Calculating The Mass Of 1.8 X 10^24 Molecules Of CO2

Understanding the relationship between the number of molecules, moles, and mass is a fundamental concept in chemistry. In this article, we will walk through the process of calculating the mass of $1.8 imes 10^{24}$ molecules of carbon dioxide ($CO_2$). We'll utilize Avogadro's number and the molar mass of $CO_2$ to perform this calculation, providing a clear and comprehensive explanation for students and chemistry enthusiasts.

Understanding the Basics: Moles, Molecules, and Molar Mass

Before we dive into the calculation, let's clarify some key concepts. Moles are the chemist's way of counting atoms and molecules. One mole is defined as the amount of a substance that contains as many entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12. This number, known as Avogadro's number, is approximately $6.022 imes 10^{23}$. So, 1 mole of any substance contains $6.022 imes 10^{23}$ entities.

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). The molar mass of a compound is calculated by summing the atomic masses of all the atoms in the chemical formula. For example, carbon dioxide ($CO_2$) has one carbon atom and two oxygen atoms. The atomic mass of carbon (C) is approximately 12.01 g/mol, and the atomic mass of oxygen (O) is approximately 16.00 g/mol. Therefore, the molar mass of $CO_2$ is:

$1 imes 12.01 ext{ g/mol (C)} + 2 imes 16.00 ext{ g/mol (O)} = 44.01 ext{ g/mol}$

This means that 1 mole of $CO_2$ has a mass of 44.01 grams. These foundational concepts are crucial for converting between the number of molecules and mass, which we will use in our calculation.

Problem Statement: Finding the Mass of $1.8 imes 10^{24}$ Molecules of $CO_2$

Our primary goal is to determine the mass in grams of $1.8 imes 10^{24}$ molecules of carbon dioxide ($CO_2$). We are given two key pieces of information:

1. The molar mass of $CO_2$ is 44.01 g/mol.
2. 1 mole of $CO_2$ contains $6.022 imes 10^{23}$ molecules (Avogadro's number).

Using these data points, we can set up a conversion strategy that will allow us to move from the number of molecules to mass. The general approach involves two steps:

1. Convert molecules to moles: Divide the given number of molecules by Avogadro's number to find the number of moles.
2. Convert moles to grams: Multiply the number of moles by the molar mass of $CO_2$ to find the mass in grams.

This step-by-step method ensures a clear and logical progression toward the solution, minimizing errors and enhancing understanding. Let’s proceed with the calculation.

Step-by-Step Calculation

Step 1: Convert Molecules to Moles

To convert the number of molecules to moles, we will divide the given number of $CO_2$ molecules by Avogadro's number ($6.022 imes 10^{23}$ molecules/mol).

$$\text{Moles of } CO_2 = \frac{\text{Number of } CO_2 \text{ molecules}}{\text{Avogadro's number}}$$

$$\text{Moles of } CO_2 = \frac{1.8 imes 10^{24} \text{ molecules}}{6.022 imes 10^{23} \text{ molecules/mol}}$$

Performing this division yields:

$$\text{Moles of } CO_2 \approx 2.99 ext{ mol}$$

So, $1.8 imes 10^{24}$ molecules of $CO_2$ is approximately equal to 2.99 moles. This conversion is a crucial intermediate step, transforming the number of individual molecules into a more manageable unit for chemical calculations. The result allows us to proceed to the next step, where we will convert moles into grams.

Step 2: Convert Moles to Grams

Now that we have the number of moles of $CO_2$, we can convert this to grams using the molar mass of $CO_2$ (44.01 g/mol). The formula to convert moles to grams is:

$$\text{Mass of } CO_2 = \text{Moles of } CO_2 imes \text{Molar mass of } CO_2$$

Plugging in the values we have:

$$\text{Mass of } CO_2 = 2.99 ext{ mol} imes 44.01 ext{ g/mol}$$

Performing this multiplication gives us:

$$\text{Mass of } CO_2 \approx 131.59 ext{ g}$$

Therefore, the mass of $1.8 imes 10^{24}$ molecules of $CO_2$ is approximately 131.59 grams. This final calculation provides the answer to our initial question, demonstrating how we can use the concepts of moles, molar mass, and Avogadro's number to convert between the number of molecules and mass.

Final Answer and Summary

In conclusion, the mass of $1.8 imes 10^{24}$ molecules of carbon dioxide ($CO_2$) is approximately 131.59 grams. This result was obtained by first converting the number of molecules to moles using Avogadro's number and then converting moles to grams using the molar mass of $CO_2$. The calculation underscores the importance of understanding these fundamental concepts in chemistry for solving quantitative problems.

To summarize, we followed these steps:

1. Identified the given information: $1.8 imes 10^{24}$ molecules of $CO_2$, molar mass of $CO_2$ (44.01 g/mol), and Avogadro's number ($6.022 imes 10^{23}$ molecules/mol).
2. Converted molecules to moles:

   $$\text{Moles of } CO_2 = \frac{1.8 imes 10^{24} \text{ molecules}}{6.022 imes 10^{23} \text{ molecules/mol}} \approx 2.99 ext{ mol}$$

3. Converted moles to grams:

   $$\text{Mass of } CO_2 = 2.99 ext{ mol} imes 44.01 ext{ g/mol} \approx 131.59 ext{ g}$$

This step-by-step approach highlights the logical progression required to solve such problems, reinforcing the critical relationship between moles, molecules, and mass in chemical calculations. Understanding these relationships is essential for further studies in chemistry and related fields.

Practical Applications and Further Learning

Understanding how to convert between the number of molecules, moles, and mass has numerous practical applications in chemistry and related fields. For instance, in a laboratory setting, chemists often need to calculate the mass of a substance required for a specific reaction based on the number of molecules or moles. This is crucial for ensuring the accuracy and efficiency of experiments.

In industrial chemistry, these calculations are vital for scaling up chemical processes. Determining the correct mass of reactants is essential for optimizing production yields and minimizing waste. In environmental science, understanding molar mass and Avogadro's number is important for quantifying pollutants and assessing their impact.

For further learning, consider exploring topics such as stoichiometry, which deals with the quantitative relationships between reactants and products in chemical reactions. Additionally, understanding the concept of limiting reactants and percent yield can provide a more comprehensive understanding of chemical calculations.

By mastering these fundamental concepts and practicing problem-solving, students and chemistry enthusiasts can build a solid foundation for more advanced topics in chemistry. The ability to confidently convert between molecules, moles, and mass is a cornerstone of chemical understanding and problem-solving.