Mixing Sulfuric Acid Solutions: A Flow Rate Calculation

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Hey guys! Let's dive into a classic chemistry problem involving the mixing of sulfuric acid solutions. This kind of problem pops up all the time in chemical engineering, and it's super important to understand the basics. We're going to break down how to determine the flow rate of one solution when it's mixed with another, given their concentrations and the concentration of the resulting mixture. It's not as scary as it sounds, I promise! We'll use a straightforward approach, and you'll be solving these problems like a pro in no time. This is a practical example of how we can use mass balance to solve real-world problems. We'll be focusing on a scenario with two sulfuric acid (Hâ‚‚SOâ‚„) solutions and figuring out the flow rate of one of them. Ready? Let's get started!

Understanding the Problem: The Setup

So, here’s the scenario, imagine we have two sulfuric acid solutions, which we'll call Solution A and Solution B. We're mixing these two to get a new solution. The fun part is figuring out how much of Solution B we need to get our desired final concentration.

Let’s get the details straight. Solution A is 40% H₂SO₄, and it’s flowing at a rate of 2,000 kg/h. This means that for every hour, 2,000 kilograms of this solution are moving through the system. We know the composition: 40% of this 2,000 kg is actually sulfuric acid, and the remaining 60% is water. Now, let’s talk about Solution B. We know it's 70% H₂SO₄, but we don’t know its flow rate. That's what we need to calculate! The last piece of the puzzle is the resulting mixture. The resulting mixture has a concentration of 50% H₂SO₄. This tells us that the combined solution of A and B is a 50/50 mix of sulfuric acid and water.

This type of problem is a classic example of a mass balance problem. We are going to apply the principle of conservation of mass, which states that mass is neither created nor destroyed in a closed system. This means that the total mass of the reactants (Solutions A and B) entering the system is equal to the total mass of the product (the resulting mixture) leaving the system. To solve this, we can set up a system of equations based on the mass of the sulfuric acid and the total mass.

To make things easier, we're assuming that there's no loss of acid during the mixing process (like, it's not reacting with something else). We're also assuming that the solutions mix perfectly. These are common assumptions that simplify the math without significantly affecting the accuracy of the result. So, the question is, how do we figure out the flow rate of Solution B? Let's break it down step by step.

Data Summary

Let's summarize the key data we have:

  • Solution A: 40% Hâ‚‚SOâ‚„, 2,000 kg/h flow rate
  • Solution B: 70% Hâ‚‚SOâ‚„, unknown flow rate (let's call it 'x' kg/h)
  • Resulting Mixture: 50% Hâ‚‚SOâ‚„

Now, let's look at how to approach the solution.

Setting Up the Equations: Mass Balance

Alright, time to get our math hats on! The core concept here is the mass balance. This means that the mass of sulfuric acid going in must equal the mass of sulfuric acid coming out. We'll also consider the total mass balance, meaning the total mass of the solutions going in equals the total mass of the resulting mixture. This gives us two equations to work with.

First, let's do a total mass balance. The total mass of Solution A plus the total mass of Solution B equals the total mass of the resulting mixture. We can write this as:

  • Mass of A + Mass of B = Mass of Mixture
  • 2,000 kg/h + x kg/h = Total Mixture Flow Rate

So, the total flow rate of the mixture is (2,000 + x) kg/h.

Next, let’s look at the sulfuric acid mass balance. The mass of H₂SO₄ in Solution A plus the mass of H₂SO₄ in Solution B equals the mass of H₂SO₄ in the resulting mixture. This gives us:

  • (0.40 × Mass of A) + (0.70 × Mass of B) = 0.50 × Mass of Mixture

  • (0.40 × 2,000 kg/h) + (0.70 × x kg/h) = 0.50 × (2,000 + x) kg/h

See? It's all about keeping track of the sulfuric acid. The key is to convert the percentages to decimals so that we can accurately calculate the mass. The equation is set up so that we are equating the mass of acid entering the system with the mass of acid leaving the system. Now we are going to solve the equation and figure out the unknown 'x' which represents the flow rate of solution B. We have everything we need to solve the equations and find out the unknown value. Let’s do it!

The Mass Balance Equations Summary

  • Total Mass Balance: 2,000 kg/h + x kg/h = Total Mixture Flow Rate
  • Hâ‚‚SOâ‚„ Mass Balance: (0.40 × 2,000 kg/h) + (0.70 × x kg/h) = 0.50 × (2,000 + x) kg/h

Solving for the Unknown: Finding the Flow Rate of Solution B

Okay, time for the magic! Now that we have our equations, we're going to solve for 'x,' which represents the flow rate of Solution B. We'll start by simplifying the Hâ‚‚SOâ‚„ mass balance equation and then solving for 'x'.

First, let’s simplify the equation: (0.40 × 2,000 kg/h) + (0.70 × x kg/h) = 0.50 × (2,000 + x) kg/h. This is:

  • 800 kg/h + 0.70x = 1000 + 0.50x

Now, let's isolate 'x.' We'll subtract 0.50x from both sides and subtract 800 from both sides.

  • 0.70x - 0.50x = 1000 - 800

  • 0.20x = 200

Finally, divide both sides by 0.20 to solve for x:

  • x = 200 / 0.20

  • x = 1,000 kg/h

So, the flow rate of Solution B is 1,000 kg/h! This means we need to add Solution B at a rate of 1,000 kg/h to get a final mixture that's 50% Hâ‚‚SOâ‚„. Congratulations, we solved it! The math might look a bit intimidating at first, but with a clear understanding of the concepts and a systematic approach, you can solve these problems with confidence.

Step-by-Step Solution

  1. Set up the mass balance equations: Total mass and Hâ‚‚SOâ‚„ mass balance.
  2. Simplify the equations: Convert percentages to decimals and perform the multiplication.
  3. Isolate 'x': Rearrange the equation to get 'x' (the flow rate of Solution B) by itself.
  4. Solve for 'x': Perform the final calculation to find the flow rate.

Conclusion: Wrapping It Up

And there you have it, guys! We successfully calculated the flow rate of Solution B needed to achieve a 50% Hâ‚‚SOâ‚„ concentration in the resulting mixture. This problem demonstrates the power of mass balance in chemical engineering. We started with the concentrations of the two solutions and the flow rate of one and, using the principle of conservation of mass, we were able to determine the unknown flow rate.

By following these steps, you can tackle similar problems. Remember, the key is to set up the equations correctly and keep track of the mass of each component. This approach is not only useful for this specific problem but also provides a solid foundation for understanding more complex chemical engineering calculations. This approach can be applied to different systems and scenarios, so keep practicing.

So, next time you come across a mixing problem like this, you'll know exactly what to do. You can apply this method to other types of mixtures. Just remember to define your system, identify the components, and set up your mass balance equations. Keep practicing, and you'll become a pro in no time! If you have any questions, feel free to ask. Cheers, and happy mixing!

Recap of the Solution

  • The flow rate of Solution B is 1,000 kg/h.
  • We used mass balance principles to solve the problem.
  • Understanding these concepts is crucial for chemical engineering applications.