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Probability Distribution Graph

Patrick Ford
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everybody. So let's go ahead and check out this problem here. So, we have a graph that shows the probability distribution for oxygen gas here. Now, remember that this probability distribution applies to a specific temperature. So there's one specific temperature of this curve right now. So, for example, if the curve looks like this, it would have a different temperature and if it looked like this, it would have an even different temperature. What we want to do is we want to figure out what's the temperature of this sample of oxygen gas, given that the probability curve looks exactly like this. And the one thing that we're told here is that the farthest tick mark on the X axis is 1200 m meters per second. We're also told that the molar mass of oxygen is. So let's go ahead and get started here. Which equation are we going to use in our probability distributions? Remember there's three of them, there is the most probable, there's the average and then there's the R. M. S. So what happens here is if you remember the shape, the average isn't exactly always in the middle or like at the at the at the peak, it's kind of like somewhere beyond the peak like this. So this would be my V average and the RMS velocity is a kind of average that favors generally higher velocity. So the average to the V. RMS would probably be somewhere over here. The problem with this is that we're kind of just guessing at this point. There's no real way to figure that out. Would actually you have to use a really, really complicated equation. What's the one velocity that we actually can absolutely figure out where it is. It's the most probable velocity because remember, the most probable velocity occurs at the peaks of the curve. So this right here is the most probable velocity, it's the very very tip of the curve. So then how do we find what this velocity is? We can actually just use our scale here for the velocity. Alright, so working backwards, if this is 1200 there is equally spaced, this is going to be 11, 10, 98765 and m per second. Right? So this is 302 101 100. So we're kind of having to work backwards to figure out what the tick marks are. But then if you look, if you go ahead you'll see that this line corresponds perfectly to the dot that I just drew. So we can tell here from this graph, is that the most probable velocity here is 400 m/s. So now that we know what this most probable velocity is, we know we're gonna be looking at this equation over here because what's the target variable again? Remember we're looking for the temperature. So we're looking for capital T and if you look at this equation, you're the most probable velocity that's going to be in our equation. Alright, so let's get started. So we're gonna we're gonna use the V. M. P. Equation. Now remember which version are we gonna use? We're gonna use the one that has mass or molar mass. Well we're told what the molar mass of oxygen is. So we're just going to use this format over here. The one with capital M. So we have that VM P. Is equal to the square root of two. R. T. Divided by big M. So all we have to do now is just go ahead and solve and isolate this tea. Now I actually know what the V. M. P. Is. Remember it's just the it's just the speed that we found which is 400 m per second. However, I'm gonna have to square this to get rid of the square roots. This is gonna be 400. We're gonna square this and then the square root comes off of this. So this is gonna be two R. T. Divided by big M. Now all we have to do is just move this guy to the other side and then divide by two are so putting this all together here we have 400 squared times and then the moller mass. Now we're just be very careful here because we were given as grams per mole. So this M. Here if we convert this is going to be 0. kg per mole. So that's the number that we have to plug into our equation here. So this is gonna be times 0.32. And then we just divide by two are so we're gonna divide by two times 8.314. This is going to give you your temperature, and if you work this out, what you're gonna get is 307.9 degrees kelvin. And that's the answer. Alright folks, so that's it for this one.

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