Problem 67 - Introductory Chemistry Atoms First

Question

Consider a sample of \(24.0 \mathrm{~g}\) of \(\mathrm{O}_{2}\) molecules. (a) How many moles of \(\mathrm{O}_{2}\) molecules are present? (b) How many moles of \(\mathrm{O}\) atoms are present?

Step-by-Step Solution

Answer

(a) There are \(0.75 \mathrm{mol}\) of O₂ molecules in the sample. (b) There are \(1.5 \mathrm{mol}\) of O atoms in the sample.

Step 1: Calculate the moles of O₂ molecules

To find out the moles of O₂ molecules, we need to use the formula: \[ moles = \frac{mass}{molar\,mass} \] The given mass of the O₂ molecules is \(24.0 \mathrm{g}\), and the molar mass of O₂ is the sum of two oxygen atoms, and as each oxygen atom has a mass of \(16.0 \mathrm{g/mol}\), the molar mass of O₂ is \(32.0 \mathrm{g/mol}\). Now we can calculate the number of moles: \[ moles\,of\,O₂ = \frac{24.0 \mathrm{g}}{32.0 \mathrm{g/mol}} = 0.75 \mathrm{mol} \]

Step 2: Calculate the moles of O atoms

Knowing the moles of O₂ molecules, we can determine the number of moles of O atoms in the sample. Since each O₂ molecule contains two O atoms, the number of moles of O atoms will be double the number of O₂ moles: \[ moles\,of\,O = 2 \times moles\,of\,O₂ \] \[ moles\,of\,O = 2 \times 0.75 \mathrm{mol} = 1.5 \mathrm{mol} \] The answer to the questions is: (a) There are \(0.75 \mathrm{mol}\) of O₂ molecules in the sample. (b) There are \(1.5 \mathrm{mol}\) of O atoms in the sample.

Key Concepts

Molar Mass

Understanding molar mass is crucial when you are dealing with chemical calculations and conversions. Molar mass is the mass of one mole of a substance (typically given in grams per mole, g/mol). For example, the molar mass of oxygen (O) is 16.0 g/mol. This value is derived from the atomic mass of oxygen, found on the periodic table. To calculate the molar mass of a compound, you sum the molar masses of the individual elements that make up that compound. For example, in an oxygen molecule (O₂), there are two oxygen atoms. Therefore, the molar mass of O₂ is calculated as:

• Molar mass of O₂ = 2 × 16.0 g/mol = 32.0 g/mol

Understanding molar mass is a foundation for converting mass to moles, an important step in many chemical calculations.

O₂ Molecules

Oxygen molecules, denoted as O₂, are diatomic, meaning they are composed of two oxygen atoms bonded together. In many chemical processes, oxygen is essential for reactions like combustion and respiration. When you are given a mass of O₂, it's often necessary to determine how many moles of O₂ molecules are present. This is done using the concept of molar mass along with mass to moles conversion, which allows us to transition from a macroscopic quantity (mass in grams) to a molecular quantity (moles). For example, if we have a 24.0 g sample of O₂, we can calculate that there are 0.75 moles of O₂ present by using the molar mass of O₂ (32.0 g/mol).

Mass to Moles Conversion

Converting mass to moles is a common task in chemistry that helps connect the macroscopic and microscopic worlds. The conversion uses the formula:\[ moles = \frac{mass}{molar\,mass} \]This formula allows you to calculate the number of moles of a substance when you know its mass and molar mass. Let's consider a sample of 24.0 g of O₂. Given that the molar mass of O₂ is 32.0 g/mol, the calculation is:

• Moles of O₂ = \(\frac{24.0\, g}{32.0\, g/mol} = 0.75\, mol\)

The result, 0.75 mol, tells us how many moles of O₂ molecules are in the sample. This conversion is vital for understanding and carrying out further chemical calculations.

Chemical Calculations

Chemical calculations often involve a series of steps to determine various quantities, such as moles, atoms, or molecules. In this context, knowing how many moles of a substance you have can help initiate other calculations, like finding the number of atoms. For example, once you know there are 0.75 moles of O₂ molecules in a sample, calculating the moles of O atoms is straightforward. Each molecule of O₂ contains two O atoms, so you multiply the moles of O₂ by 2:

• Moles of O atoms = 2 × Moles of O₂ = 2 × 0.75 mol = 1.5 mol

Understanding these calculations opens the door to solving more complex problems in chemistry, such as predicting product yields in reactions or balancing chemical equations.