Gas Pressure And Temperature: Solving For New Temperature
Hey everyone! Let's dive into a fun physics problem today involving gases, pressure, and temperature. We're going to use a handy formula to figure out how the temperature of a gas changes when the pressure changes, assuming the volume stays the same. This is a classic example of Gay-Lussac's Law, which describes the relationship between pressure and temperature of a gas at constant volume. So, let's get started and break down this problem step by step. Understanding these concepts is crucial for anyone studying thermodynamics or even just curious about how the world around them works. Stick with me, and we'll make this physics problem a piece of cake!
Understanding the Problem
So, let's break down the problem we're tackling. We have a gas chilling in a cylinder, and it's got a pressure of 115 mmHg when the temperature is 180 Kelvin. Now, imagine we change things up a bit and the pressure drops to 100 mmHg. The big question is: what's the new temperature of the gas? Here's the catch β the volume of the cylinder isn't changing, it remains constant. This is super important because it allows us to use a specific gas law. We're given the formula P1/T1 = P2/T2, which directly links the initial and final pressures and temperatures. This formula is our key to unlocking the solution. Thinking about the scenario, you might intuitively guess that if the pressure goes down, the temperature probably does too. But let's use the formula to figure out exactly how much. This kind of problem pops up all the time in chemistry and physics, so mastering it is a real win!
Key Concepts: Pressure, Temperature, and Volume
Before we jump into calculations, let's quickly recap what pressure, temperature, and volume actually mean in the context of gases. Pressure, in simple terms, is the force that the gas molecules are exerting on the walls of the cylinder. Think of it like tiny little bouncy balls constantly hitting the walls β the harder they hit, the higher the pressure. It's usually measured in units like mmHg (millimeters of mercury), atmospheres (atm), or Pascals (Pa). Temperature, on the other hand, is a measure of how much the gas molecules are jiggling around, the average kinetic energy in other words. The faster they're moving, the higher the temperature. We often use Kelvin (K) as our temperature unit in these kinds of calculations because it's an absolute scale, meaning it starts at absolute zero (the coldest possible temperature). Volume, well, that's just the amount of space the gas is taking up inside the cylinder. If the volume stays constant, it simplifies things a lot because we don't have to worry about it changing in our calculations. Getting these basics down is essential for understanding gas laws and solving problems like this one. Without knowing the concepts, you will never figure out the formula.
The Formula: P1/T1 = P2/T2
Okay, let's zoom in on this formula: P1/T1 = P2/T2. This is the heart and soul of solving this problem. It's a simplified version of the ideal gas law, but because the volume and the number of moles of gas are constant, we can use this shorter version. P1 and T1 represent the initial pressure and temperature of the gas, respectively, while P2 and T2 are the final pressure and temperature. The cool thing about this formula is that it shows a direct relationship between pressure and temperature when the volume is constant. What does "direct relationship" mean? It means that if you increase the pressure, the temperature will increase proportionally, and if you decrease the pressure, the temperature will decrease proportionally. It's like a seesaw β when one side goes up, the other goes up too. This makes intuitive sense if you think about it: if the gas molecules are hitting the walls of the container less frequently (lower pressure), they must be moving slower (lower temperature). This formula is super useful because it lets us predict how gases will behave under different conditions, which is pretty awesome. Make sure you keep this formula in mind.
Solving the Problem Step-by-Step
Alright guys, let's get our hands dirty and actually solve this problem step-by-step. This is where we turn theory into practice, and it's super satisfying when you see everything come together. Grab your calculators, and let's go!
1. Identify the Given Values
First things first, we need to figure out what information the problem has already given us. This is like gathering your ingredients before you start baking. So, let's go back to the problem statement and fish out those values.
- Initial Pressure (P1): 115 mmHg
- Initial Temperature (T1): 180 Kelvin
- Final Pressure (P2): 100 mmHg
The question is asking us to find the final temperature (T2). We've got three pieces of the puzzle, and we're hunting for the fourth. Think of it like a treasure hunt, but the treasure is the temperature! Making sure you've correctly identified the given values is crucial, because if you start with the wrong numbers, the whole calculation will be off. It's like putting the wrong ingredients in a cake β it just won't turn out right. So, double-check your work and make sure you've got everything in the right place. Pay attention to the units, too!
