A very small amount of hydrogen gas is released into the air. ...

Question

A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for an H₂ molecule. What assumptions did you make?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem in a container full of chlorine gas. A small amount of helium was released, the effective radii of the chlorine molecules and the helium atoms are 1.8 multiplied by 10 to the power negative 10 m and 1.4 multiplied by 10 to the power negative 10 m. Respectfully, if the temperature and the pressure inside the container are at 25 °C and one atmosphere respectfully approximate the mean free path of the helium molecules inside the container. Also determine what assumptions are necessary for this approximation to be made. Hint the molecular weights of chlorine and helium are 71 U and four U respectfully where U is the atomic units. Awesome. So now it's important to note that we're ultimately trying to solve for the approximate mean free path of the helium molecules inside of this particular ca container. And that's our final answer that we're ultimately trying to solve for. We're also trying to solve for an assumption for this particular problem in order to make this approximation valid. So we know we'll solve for this problem completely when we have two separate answers, when we have an answer for I subscript capital M, which that denotes our mean free path for the helium molecules, which are all gonna be for all of our multiple choice answers, as we can see here are in units of meters for the mean free path. And then we also have an assumption where it says assumption colon and then some assumption. So let's read off all of our multiple choice answers now to see which answer pair might be our final answer. And A is 1.3 multiplied by 10 to the power of negative seven. And the target chlorine molecules are stationary B is 3.3 multiplied by 10 to the power of negative seven. And the helium atoms are stationary C is 4.2 multiplied by 10 to the power of negative seven. And the pressure is negligible. D is 4.2 multiplied by 10 to the power of negative seven. And the mass of helium atom is negligible. OK. So moving right along here, first off, let us compare the root mean square speeds, which we can shorten and denote root mean square as R MS speeds of the two gasses. And we should be able to recall and use the following equation which states that V R MS for helium divided by V R MS for chlorine. So R MS chlorine is equal to the square root of three multiplied by K multiplied by T divided by the mass of helium, divided by three multiplied by K multiplied by T divided by the mass of chlorine. Awesome. So with that in mind, we can simplify this equation too. Let's call this equation one, we can simplify to state that VR MS he or he is helium divided by VR MS Cl where cl once again is chlorine is equal to the square root of the mass of chlorine divided by the mass of helium, which is equal to when we plug in our known variables, 71 U divided by four U which will mean that this is all equal to 4.213. Let me plug that into our calculator. So 4.213 fantastic. So now at this point, we need to note that helium in this case, has such a great velocity. In comparison with the chlorine, we also can assume that the target chlorine can be stationary. So let us also note that since helium is an inert gas, it doesn't form vol like doesn't form molecules. So since it's an inert gas, inert gas, as we should recall, it means that it will not form molecules. Thus, its atoms will move freely. Therefore, we will consider its atomic radius and the chlorine's molecular radius radius. Awesome. So once again, since we know it's a helium and inert gas and that the molecules will not form, we can therefore consider that the atomic radius and chlorine molecular radius in order to help us solve for this particular problem. Thus, with that in mind, we need to recall and use the following equation which states that P multiplied by V is equal to capital N multiplied by K multiplied by capital T. Now, we need to rearrange this equation to isolate and solve for N divided by V to get. And let's call this equation two. So NV is equal to, so we use a little bit of algebra to shift things around. You will get that P divided by K multiplied by T is equal to N divided by V. So now at this point, we can start to solve for the mean free path, which is denoted as I subscript M by recalling and using the following equation which states that im the mean free path is equal to one divided by pi multiplied by the radius of chlorine plus the radius of helium squared multiplied by N divided by V Awesome. So this is also equal to a multiplied by T divided by pi multiplied by the radius of chlorine plus the radius of helium squared multiplied by P. And let us call this equation three Awesome. So this is the radius of the radii for CL and helium. So that's the radius of the radii. OK. Moving right along here. So now we must quickly convert the temperature from degrees Celsius to Calvin. So let us note and let's make this little side note in blue here that the temperature T is equal to 25 °C. But we need to take 25 using our conversion factor which is 273. And when we add 273 plus 25 we'll get 298 Calvin. Fantastic. So moving right along here, we can now plug in all of our known variables to solve for IM which is our final answer. So let's plug in all of our known variables. So I am the mean free path is equal to Boltzmann's constant, which is K. So K denotes Boltzmann's constant, which we should be able to recall this constant to be 1.38 multiplied by 10 to the power of negative 23 jewels Calvin. And we need to multiply it by our temperature which we just did the conversion. So we got 298 Kelvin divided by pi multiplied by. And let's use a bracket the radius of the radii or chlorine which is 1.8 multiplied by 10 to the power of negative 10. Awesome. And don't forget that the units are in meters plus the radius of the radii for helium, which is 1.4 multiplied by 10 to the power of negative 10. Awesome. So we need to make sure we write the units down for meters and then we need to make sure that this value which is the radii radiuses or plus. So the radius of chlorine plus the radius appealing is squared. And then we need to multiply it by our pressure P which we need to note that in one atmosphere, we have to do a little bit of dimensional analysis here multiplied by, we need to note that there's one so 10, one so 101,325 ask scales in one atmosphere. Fantastic. OK. So with that in mind, when we plug that into our calculator, we will find that I am is equal to 1.3 multiplied by 10 to the power of negative seven meters when we round to one decimal place. And that's it we've solved for our first answer. Hooray. We did it. Now we must figure out our assumption. So we can therefore assume that the target chlorine molecules are stationary. So with that in mind, let's look at our multiple choice answers to see which answer corresponds with the answers that we discovered together. It appears that multiple choice answer letter A is our final answer. And A states that im is equal to 1.3 multiplied by 10 to the power of negative 7 m. And the assumption is the target chlorine molecules are stationary. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for an H₂ molecule. What assumptions did you make?

Key Concepts

Mean Free Path

Kinetic Theory of Gases

Assumptions in Gas Behavior

Watch next

A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for an H2 molecule. What assumptions did you make?

Key Concepts

Mean Free Path

Kinetic Theory of Gases

Assumptions in Gas Behavior

Watch next

A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for an H₂ molecule. What assumptions did you make?