What Is The Mass Of A Sample Of NH3 Containing 7.20 X 10^24 Molecules?

Hey guys! Ever stumbled upon a chemistry question that just makes you scratch your head? Well, you're not alone! Let's dive into a fascinating problem today: figuring out the mass of a sample of ammonia ($NH_3$) containing a whopping $7.20 imes 10^{24}$ molecules. Sounds intimidating? Don't worry, we'll break it down step by step, making it super easy to understand. We'll explore the concepts, do some calculations, and by the end, you'll be a pro at solving similar problems. So, buckle up and let's get started on this chemical journey!

Understanding the Fundamentals

Before we jump into the calculations, let’s make sure we've got a solid grasp of the foundational concepts. This is where the magic of chemistry truly begins, and understanding these basics is key to nailing this problem. We're talking about moles, molar mass, and Avogadro's number – the superheroes of chemical calculations! Without these, we'd be lost in a sea of molecules. So, let's put on our thinking caps and dive into these essential ideas. Grasping these concepts will not only help us solve this specific problem but will also make you a more confident chemist overall. Trust me, once you get these down, you'll be able to tackle all sorts of stoichiometry questions with ease!

The Mole Concept: Chemistry's Counting Unit

Think of a mole as a chemist's way of counting atoms and molecules. Just like we use 'dozen' to represent 12 items, a mole represents a specific number of particles. It's a crucial concept because it allows us to relate the microscopic world of atoms and molecules to the macroscopic world that we can measure in the lab. It’s kind of like having a universal translator between the tiny world we can’t see and the larger world we can. The 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. It might sound a bit technical, but the key takeaway is that the mole provides a consistent way to count particles in chemistry. This consistency is what makes it so invaluable for calculations and experiments. Without the mole, we'd be stuck trying to count individual atoms, which is, let’s face it, impossible!

Molar Mass: The Bridge Between Mass and Moles

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It's like the weight of a dozen eggs, but for molecules! Each substance has its own unique molar mass, which is determined by the atomic masses of the elements it contains. For example, to find the molar mass of ammonia ($NH_3$), we add the atomic masses of one nitrogen atom and three hydrogen atoms. This concept is super important because it provides a direct link between the mass of a substance (which we can measure) and the number of moles (which tells us how many particles we have). Think of molar mass as the bridge that allows us to cross from the world of grams to the world of moles, and vice versa. It’s the key to converting between mass and amount in chemical reactions, making it an indispensable tool for any chemist. Understanding molar mass unlocks a whole new level of understanding in chemical calculations.

Avogadro's Number: The Magic Number

Avogadro's number, approximately $6.022 imes 10^{23}$, is the number of entities (atoms, molecules, ions, etc.) in one mole of a substance. This is a colossal number, reflecting just how tiny atoms and molecules are. Imagine trying to count that many grains of sand – it's mind-boggling! Avogadro's number is like the secret ingredient that ties the mole concept to the actual number of particles. It's the conversion factor that allows us to switch between moles and the number of individual atoms or molecules. Knowing Avogadro's number helps us appreciate the sheer magnitude of the microscopic world and how many particles are packed into even the smallest sample of matter. It's a testament to the scale of the universe and the incredible precision of chemical measurements. This number is fundamental to understanding quantitative chemistry.

Calculating the Molar Mass of $NH_3$

Now, let's get down to business and calculate the molar mass of ammonia ($NH_3$). This is a critical step in solving our problem, as the molar mass acts as our conversion factor between mass and moles. We’ll need the periodic table for this, our trusty guide to the elements! The periodic table gives us the atomic masses of each element, which are the building blocks for calculating molar mass. Don't worry, it's not as daunting as it sounds. It’s actually quite straightforward once you know the atomic masses of the elements involved. Think of it as assembling a puzzle – each element contributes its atomic mass, and together they form the molar mass of the compound. Once we have the molar mass, we’re one step closer to figuring out the mass of our ammonia sample. So, let’s grab our periodic table and get started on this essential calculation.

Step-by-Step Calculation

1. Identify the elements: Ammonia ($NH_3$) consists of one nitrogen (N) atom and three hydrogen (H) atoms.

2. Find the atomic masses: Look up the atomic masses of nitrogen and hydrogen on the periodic table. The atomic mass of nitrogen (N) is approximately 14.01 g/mol, and the atomic mass of hydrogen (H) is approximately 1.01 g/mol.

3. Calculate the molar mass: To calculate the molar mass of $NH_3$, we add the atomic mass of nitrogen to three times the atomic mass of hydrogen:

   Molar mass of $NH_3$ = (1 × Atomic mass of N) + (3 × Atomic mass of H) Molar mass of $NH_3$ = (1 × 14.01 g/mol) + (3 × 1.01 g/mol) Molar mass of $NH_3$ = 14.01 g/mol + 3.03 g/mol Molar mass of $NH_3$ = 17.04 g/mol

So, the molar mass of ammonia ($NH_3$) is approximately 17.04 g/mol. This means that one mole of $NH_3$ weighs 17.04 grams. This value is essential for our next steps, as it allows us to convert between the number of moles and the mass of ammonia. It’s like having the key to unlock the relationship between the amount of substance and its weight. With the molar mass in hand, we're well-equipped to tackle the rest of the problem and find the mass of our sample containing $7.20 imes 10^{24}$ molecules.