2. Rearrange the Formula to Solve for T2
Now that we have our ingredients, it's time to cook up a solution! Remember that formula we talked about earlier, P1/T1 = P2/T2? We need to tweak it a little bit to get it in the right shape for finding T2. Right now, T2 is hanging out in the denominator on the right side, which isn't very helpful. We want to get it all by itself on one side of the equation.
Here's how we can do it:
- Cross-multiply: Multiply both sides of the equation by T2 and by T1. This gets rid of the fractions and gives us P1 * T2 = P2 * T1.
- Isolate T2: Divide both sides of the equation by P1. This leaves us with T2 = (P2 * T1) / P1.
Ta-da! We've got our formula rearranged and ready to go. It's like having a perfectly sharpened knife for slicing and dicing β it makes the whole process much smoother. Rearranging formulas is a fundamental skill in physics and math, so make sure you're comfortable with it. It's like learning a new language β once you get the grammar down, you can express yourself much more clearly.
3. Plug in the Values and Calculate
Okay, the moment we've been waiting for! Time to plug in those numbers we identified earlier and see what we get. This is like putting the ingredients into the mixing bowl β we're getting close to the final product!
Our rearranged formula is T2 = (P2 * T1) / P1. Let's substitute our values:
T2 = (100 mmHg * 180 K) / 115 mmHg
Now, grab your calculator and do the math. First, multiply 100 by 180, which gives us 18000. Then, divide that by 115. What do you get?
T2 β 156.52 K
There it is! Our final temperature is approximately 156.52 Kelvin. Plugging in the values correctly is super important, because a small mistake here can throw off the whole answer. It's like accidentally adding salt instead of sugar to your cake β not a good result! So, double-check your numbers and make sure you're using the correct units.
4. State the Answer with Units
We've done the calculations, but we're not quite finished yet. In physics (and in life!), it's crucial to state your answer clearly and with the correct units. It's like putting the frosting and decorations on your cake β it makes the final product look polished and professional.
So, our answer is:
The final temperature of the gas is approximately 156.52 Kelvin.
See how we included the units (Kelvin)? That's essential! A number without units is like a word without context β it doesn't really mean anything. Always, always, always include the units in your final answer. It shows that you understand what you're measuring and it prevents confusion. It's a simple thing, but it makes a big difference in how your answer is perceived.
Checking Your Work
Before we declare victory, let's take a moment to check our work. This is like taste-testing your cake before you serve it β you want to make sure it's perfect! It's always a good idea to double-check your calculations and make sure your answer makes sense in the context of the problem.
Sanity Check: Does the Answer Make Sense?
First, let's do a quick sanity check. We know that the pressure decreased from 115 mmHg to 100 mmHg. Since pressure and temperature are directly proportional when the volume is constant, we'd expect the temperature to decrease as well. Our calculated final temperature of 156.52 K is indeed lower than the initial temperature of 180 K, so that's a good sign. It means we're on the right track!
Double-Check the Calculations
Next, let's double-check our calculations. Go back to the formula we used, T2 = (P2 * T1) / P1, and plug in the values again. Use your calculator and make sure you get the same answer. It's easy to make a small mistake when you're doing calculations, so it's always worth taking the extra time to double-check. If you get a different answer the second time, don't panic! Just go through each step carefully and see where you went wrong.
Checking your work is a crucial skill in problem-solving. It's like having a safety net β it catches you if you make a mistake. By doing a sanity check and double-checking your calculations, you can be confident that your answer is correct.
Conclusion
Awesome job, guys! We've successfully solved a physics problem involving gas pressure and temperature. We took a problem statement, identified the given values, rearranged the formula, plugged in the numbers, and calculated the final temperature. And we even checked our work to make sure we got it right! This is how you tackle physics problems like a pro. So, to recap;
- We started by understanding the problem and identifying the given information.
- Then, we recalled the formula P1/T1 = P2/T2 and rearranged it to solve for T2.
- Next, we plugged in the values and calculated the final temperature.
- Finally, we stated our answer with the correct units and checked our work to make sure it made sense.
These steps can be applied to other gas law problems as well. Understanding the concepts, knowing the formulas, and practicing problem-solving are the keys to success in physics. Keep up the great work, and you'll be a physics whiz in no time!