A cylinder of nitrogen and a cylinder of neon are at the same temperature and pressure. The mean free path of a nitrogen molecule is 150 nm. What is the mean free path of a neon atom?

Question

Accepted Answer

Hi everyone. In this practice problem, we are being asked to calculate the mean free pack of an organ atom at a given temperature and pressure. The mean free pad of a helium atom in a gas sample is 300 nanometer. If the gas sample is replaced with an Argan gas at the same temperature and pressure, what will be the new mean free path of the Argan atom? The options given are a 450 nanometer B 600 nanometer C 300 nanometer and lastly D 900 nanometer. So the two gasses in the problem are going to be at the same temperature and at the same pressure. Therefore, they will have the same number of particles per unit volume. The formula that we need to use for this particular practice problem is the formula for the mean free path, which is going to be represented by lambda. Lambda will be equals to one divided by four square root of two pi open parentheses and divided by V and R squared for helium. So lambda of helium, then our equation will come out to be one divided by four, multiplied by squared of two multiplied by pi multiply by and divided by V multiplied by R of helium squared. So the only thing that is specific to helium is the R of helium or, or the radius of helium. So the lambda of Argan similarly is going to be equals to one divided by four, multiplied by squared of two multiplied by pi multiplied by N divided by V multiplied by R of Argan squared. So if you guys notice in this two equation, the N divided by V are going to be the same. And the only difference is just the radius for helium. And for Argan, as I have described previously, this is the result of the two gasses having the same temperature and the same pressure. So we wanna then next combine these two equation and divide them so that we get an equation for lambda of organ divided by a lambda of helium. And if you guys are looking at this two equation, then doing this uh division here, we can notice that the result will just come out to be the radius of helium squared divided by the radius of organ squared. So just the reverse of the radiuses. So what we are looking for is the mean free path of the Argan atom. So that is going to be the lambda of A R. So finding an equation for lambda of A R, we can then get the radius of helium squared divided by the radius of organ squared multiplied by lambda of helium. So helium and Argan are both mono mono atomic molecules and four mono atomic molecules are will be approximately equal to 0.5 times 10 to the power of negative m. Since both organ and helium are mono atomic. So helium and Argan mono atomic, since both are mono atomic, then R of helium will be approximately the same as R of organ which will be approximately 0.5 times 10 to the power of negative 10 m. So since both R are going to be approximately the same, so this uh fraction right here will come out to a value of one or close to one. So lambda of a or the mean free path of the Argan atom will actually be equal to lambda of helium, which will come out to a value of 300 nanometer, which is given in the problem statement. So throughout all this stuff, we found out that the mean free path for the helium is going to be the same for the organ atom. And the answer to this practice problem then will come up to be option C. So option C will be the answer to this particular practice problem and that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our other lesson videos on our website and that will be it for this video. Thank you.

A cylinder of nitrogen and a cylinder of neon are at the same temperature and pressure. The mean free path of a nitrogen molecule is 150 nm. What is the mean free path of a neon atom?

Verified Solution

Key Concepts

Mean Free Path

Kinetic Theory of Gases

Gas Properties and Molecular Size