Calculate Atoms In 48.60 G Of Magnesium (Mg)

#seo-title: Calculating Atoms in Magnesium A Step-by-Step Guide

In the realm of chemistry, a fundamental concept involves determining the number of atoms present in a given mass of a substance. This article delves into a step-by-step guide on how to calculate the number of atoms in a specific mass of an element, using magnesium (Mg) as our example. We will explore the necessary concepts, formulas, and calculations to arrive at the solution. This guide aims to provide a clear and comprehensive understanding of the process, making it accessible to students, chemistry enthusiasts, and anyone seeking to enhance their knowledge in this area.

Understanding the Core Concepts

Before diving into the calculations, it's crucial to grasp the key concepts involved. These include molar mass, Avogadro's number, and the relationship between mass, moles, and the number of atoms.

Molar Mass

The molar mass of an element is the mass of one mole of that element. It is numerically equal to the element's atomic mass expressed in grams per mole (g/mol). For magnesium (Mg), the molar mass is given as 24.30 g/mol. This means that 24.30 grams of magnesium contain one mole of magnesium atoms. The molar mass is a critical conversion factor in stoichiometry, allowing us to move between mass and moles, which are essential for atom calculations.

Avogadro's Number

Avogadro's number, denoted as $6.022 imes 10^{23}$, represents the number of atoms, molecules, or other entities in one mole of a substance. This constant is a cornerstone of chemistry, linking the macroscopic world (grams) to the microscopic world (atoms). In the context of our problem, Avogadro's number tells us that there are $6.022 imes 10^{23}$ magnesium atoms in one mole of magnesium. Understanding Avogadro's number is paramount for converting moles to the number of atoms and vice versa.

The Mole Concept

The mole is the SI unit for the amount of a substance. It provides a bridge between the mass of a substance and the number of particles it contains. One mole of any substance contains Avogadro's number of particles. The concept of the mole is fundamental in quantitative chemistry, enabling us to perform calculations involving chemical reactions, stoichiometry, and, as in our case, determining the number of atoms in a given mass. The mole concept simplifies complex calculations by providing a standardized unit for measuring amounts of substances.

Step-by-Step Calculation

Now, let's apply these concepts to calculate the number of atoms in 48.60 g of Mg.

Step 1: Convert grams to moles

To begin, we need to convert the given mass of magnesium (48.60 g) into moles. We use the molar mass of magnesium (24.30 g/mol) as a conversion factor. The formula for this conversion is:

$$\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}$$

Plugging in the values:

$$\text{Moles of Mg} = \frac{48.60 \text{ g}}{24.30 \text{ g/mol}} = 2 \text{ mol}$$

Therefore, 48.60 g of magnesium is equivalent to 2 moles of magnesium. This conversion is the first crucial step in determining the number of atoms, as it allows us to relate the macroscopic mass to the microscopic quantity of moles.

Step 2: Convert moles to the number of atoms

Next, we convert moles of magnesium to the number of atoms using Avogadro's number ($6.022 imes 10^{23}$ atoms/mol). The formula for this conversion is:

$$\text{Number of Atoms} = \text{Moles} imes \text{Avogadro's Number}$$

Substituting the values:

$$\text{Number of Mg atoms} = 2 \text{ mol} imes 6.022 imes 10^{23} \text{ atoms/mol} = 1.2044 imes 10^{24} \text{ atoms}$$

Thus, 48.60 g of magnesium contains approximately $1.2044 imes 10^{24}$ magnesium atoms. This step directly answers our initial question by translating the moles of magnesium into the actual number of atoms present, utilizing the fundamental relationship defined by Avogadro's number.

Step 3: Rounding to significant figures

In scientific calculations, it's important to consider significant figures. The given mass (48.60 g) and molar mass (24.30 g/mol) both have four significant figures. Therefore, our final answer should also be rounded to four significant figures. The calculated number of atoms, $1.2044 imes 10^{24}$, is already expressed with five significant figures, so we round it to four:

$$\text{Number of Mg atoms} \approx 1.204 \times 10^{24} \text{ atoms}$$

Rounding to significant figures ensures that our answer accurately reflects the precision of the measurements used in the calculation. This is a critical aspect of scientific reporting, maintaining the integrity and reliability of the results.

The Final Answer

Based on our calculations, there are approximately $1.20 imes 10^{24}$ atoms in 48.60 g of magnesium. This result aligns with option C in the provided choices.

