Water Mixing Problem: Calculating Final Temperature

by TextBrain Team 52 views

Hey guys! Today, we're diving into a classic physics problem involving heat transfer and calorimetry. We're going to figure out how much hot water you need to add to some cold water to reach a specific temperature. This is a super practical application of thermodynamics, and understanding it can help you with everything from making the perfect cup of tea to understanding industrial cooling processes. So, let's break it down step by step!

Understanding the Problem

Let's start by restating the problem in a clear and understandable way. Imagine you have a calorimeter, which is basically a fancy insulated container used for measuring heat changes. Inside this calorimeter, we've got 300 grams of water sitting at a cozy 20°C. Now, we want to add some hot water, which is at 70°C, to this cold water. The big question is: how much of this hot water do we need to add so that the final temperature of the mixture reaches a certain point? To solve this, we'll use the principle of heat exchange, which states that the heat lost by the hot water will be equal to the heat gained by the cold water, assuming no heat is lost to the surroundings (which is the ideal scenario in a calorimeter).

Key Concepts

Before we jump into the calculations, let's make sure we're all on the same page with the key concepts:

  • Heat (Q): Heat is the energy transferred between objects or systems due to a temperature difference. It's measured in Joules (J) or calories (cal).
  • Specific Heat Capacity (c): This is the amount of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius (°C). For water, the specific heat capacity (c) is approximately 4.186 J/g°C or 1 cal/g°C. This is a crucial value for our calculations!
  • Calorimetry: Calorimetry is the science of measuring heat. A calorimeter is the device used for these measurements, designed to minimize heat exchange with the surroundings.
  • Heat Exchange Principle: In a closed system (like our calorimeter), the total heat remains constant. This means that the heat lost by the hotter object(s) is equal to the heat gained by the colder object(s).

The Formula

The fundamental formula we'll be using is: Q = mcΔT

Where:

  • Q is the heat transferred (in Joules or calories)
  • m is the mass of the substance (in grams)
  • c is the specific heat capacity of the substance (in J/g°C or cal/g°C)
  • ΔT is the change in temperature (in °C), which is the final temperature (Tf) minus the initial temperature (Ti).

Setting Up the Problem

Okay, now that we've got the basics down, let's get this problem set up for success. We need to identify our knowns and unknowns. Think of it like gathering your ingredients before you start baking – you want to make sure you have everything you need!

  • Knowns:
    • Mass of cold water (mc) = 300 g
    • Initial temperature of cold water (Tic) = 20°C
    • Initial temperature of hot water (Tih) = 70°C
    • Specific heat capacity of water (c) = 4.186 J/g°C (or 1 cal/g°C)
    • We need to decide on a final temperature (Tf) to target. For this example, let's say we want the final temperature to be 40°C. This is a reasonable temperature to aim for when mixing hot and cold water.
  • Unknown:
    • Mass of hot water (mh) = ? This is what we're trying to find!

The Heat Exchange Equation

Now, let's write down the heat exchange equation. Remember, the heat lost by the hot water is equal to the heat gained by the cold water. So, we can write:

Q_lost = Q_gained

Using our Q = mcΔT formula, we can expand this to:

mh * c * (Tih - Tf) = mc * c * (Tf - Tic)

Notice that the change in temperature (ΔT) is calculated differently for the hot and cold water. For the hot water, it's the initial temperature minus the final temperature (Tih - Tf) because the hot water is cooling down. For the cold water, it's the final temperature minus the initial temperature (Tf - Tic) because the cold water is heating up.

Since the specific heat capacity (c) is the same for both hot and cold water, we can actually cancel it out from both sides of the equation, which simplifies things a bit:

mh * (Tih - Tf) = mc * (Tf - Tic)

Solving for the Unknown

Alright, the hard part is over! We've got our equation set up, and now it's just a matter of plugging in the numbers and solving for the mass of hot water (mh). This is where the algebra comes in, but don't worry, we'll walk through it together.

Let's plug in our known values:

mh * (70°C - 40°C) = 300 g * (40°C - 20°C)

Simplify the equation:

mh * (30°C) = 300 g * (20°C)

mh * 30°C = 6000 g°C

Now, to isolate mh, we'll divide both sides of the equation by 30°C:

mh = (6000 g°C) / (30°C)

mh = 200 g

So, there you have it! We've calculated that you need to add 200 grams of hot water at 70°C to the 300 grams of cold water at 20°C to reach a final temperature of 40°C. Awesome!

Checking Your Work

It's always a good idea to double-check your work to make sure your answer makes sense. One way to do this is to think about the proportions. We added 200g of hot water to 300g of cold water. The final temperature (40°C) is closer to the initial temperature of the cold water (20°C) than the hot water (70°C), which makes sense since we have more cold water than hot water. If our final temperature was closer to 70°C, we'd know something went wrong!

Real-World Applications

This type of calculation isn't just a theoretical exercise; it has tons of real-world applications! Think about:

  • Heating and Cooling Systems: Engineers use these principles to design efficient heating and cooling systems for buildings.
  • Cooking: When you're cooking, you're constantly dealing with heat transfer, whether you're boiling water, baking a cake, or searing a steak.
  • Industrial Processes: Many industrial processes involve heating and cooling materials, and understanding heat transfer is crucial for optimizing these processes.
  • Climate Science: Scientists use similar calculations to model how heat is distributed in the Earth's atmosphere and oceans.

Conclusion

So, guys, we've tackled a classic calorimetry problem and learned how to calculate the amount of hot water needed to reach a specific final temperature when mixed with cold water. We covered the key concepts of heat, specific heat capacity, and the heat exchange principle. Remember, the formula Q = mcΔT is your friend! By understanding these principles, you're one step closer to mastering the world of thermodynamics. Keep practicing, and you'll be solving these problems like a pro in no time!

If you have any questions or want to try another example, let me know in the comments below. And don't forget to share this article with your friends who are also studying physics. Happy calculating!