Calculating Molecules of Carbon Dioxide Containing 8g of Oxygen

The problem of calculating the number of molecules of carbon dioxide, (CO_2), that contain 8 grams of oxygen can be solved using unit cancellation and the principles related to Avogadro’s number. This article will guide you through the process step by step, ensuring you understand the underlying concepts and their application.

Step-by-Step Solution

First, let's break down the problem into smaller, manageable steps. We will start by converting the given mass of oxygen into moles, and then we will determine the corresponding number of (CO_2) molecules and finally, the number of oxygen atoms within those molecules.

Step 1: Converting Mass to Moles of Oxygen

The molar mass of oxygen, (O), is approximately 16 grams per mole. Therefore, for 8 grams of oxygen, we can calculate the number of moles of oxygen as follows:

Number of moles of (O) (frac{8 , text{g}}{16 , text{g/mol}} 0.5 , text{moles of } O)

Step 2: Determining Moles of (CO_2) from Oxygen

The balanced chemical reaction for the formation of (CO) from (O_2) is:

(C O_2 rightarrow CO_2)

However, to directly find the moles of (CO_2) containing the given oxygen, we can use the ratio of oxygen to carbon dioxide. Each mole of (O_2) (dimer of oxygen) produces one mole of (CO_2). Since we have 0.5 moles of (O_2), we have:

Number of moles of (CO_2) 0.5 moles of (O_2) 0.5 moles of (CO_2)

Step 3: Calculating the Number of (CO_2) Molecules

To convert moles of (CO_2) to the number of molecules, we use Avogadro's number, which is approximately (6.022 times 10^{23}) molecules per mole.

Number of (CO_2) molecules 0.5 moles of (CO_2) (times) (6.022 times 10^{23} , text{molecules/mol}) (3.011 times 10^{23} , text{molecules of } CO_2)

Step 4: Number of Oxygen Atoms in (CO_2) Molecules

Each molecule of (CO_2) contains two oxygen atoms. Therefore, the total number of oxygen atoms in the (CO_2) molecules is:

Number of oxygen atoms 2 (times) (3.011 times 10^{23}) (6.022 times 10^{23} , text{oxygen atoms})

Unit Cancellation and Dimensional Analysis

To work through these problems efficiently, it is essential to master the technique of unit cancellation and dimensional analysis. Here’s a quick reference for the conversion of 8.8 grams of (CO_2) into oxygen atoms:

(8.8 , text{g } CO_2 times frac{1 , text{mol} , CO_2}{44.01 , text{g}} times frac{2 , text{mol} , O}{1 , text{mol} , CO_2} times frac{6.022 times 10^{23} , text{atoms} , O}{1 , text{mol} , O} approx 2.4088 times 10^{23} , text{atoms} , O)

Summary

The number of molecules of carbon dioxide that contain 8 grams of oxygen gas is approximately (3.011 times 10^{23}). This solution demonstrates the application of Avogadro’s number, unit cancellation, and dimensional analysis to solve such problems.