Alternative Methods and Considerations

While the step-by-step method outlined above is the most direct approach, it's worth exploring alternative methods and considerations that can enhance our understanding and problem-solving skills in similar scenarios.

Dimensional Analysis

Dimensional analysis is a powerful technique for solving chemistry problems, particularly those involving unit conversions. It involves setting up the calculation so that the units cancel out, leaving the desired unit in the final answer. For our problem, we can set up the calculation as follows:

$$48.60 \text{ g Mg} imes \frac{1 \text{ mol Mg}}{24.30 \text{ g Mg}} imes \frac{6.022 imes 10^{23} \text{ atoms Mg}}{1 \text{ mol Mg}}$$

Notice how the units (g Mg) and (mol Mg) cancel out, leaving us with atoms Mg. Performing the calculation yields the same result: $1.2044 imes 10^{24}$ atoms, which we round to $1.20 imes 10^{24}$ atoms.

Dimensional analysis is not just a computational tool; it's a method that ensures the logical consistency of the calculation by tracking units. This approach can help prevent errors and deepen understanding of the relationships between different quantities.

Conceptual Understanding

Beyond the mathematical calculations, a strong conceptual understanding of the underlying principles is crucial. This includes grasping the significance of molar mass and Avogadro's number as bridges between the macroscopic and microscopic worlds. Understanding that one mole of any substance contains the same number of particles (Avogadro's number) allows us to relate different elements and compounds quantitatively.

For instance, knowing that the molar mass of magnesium is 24.30 g/mol means that for every 24.30 grams of magnesium, there are $6.022 imes 10^{23}$ atoms. This direct proportionality allows for quick estimations and a deeper intuition about the quantities involved in chemical calculations.

Common Mistakes to Avoid

When tackling problems like this, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them.

Incorrect Unit Conversions

One frequent error is using the wrong conversion factors or setting up the calculation incorrectly. For example, mistakenly dividing by Avogadro's number instead of multiplying, or using the atomic mass directly without converting to moles, can lead to drastically wrong results. Always double-check the units and ensure they cancel out correctly in your calculation.

Misunderstanding Molar Mass

Another common mistake is misunderstanding the concept of molar mass. The molar mass is specific to each element and compound and must be used accurately. Confusing molar mass with atomic mass or using the molar mass of a different element will lead to incorrect calculations.

Neglecting Significant Figures

Failing to consider significant figures can result in an answer that is not scientifically accurate. The number of significant figures in the final answer should reflect the precision of the measurements used in the calculation. Rounding errors can accumulate if significant figures are not properly accounted for throughout the calculation.

Calculation Errors

Simple arithmetic errors can also lead to incorrect answers. It's always a good practice to double-check your calculations, especially when dealing with large numbers and exponents. Using a calculator carefully and systematically can help minimize these errors.

Practice Problems

To solidify your understanding, let's consider a couple of practice problems.

Practice Problem 1

How many atoms are present in 100.0 g of gold (Au), given that the molar mass of gold is 196.97 g/mol?

Solution

1. Convert grams to moles:

   $$\text{Moles of Au} = \frac{100.0 \text{ g}}{196.97 \text{ g/mol}} \approx 0.5077 \text{ mol}$$

2. Convert moles to the number of atoms:

   $$\text{Number of Au atoms} = 0.5077 \text{ mol} imes 6.022 \times 10^{23} \text{ atoms/mol} \approx 3.057 \times 10^{23} \text{ atoms}$$

Therefore, there are approximately $3.057 imes 10^{23}$ atoms in 100.0 g of gold.

Practice Problem 2

What is the mass of $3.011 imes 10^{23}$ atoms of carbon (C), given that the molar mass of carbon is 12.01 g/mol?

Solution

1. Convert the number of atoms to moles:

   $$\text{Moles of C} = \frac{3.011 \times 10^{23} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}} = 0.5 \text{ mol}$$

2. Convert moles to grams:

   $$\text{Mass of C} = 0.5 \text{ mol} imes 12.01 \text{ g/mol} = 6.005 \text{ g}$$

Thus, the mass of $3.011 imes 10^{23}$ atoms of carbon is approximately 6.005 g.

Conclusion

Calculating the number of atoms in a given mass of an element is a fundamental skill in chemistry. By understanding the concepts of molar mass, Avogadro's number, and the mole, we can confidently perform these calculations. This guide has provided a step-by-step approach, along with alternative methods, common mistakes to avoid, and practice problems, to help you master this essential skill. With practice and a solid understanding of the underlying principles, you can confidently tackle similar problems and deepen your understanding of chemistry.