use the molar mass of carbon, which is approximately 12.01 g/mol, to find the moles of carbon:\[\text{Moles of C} = \frac{1.20 \text{ g}}{12.01 \text{ g/mol}} \approx 0.100 \text{ mol}\]

Step 2: Calculate atoms of Iron

The number of atoms in 0.250 mol of Iron (Fe) can be calculated using Avogadro's number, \(6.022 \times 10^{23}\) atoms/mol:\[\text{Atoms of Fe} = 0.250 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 1.51 \times 10^{23} \text{ atoms}\]

Step 3: Calculate atoms of Carbon

Using the previously calculated moles of carbon, we calculate the number of carbon atoms:\[\text{Atoms of C} = 0.100 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 6.02 \times 10^{22} \text{ atoms}\]

Step 4: Calculate total number of atoms

To find the total number of atoms in the mixture, add the number of iron atoms and carbon atoms:\[\text{Total atoms} = 1.51 \times 10^{23} + 6.02 \times 10^{22} \approx 2.11 \times 10^{23} \text{ atoms}\]

Key Concepts

Moles Calculation

When working with chemical reactions and mixtures, converting mass to moles is an essential step. Moles provide a bridge between the atom-scale realities of chemistry and the macroscopic quantities we can measure in the lab. In the exercise, we were given the mass of carbon, which amounted to 1.20 grams. The concept of moles allows us to convert this mass into a more useful quantity: number of atoms.
To perform this conversion, we rely on the molar mass of an element. For carbon, the molar mass is approximately 12.01 g/mol. To calculate the moles of carbon:

• Divide the mass of carbon by its molar mass: \[\text{Moles of C} = \frac{1.20 \text{ g}}{12.01 \text{ g/mol}} \approx 0.100 \text{ mol}\]

This equation tells us that 1.20 grams of carbon is equivalent to 0.100 moles. Converting mass to moles lets us delve deeper and determine the number of particles involved, which is where Avogadro's Number becomes so crucial.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry. It is used to convert between the number of moles of a substance and the number of individual atoms or molecules present in a sample. The value of Avogadro's Number is approximately\[6.022 \times 10^{23} \text{ atoms/mol}\].
This means that one mole of any substance contains exactly\[6.022 \times 10^{23} \text{ atoms, molecules, or particles}\].
In our exercise, we used Avogadro's Number to determine how many atoms were in a given amount of substance. With 0.250 mol of iron, the calculation would be:

• Using Avogadro's Number to find the atoms in 0.250 mol:\[\text{Atoms of Fe} = 0.250 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 1.51 \times 10^{23} \text{ atoms}\]

This large number illustrates why using the concept of moles and Avogadro's Number is so beneficial in chemistry, as dealing with such massive quantities in their individual form is not practical.

Atoms Calculation

Once we have converted everything to moles, calculating the total number of atoms becomes a straightforward task, thanks to Avogadro's Number. In the exercise, we calculated the moles of carbon and iron separately, and then used those values to find the total number of atoms.
For the carbon, with 0.100 moles calculated, its atom count is:

• \[\text{Atoms of C} = 0.100 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 6.02 \times 10^{22} \text{ atoms}\]
• For the total number of atoms in the mixture, we simply add the atoms from each element:\[\text{Total atoms} = 1.51 \times 10^{23} + 6.02 \times 10^{22} \approx 2.11 \times 10^{23} \text{ atoms}\]

This calculation shows how the individual parts contribute to the whole. Understanding these steps is crucial in stoichiometry, enabling accurate predictions of substance quantities involved in chemical reactions.