Atoms In 48.60g Mg: A Chemistry Calculation

Hey there, science enthusiasts! Today, we're diving into the fascinating world of chemistry to tackle a question about magnesium (Mg) and its atomic composition. We'll be using concepts like molar mass and Avogadro's number to figure out just how many atoms are packed into a given sample of magnesium. So, buckle up and let's get started!

Understanding the Basics: Molar Mass and Avogadro's Number

Before we jump into the problem, let's quickly review some key concepts. Molar mass is essentially the mass of one mole of a substance. Think of a mole as a chemist's special unit for counting atoms or molecules – it's a huge number! Specifically, one mole contains approximately $6.02 imes 10^{23}$ particles, whether they're atoms, molecules, or anything else. This magical number is known as Avogadro's number, and it's a cornerstone of chemistry.

In our case, we're told that the molar mass of magnesium (Mg) is $24.30 g/mol$. This means that if you have 24.30 grams of magnesium, you've got yourself one mole of magnesium atoms. And remember, that one mole contains Avogadro's number of atoms! So, we already know that 24.30 grams of Mg contains $6.02 imes 10^{23}$ atoms. Now, the real question is: what if we have a different amount of magnesium? That's where we'll put our knowledge to the test.

Delving Deeper into Molar Mass: The molar mass is a fundamental property of any element or compound. It acts as a bridge between the macroscopic world (grams, which we can measure in the lab) and the microscopic world (atoms and molecules, which are too tiny to see). The molar mass is numerically equivalent to the atomic mass (for elements) or the molecular mass (for compounds) expressed in grams per mole (g/mol). You can find the atomic mass of an element on the periodic table – it's the number usually listed below the element's symbol. For magnesium, that number is approximately 24.30, hence its molar mass of 24.30 g/mol. Understanding molar mass is crucial for performing stoichiometric calculations, which allow us to predict the amounts of reactants and products involved in chemical reactions. It's like having a recipe for chemical reactions – the molar mass helps us determine how much of each ingredient we need!

Avogadro's Number: The Chemist's Counting Unit: Avogadro's number ($6.02 imes 10^23}$) is an incredibly large number, and it might seem a bit abstract at first. But it's essential for connecting the number of particles (atoms, molecules, ions, etc.) to the amount of substance in moles. Think of it like this just as we use "dozen" to represent 12 items, chemists use "mole" to represent $6.02 imes 10^{23$ particles. This allows us to work with manageable numbers in our calculations, even though we're dealing with astronomical quantities of atoms and molecules. Avogadro's number is not just a random value; it's experimentally determined and reflects the relationship between the atomic mass unit (amu) and the gram. One atomic mass unit is defined as 1/12 the mass of a carbon-12 atom, and Avogadro's number links this microscopic mass scale to the macroscopic scale of grams. The concept of the mole and Avogadro's number is the bedrock of quantitative chemistry, enabling us to accurately measure and manipulate chemical substances.

Solving the Problem: How Many Atoms in 48.60 g of Mg?

Now, let's tackle the question at hand: How many atoms are present in 48.60 g of Mg? We know that 24.30 g of Mg contains $6.02 imes 10^{23}$ atoms. We can use this information to set up a proportion or use the concept of dimensional analysis to find the number of atoms in 48.60 g of Mg.

Here's how we can solve it using a proportion:

$$\frac{24.30 \text{ g Mg}}{6.02 \times 10^{23} \text{ atoms}} = \frac{48.60 \text{ g Mg}}{x \text{ atoms}}$$

Where x is the number of atoms we're trying to find. To solve for x, we can cross-multiply:

$$24.30 \text{ g Mg} \cdot x \text{ atoms} = 48.60 \text{ g Mg} \cdot 6.02 \times 10^{23} \text{ atoms}$$

Now, divide both sides by 24.30 g Mg:

$$x \text{ atoms} = \frac{48.60 \text{ g Mg} \cdot 6.02 \times 10^{23} \text{ atoms}}{24.30 \text{ g Mg}}$$

Notice that the grams of Mg units cancel out, leaving us with atoms:

$$x \text{ atoms} = \frac{48.60}{24.30} \cdot 6.02 \times 10^{23} \text{ atoms}$$

$$x \text{ atoms} = 2 \cdot 6.02 \times 10^{23} \text{ atoms}$$

$$x \text{ atoms} = 1.204 \times 10^{24} \text{ atoms}$$

So, there are approximately $1.204 imes 10^{24}$ atoms in 48.60 g of Mg.

Alternative Approach: Dimensional Analysis: Another way to solve this problem is using dimensional analysis, which involves tracking units to ensure we're performing the correct calculations. We start with the given quantity (48.60 g Mg) and multiply it by conversion factors to get the desired units (atoms):

$$48.60 \text{ g Mg} \times \frac{1 \text{ mol Mg}}{24.30 \text{ g Mg}} \times \frac{6.02 \times 10^{23} \text{ atoms Mg}}{1 \text{ mol Mg}}$$

Notice how the units cancel out (g Mg and mol Mg), leaving us with atoms Mg. Performing the calculation, we get:

$$48.60 \times \frac{1}{24.30} \times 6.02 \times 10^{23} \text{ atoms Mg} = 1.204 \times 10^{24} \text{ atoms Mg}$$

This gives us the same answer as before: there are approximately $1.204 imes 10^{24}$ atoms in 48.60 g of Mg. Dimensional analysis is a powerful technique for solving many types of chemistry problems, as it helps ensure that you're using the correct conversion factors and units.

Interpreting the Result: The result, $1.204 imes 10^{24}$ atoms, is a tremendously large number, which underscores the incredibly small size of individual atoms. Even in a relatively small sample of magnesium (48.60 g), there are over a quadrillion atoms! This immense number highlights the power of Avogadro's number in bridging the gap between the macroscopic and microscopic worlds. It also reinforces the idea that matter is composed of discrete particles (atoms) and that these particles are incredibly numerous. The ability to calculate the number of atoms in a given mass of a substance is a fundamental skill in chemistry, with applications ranging from stoichiometry to materials science.

The Correct Answer

Looking at the options provided in the original problem, none of them match our calculated answer of $1.204 imes 10^{24}$ atoms. The closest answer would be if the option included $1.204 imes 10^{24}$, which isn't present. This highlights the importance of performing the calculation yourself and not just relying on the given options. It's always a good idea to double-check your work and ensure that your answer makes sense in the context of the problem.

Key Takeaways

• Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol).
• Avogadro's number ($6.02 imes 10^{23}$) is the number of particles in one mole.
• We can use molar mass and Avogadro's number to convert between mass and the number of atoms or molecules.
• Dimensional analysis is a useful technique for solving chemistry problems by tracking units.

I hope this explanation has helped you understand how to calculate the number of atoms in a given mass of magnesium! Keep practicing, and you'll become a chemistry whiz in no time!