Finding The Unknown Gas If The Ratio Of Helium And The Unknown Gas Is 1 2 And The Average Molar Mass Is 12 G/mol

Hey there, chemistry enthusiasts! Ever been faced with the challenge of identifying an unknown gas? It's like being a detective, piecing together clues to solve a mystery. In this comprehensive guide, we'll dive into a fascinating problem where we need to find an unknown gas given the ratio of Helium (He) to the unknown gas and their average molar mass. So, grab your lab coats and let's get started!

Understanding the Problem

In this section, let's break down the problem statement and understand the key concepts involved. We are given that the ratio of Helium (He) to the unknown gas is 1:2. This means that for every 1 mole of Helium, there are 2 moles of the unknown gas. We are also given that the average molar mass of the mixture is 12 g/mol. Our goal is to identify the unknown gas. This requires us to use our knowledge of molar mass, mole ratios, and average molar mass calculations. Think of it like this, guys: we're given the recipe for a gas mixture, and we need to figure out one of the ingredients!

The Importance of Molar Mass

Molar mass, a fundamental concept in chemistry, is the mass of one mole of a substance. It's like the fingerprint of a molecule, uniquely identifying it. Understanding molar mass is crucial for solving this problem because it allows us to relate the mass of a substance to the number of moles present. For instance, the molar mass of Helium (He) is approximately 4 g/mol. This means that 1 mole of Helium weighs 4 grams. When dealing with gas mixtures, molar mass becomes even more critical. The average molar mass of a mixture depends on the molar masses of the individual gases and their respective mole fractions. This is where the 1:2 ratio comes into play, as it tells us about the relative amounts of Helium and the unknown gas.

Mole Ratios and Their Significance

Mole ratios, like the 1:2 ratio given in our problem, are essential for understanding the composition of a mixture. The mole ratio tells us the relative number of moles of each component in the mixture. In our case, the 1:2 ratio of Helium to the unknown gas means that for every mole of Helium, there are two moles of the unknown gas. This information is vital because it allows us to calculate the mole fractions of each gas in the mixture. Mole fraction, defined as the ratio of the number of moles of a component to the total number of moles in the mixture, is a crucial factor in determining the average molar mass. Think of mole ratios as the proportions in our gas mixture recipe – they tell us how much of each ingredient we need!

Average Molar Mass Demystified

Average molar mass is the weighted average of the molar masses of all the components in a mixture. It takes into account both the molar masses of the individual gases and their mole fractions. The formula for average molar mass ( M_\text{avg} ) is:

M_\text{avg} = x_1M_1 + x_2M_2 + ... + x_nM_n

Where:

M_\text{avg} is the average molar mass of the mixture. *
x_i is the mole fraction of component i. *
M_i is the molar mass of component i.

In our problem, we are given the average molar mass (12 g/mol) and need to find the molar mass of the unknown gas. To do this, we'll use the mole ratio to find the mole fractions of Helium and the unknown gas, and then apply the average molar mass formula. It's like having the final dish and some of the ingredients – we need to work backward to find the missing ingredient!

Setting Up the Equations

Now that we understand the key concepts, let's set up the equations we'll use to solve the problem. This involves translating the given information into mathematical expressions that we can work with. This is where the puzzle pieces start to come together, guys!

Defining Variables

First, we need to define our variables. Let's use the following:

M_\text{He} : Molar mass of Helium (approximately 4 g/mol) *
M_\text{unknown} : Molar mass of the unknown gas (what we want to find) *
x_\text{He} : Mole fraction of Helium *
x_\text{unknown} : Mole fraction of the unknown gas *
M_\text{avg} : Average molar mass of the mixture (12 g/mol)

Defining variables helps us keep track of what we know and what we need to find. It's like labeling the different parts of our equation machine!

Mole Fractions from the Mole Ratio

We know the ratio of Helium to the unknown gas is 1:2. This means that if we have 1 mole of Helium, we have 2 moles of the unknown gas. The total number of moles in the mixture is then 1 + 2 = 3 moles. We can use this information to calculate the mole fractions:

• Mole fraction of Helium ( x_\text{He} ) = (Moles of Helium) / (Total moles) = 1 / 3
• Mole fraction of the unknown gas ( x_\text{unknown} ) = (Moles of unknown gas) / (Total moles) = 2 / 3

These mole fractions tell us the proportion of each gas in the mixture. It's like knowing the percentage of each ingredient in our gas mixture recipe!

