At constant pressure, the mean free path 1l2 of a gas molecule is directly proportional to temperature. At constant temperature, l is inversely proportional to pressure. If you compare two different gas molecules at the same temperature and pressure, l is inversely proportional to the square of the diameter of the gas molecules. Put these facts together to create a formula for the mean free path of a gas molecule with a proportionality constant (call it Rmfp, like the ideal-gas constant) and define units for Rmfp.

Video transcript

Welcome back everyone in this example, we're using concepts from kinetic molecular theory. We're told that the pressure of a gas is defined as pressure equal to N. M V squared divided by V where N is the number of moles of gas and is the molar mass of our gas. And v squared is the mean of square speeds. We're told in three dimensions, since gas molecules are moving randomly, the root mean square speed is divide defined as the root mean square speeds equal to three times the mean of square speed to the one half power. And from this, we need to derive the formula for the root mean square speed of an ideal gas. So our first step is to outline the first the first two equations given in our prompt here. So for equation one were given that pressure is equal to the molds of our gas times at smaller mass multiplied by the mean of square speeds divided by our gas is volume. And then for equation too, from the prompt were given our root mean square speed, V r M s is equal to three, multiplied by the mean of square speed. So v squared which is then all raised to the one half power. Now our first step is to go ahead and isolate equation two for the mean of square speeds term and so to do so we're going to first get rid of that one half power by taking both sides to the reciprocal, so we're just raising both sides to the second power. So that's going to cancel that one half term out in the exponents. And we will then have that are Root mean square speed squared is equal to three times our mean of square speeds. And now to isolate for that we're going to divide both sides by three which will then give us our mean of square speeds equal to our root mean square speed. So V R M S squared divided by three. And sorry, this cancels out here this three term. So in isolating for Armenia score speed, we have now come up with what we can say is equation three. Where because ultimately for our final answer, we want the formula for the root mean square speed meaning for this term here V R M S. We want to go ahead and recall our ideal gas equation which we should recognize utilizes the terms pressure and most of our gas from equation one. And so we're going to plug in equation three in equation one and so doing that below. So equation three, going into equation one, We would come up with now a 4th equation where we can say that the pressure of our gas is equal to from equation one, the molds of our gas times the molar mass of our gas. And then from equation to we have V R M S squared in the numerator. So we plug that in so V R M S squared and then from our denominator in equation one, we have the volume term which is multiplied by three from our denominator in equation three. And so this is what equation for us to find as. And now that we have equation four and again we referenced our ideal gas equation. We want to make note of our ideal gas equation. So writing that out, we should recall that our ideal gas equation which is now going to be our fifth equation, is our pressure of our gas time, star volume of our gas equal to the molds of our gas times, the gas constant R times the temperature of our gas in kelvin. And because we have came up with an interpretation for pressure being our equation for here, We can actually substitute our term for pressure in our ideal gas law, from our term for pressure in equation four. So we would say that because we understand what pressure is defined as an equation for. We would say that therefore, and we'll use the color green now, so therefore based on equation four, we can say equation six is going to be again substituting this for our term pressure here in our ideal gas law. So we would say equation six is the most of our gas time. Similar mass of our gas times of root mean square speed of our gas divided by we would just say three because our volume term from our ideal gas equation, equation five is in the numerator and it's in the denominator in equation four. So the volume term cancels out. So now that we substituted that for pressure and have canceled out volume. We would set this equal to the right hand side of our ideal gas equation. So N. R. T. And now as the prompt states, we need to derive an equation for our ideal gas for the root mean square speed of our gas. So for V. RMS meaning we need to isolate for V. RMS. And so in doing so, we first want to make the step of multiplying both sides by three to get rid of that denominator there. So this would cancel out the three here. And this would now give us the most of our gas times its molar mass times the root mean square speed of our gas squared equal to three, multiplied by the molds of our gas times, R times T continuing to isolate, we want to get rid of that square term on the exponents. So we're going to take both sides of our equation to the one half power. Or you can just take the square root which is the same thing. And what you would get is the moles of our gas times the molar mass of our gas times the root. Mean square speed is equal to we had, we would have three times the most of our gas times the gas constant r times temperature and this is all raised now to the one half power and again this square term cancels out. So now furthering to isolate for our root mean square speed of our gas. We want to get rid of the N and M. Term. So we're going to divide both sides by the mold of our gas times their molar mass. So both sides divided by N times M. That would cancel out these terms here on the left hand side. And now we would have successfully isolated for the root mean square speed of our gas, which we can say is equal to three, multiplied by and on the right hand side, the end term cancels out. So the most of our gas. So we have three multiplied by R gas constant R. Times our temperature in kelvin. This is all divided by our molar mass of our guests and this entire quotient here is raised to the one half power or under a square root. And this would be our final answer. To complete this example as our formula for the root, mean square speed of an ideal gas. So I hope that everything I reviewed was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video