Integrated circuits are manufactured in vacuum chambers in which ...

Question

Integrated circuits are manufactured in vacuum chambers in which the air pressure is 1.0 x 10⁻¹⁰ of Hg. What are (a) the number density and (b) the mean free path of a molecule? Assume T = 20℃.

Accepted Answer

Hi everyone in this practice problem, we're being asked to calculate the number density and the mean free path of a molecule where we'll have a vacuum chamber with the air pressure being 1.5 times 10 to the power of 12 millimeters of mercury at a temperature of 25 degrees Celsius. Assuming that the gas in the chamber is mono atomic, we're being asked to calculate first the number density and second the mean free path of a molecule. And the options given are a 4.68 times 10 to the power of 10 per meter cube and 4.36 times 10 to the power of eight m B 4.86 times 10 to the power of 10 me per meter cube and 4.63 times 10 to the power of eight m C 4.86 times 10 to the power of six per meter cube and 4.63 times to the power of 10 m and D 4.68 times 10 to the power of six per meter cube and 4.36 times 10 to the power of 10 m. So in order to calculate first, the number density, we want to use the uh ideal gas law. So for the first part, we're going to use the equation where P P will be equals to N multiplied by K B multiplied by T. So the number density is represented in the N divided by V which through rearrangement, we can get N divided by V by taking P and dividing that with K B and T P is going to be the pressure K B is the Bosman constant and T is the temperature. So in the problem statement, it is given that the P is going to be equal to one point 5th, 1.5 times 10 to the power of negative 12 M mh G or millimeter of mercury. And then the K B is essentially just a constant where it is 1.38 times 10 to the power of um negative 23 divided by Kelvin. And the temperature is given to be 25 degrees Celsius which has to be converted into Kelvin. So we have to uh add 273 Kelvin and that will give us the temperature to be 298. Kelvin. The pressure in millimeters of mercury will have to be converted into si by multiplying it with one point oh 13 times 10 to the power of five pascals, which I'm just going to be uh doing that inside of the equation itself. So let's actually uh substitute all of our value into the equation. So we have N divided by V will be equal to pressure, which is 1.5 times 10 to the power of negative 12 M. Mh G as I have described, we have to multiply this with one point oh 13 times 10 to the power of five pascals and then dividing that with 760 millimeter mercury and then divide the pressure with the Bosman constant, which is 1.38 times 10 to the power of negative Joles per Kelvin. And I multiply that with the temperature which is 298 Kelvin. So that will then give us the N divided by V to be 4.86 times 10 to the power of per meter cube. Awesome. So now that we have solved the first part of the problem statement, which is the number density we will move on to the mean free path of the molecule or the second part of the problem statement. So we have the formula for the mean free path to be equals to lambda to be equal to one divided by four, multiplied by squared of two, multiplied by pi multiplied by open parentheses and divided by V multiplied by R squared. So let's substitute all of our values in the problem statement which is the N divided by V uh which we have obtained previously from the first part. And the R squared, which is given in the problem statement by the hint that the gas in the chamber is going to be mono atomic. So because it is uh mentioned that it is going to be mono atomic, we know that R of mono atomic gasses will be approximately equals to 0.5 times 10 to the power of negative 10 m. So let's substitute off that value. So we have the lambda to then be equals to one divided by uh four multiplied by squared of two multiplied by pi multiplied by N divided by V which is 4.86 stamps 10 to the power of per meter cube multiplied that by our uh R squared which is 0.5 times 10 to the power of negative 10 m squared just like. So, so calculating off this, we will then get our lambda value to then be equal to 4.63 times 10 to the power of eight m. So there we have it. First, we have the number density which is 4.86 times 10 to the power of per meter cube. And second, we have the mean free path of the molecule which is lambda equals to 4.63 times 10 to the power of eight m. And all of that will correspond to option B in our answer choices. So option B 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 adolescent videos on our website and that will be it for this one. Thank you.

Integrated circuits are manufactured in vacuum chambers in which the air pressure is 1.0 x 10⁻¹⁰ of Hg. What are (a) the number density and (b) the mean free path of a molecule? Assume T = 20℃.

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