The Average Molar Mass Equation

We already know the formula for average molar mass:

M_\text{avg} = x_\text{He}M_\text{He} + x_\text{unknown}M_\text{unknown}

We can plug in the values we know:

12 \text{ g/mol} = (1/3)(4 \text{ g/mol}) + (2/3)M_\text{unknown}

This equation is the key to solving our problem. It relates the average molar mass, the molar masses of the individual gases, and their mole fractions. It's like the master formula that will unlock the mystery gas!

Solving for the Unknown

Now comes the exciting part: solving the equation for the molar mass of the unknown gas. This involves some basic algebra, but don't worry, we'll take it step by step. Think of this as the final stage of our detective work, where we put all the clues together to reveal the answer!

Isolating the Unknown Molar Mass

Our equation is:

12 \text{ g/mol} = (1/3)(4 \text{ g/mol}) + (2/3)M_\text{unknown}

First, let's simplify the equation:

12 = \frac{4}{3} + \frac{2}{3}M_\text{unknown}

Next, we want to isolate the term with the unknown molar mass. To do this, we subtract \frac{4}{3} from both sides:

12 - \frac{4}{3} = \frac{2}{3}M_\text{unknown}

\frac{36}{3} - \frac{4}{3} = \frac{2}{3}M_\text{unknown}

\frac{32}{3} = \frac{2}{3}M_\text{unknown}

Calculating the Molar Mass

Now, to solve for M_\text{unknown} , we multiply both sides of the equation by \frac{3}{2} :

\frac{3}{2} \times \frac{32}{3} = M_\text{unknown}

\frac{32}{2} = M_\text{unknown}

M_\text{unknown} = 16 \text{ g/mol}

So, the molar mass of the unknown gas is 16 g/mol. We've cracked the code! This molar mass is the key piece of evidence we need to identify the gas.

Identifying the Gas

We've calculated that the molar mass of the unknown gas is 16 g/mol. Now, the final step is to identify which gas has this molar mass. This is where our knowledge of the periodic table comes into play. It's like using a chemical fingerprint database to match the suspect to the crime scene!

The Periodic Table to the Rescue

The periodic table is our best friend when it comes to identifying elements and compounds based on their molar masses. We can look up the molar masses of common gases and see which one matches our calculated value of 16 g/mol. Remember, the molar mass of an element is approximately equal to its atomic mass, which is listed on the periodic table.

Finding the Match

If we look at the periodic table, we'll find that oxygen (O) has an atomic mass of approximately 16 g/mol. However, oxygen exists as a diatomic molecule (O₂), so its molar mass is 2 * 16 = 32 g/mol. This isn't our match.

Let's think about other common gases. Nitrogen (N) has an atomic mass of approximately 14 g/mol, so N₂ would have a molar mass of 28 g/mol. Still not our gas.

But wait! What about a compound? A molar mass of 16 g/mol is a perfect match for methane (CH₄). Methane consists of one carbon atom (approximately 12 g/mol) and four hydrogen atoms (each approximately 1 g/mol), so its molar mass is 12 + (4 * 1) = 16 g/mol.

The Verdict: Methane!

Therefore, the unknown gas is likely methane (CH₄). We've successfully identified the gas using the given information and our knowledge of chemistry. High five, guys! We solved the mystery!

Conclusion

In this guide, we've walked through the process of identifying an unknown gas given the ratio of Helium to the unknown gas and their average molar mass. We've seen how important it is to understand molar mass, mole ratios, and average molar mass calculations. By setting up the equations and solving for the unknown, we were able to determine that the gas was likely methane (CH₄).

This problem illustrates the power of chemistry in solving real-world mysteries. It's like being a detective, using scientific principles to uncover the truth. So, next time you encounter a similar problem, remember the steps we've discussed, and you'll be well on your way to solving it. Keep exploring, keep questioning, and keep learning, guys! Chemistry is an amazing world, and there's always something new to discover. And remember, practice makes perfect, so keep those problem-solving skills sharp!

Practice Problems

To solidify your understanding, here are a few practice problems similar to the one we just solved:

1. A mixture of gas contains Helium (He) and an unknown gas in a ratio of 1:3. The average molar mass of the mixture is 10 g/mol. What is the unknown gas?
2. A gas mixture is composed of Nitrogen (N₂) and an unknown gas in a 2:1 ratio. The average molar mass of the mixture is 20 g/mol. Identify the unknown gas.
3. A gas sample contains Argon (Ar) and an unknown gas. The ratio of Argon to the unknown gas is 1:1, and the average molar mass is 30 g/mol. Determine the unknown gas.

Try solving these problems using the steps we discussed. Don't be afraid to make mistakes – that's how we learn! And if you get stuck, revisit the concepts and examples we covered. Happy solving, future gas detectives!