Converting Molecules to Moles

Now that we know the molar mass of ammonia, the next crucial step is converting the given number of molecules into moles. This conversion is essential because the mole is the bridge between the microscopic world of molecules and the macroscopic world of grams. We can't directly convert molecules to grams, but we can use the mole as an intermediary unit. This is where Avogadro's number comes into play, acting as our magic conversion factor. Remember, Avogadro's number ($6.022 imes 10^{23}$) tells us how many molecules are in one mole. By using this number, we can easily convert our massive number of molecules ($7.20 imes 10^{24}$) into a more manageable number of moles. It’s like translating a foreign language – Avogadro’s number is our dictionary, allowing us to understand the quantity of ammonia in moles, which we can then relate to mass. So, let’s put this knowledge into action and convert those molecules into moles!

Applying Avogadro's Number

To convert the number of molecules to moles, we use the following formula:

Moles = (Number of molecules) / (Avogadro's number)

Given that we have $7.20 imes 10^{24}$ molecules of $NH_3$ and Avogadro's number is $6.022 imes 10^{23}$ molecules/mol, we can plug in these values:

Moles of $NH_3$ = ($7.20 imes 10^{24}$ molecules) / ($6.022 imes 10^{23}$ molecules/mol)

Moles of $NH_3$ ≈ 11.96 moles

Therefore, $7.20 imes 10^{24}$ molecules of $NH_3$ is approximately equal to 11.96 moles. This number is significant because it represents the amount of ammonia we have in a unit that we can easily relate to mass. It’s like converting kilometers to miles – once we have the distance in a familiar unit, we can better understand the magnitude. Now that we know the number of moles, we’re just one step away from finding the mass of our sample. With this key piece of information, we can use the molar mass we calculated earlier to complete the puzzle. So, let’s move on to the final step and convert those moles into grams!

Converting Moles to Mass

We've done the hard part – now comes the payoff! We've successfully converted molecules to moles, and we have the molar mass of ammonia. The final step is to convert these moles into grams, giving us the mass of our sample. This is where our molar mass truly shines, acting as the bridge between moles and grams. Remember, molar mass tells us the mass of one mole of a substance. So, if we know the number of moles and the mass of one mole, we can easily calculate the total mass. It’s like knowing the weight of one apple and wanting to find the weight of a dozen – we simply multiply! This conversion is crucial for practical applications, as we often need to measure substances in grams in the lab. So, let’s put our knowledge to the test and convert those moles of ammonia into grams, finally answering our initial question!

Using the Molar Mass

To convert moles to mass, we use the following formula:

Mass = (Moles) × (Molar mass)

We have 11.96 moles of $NH_3$ and the molar mass of $NH_3$ is 17.04 g/mol. Plugging these values into the formula:

Mass of $NH_3$ = (11.96 moles) × (17.04 g/mol) Mass of $NH_3$ ≈ 203.86 grams

Rounding to the nearest whole number, the mass of the $NH_3$ sample is approximately 204 grams. So, the correct answer is C. 203 grams (closest option). This result is significant because it answers the original question and demonstrates the power of stoichiometry. We started with a huge number of molecules and, by using the mole concept, molar mass, and Avogadro's number, we were able to determine the mass of the sample in grams. It’s like solving a complex puzzle – each step builds on the previous one, leading us to the final solution. Now you can confidently say you know how to convert molecules to mass!

Conclusion

Woohoo! We did it! We successfully calculated the mass of a sample of $NH_3$ containing $7.20 imes 10^{24}$ molecules. By understanding the concepts of moles, molar mass, and Avogadro's number, we were able to navigate this problem with confidence. This journey through the world of chemical calculations highlights the importance of these fundamental concepts in chemistry. Remember, guys, practice makes perfect! The more you work with these ideas, the more comfortable you'll become with them. So, keep exploring, keep questioning, and keep learning! Chemistry is a fascinating field, and with a solid grasp of the basics, you can unlock all sorts of amazing discoveries. Keep up the great work, and who knows, maybe you'll be the next great chemist!

This problem is a fantastic example of how stoichiometry works in practice. It shows how we can relate the microscopic world of atoms and molecules to the macroscopic world of grams, which we can measure in the lab. It also underscores the importance of paying attention to units and using conversion factors correctly. Each step in the calculation builds upon the previous one, and a small error in any step can lead to a wrong answer. So, accuracy and attention to detail are key. By mastering these types of calculations, you'll be well-prepared for more advanced topics in chemistry. Remember, chemistry is all about understanding the relationships between different quantities, and stoichiometry is the foundation for that understanding. So, embrace the challenge, practice consistently, and you'll become a stoichiometry superstar!