Mean Free Path (Introductory Thermal Physics)

[warhammer]

Homework Statement

A one litre bulb at room temperature contains hydrogen gas at a pressure of 10^-4 torr. At t=0, a filament of area 0.2 cm² is suddenly heated to incandescence. Under these conditions hydrogen molecules striking the filament are dissociated. Neutral hydrogen atoms that are produced stick to the walls of the bulb on striking. How long is the mean free path for hydrogen molecules at the initial pressure?

(Ans; λ=2.3 m)

Relevant Equations

Mean Free Path λ= 0.75*k(B)*T/(π*d^2*pressure)

We have the area of incandescence. Using that we can find the radius and subsequently the diameter.

A=π* r^2 -----> r= 0.0025m so d=0.005m

Using the formula (given by Clausius as we are not specified in question whether it's a Maxwellian distribution or not)

Mean Free Path λ= 0.75*k(B)*T/(π*d^2*pressure) where we take T (room temperature)=300K and Pressure=1333 Pa after conversion from Torr.

Now after plugging in the values, I am getting an incorrect answer. My answer is 2.96*10^-20.

Request someone to guide me and check if my method is correct or not.

 

[mjc123]

When it says "at the initial pressure" I assume that means before the filament is switched on, and the filament is irrelevant. (Are there any more parts to this question?) d in the equation is the diameter of the molecules, not the filament.
And the area of the filament is the surface area of a cylinder, A = 2πrL, where we don't know L or r.

 

[haruspex]

Quoting an equation achieves nothing if you cannot say in what context the equation applies or what the variables mean in that context.

 

[warhammer]

When it says "at the initial pressure" I assume that means before the filament is switched on, and the filament is irrelevant. (Are there any more parts to this question?) d in the equation is the diameter of the molecules, not the filament.
And the area of the filament is the surface area of a cylinder, A = 2πrL, where we don't know L or r.

I also thought the same but the question referred to the filament being switched on at t=0 itself so I assumed it's when the system is just at the verge of having its pressure value changed. No there aren't any more parts.

Actually I realized that and set out on a different approach. I calculated no. of moles in the bulb whose volume is given at 1L. So I calculated the number of moles the bulb contains (and consequently no. of molecules that are contained upon Multiplication with Avogadro's Constant). Post that, I divided the area of the filament (=0.2 cm^2) by the no. of molecules that I calculated (assuming they're all congregated there due to lack of anymore inputs from the question) in order to get the diameter of each molecule. The diameter value I plugged into the mean free path equation for first order but again my answer was wildly different from the one that is specified. I'm unsure whether this novel approach is also incorrect like the previous one.

 

[Steve4Physics]

I will also chip in, in case the following points aren’t clear...

- the information about the filament is totally irrelevant to finding the mean free path;

- 'd' in your equation is the diameter of a hydrogen molecule and has nothing whatsoever to do with the filament;

- you said 'My answer is 2.96*10^-20', so lose 1 penalty-point straight away, for forgetting the unit!

 

[warhammer]

I will also chip in, in case the following points aren’t clear...

- the information about the filament is totally irrelevant to finding the mean free path;

- 'd' in your equation is the diameter of a hydrogen molecule and has nothing whatsoever to do with the filament;

- you said 'My answer is 2.96*10^-20', so lose 1 penalty-point straight away, for forgetting the unit!

Okay. Wrt points 1 & 2 I sense my follies, was trying to unnecessarily rationalise the irrelevant points intro trying to get through the question.

And sorry, I shouldn't have forgotten to specify the unit (the incorrect answer that I got was 2.96*10^-20 m).

By the way on another note, am beginning to feel there isn't any other probable way to deduce the diameter value, and it's value was supposed to be given in the question itself...

 

[Steve4Physics]

(the incorrect answer that I got was 2.96*10^-20 m).

By the way on another note, am beginning to feel there isn't any other probable way to deduce the diameter value, and it's value was supposed to be given in the question itself...

Note the pressure is given as an order of magnitude value. That means both your (incorrect) answer and the official answer have too many significant figures. An order of magnitude value for the answer (or 1 sig. fig. max.) would be better.

I agree about the value of d. You could look-up the value. Your written answer should then include a reference to your source of information.

Or, given the official answer, you could work backwards from it and find the value of d you are supposed to use! But that is just creating a self-fulfilling prophecy!

 

[haruspex]

at a pressure of 10^-4 torr.
...
Pressure=1333 Pa after conversion from Torr

Try that again.

 

[Charles Link]

The formula you are using for the mean free path, within a factor on the order of unity, comes from a more commonly used expression for the mean free path: $\lambda=1/ (n \sigma)$, where $n$ is the number of scatterers per unit volume, and $\sigma$ is the scattering cross section. It also uses $PV=N k_b T$, where $n=N/V$.

The scattering cross section (area) can be estimated as $\sigma=\pi d^2/4$, where $d$ is also an estimate. Typical diatomic molecules have $d=2.0$ E-10 m. (Note: The argument can be made that a collision will occur if the centers of the molecules are separated by a distance $d$ or less, so that perhaps $\sigma \approx \pi d^2$ is a better number than $\pi d^2/4$. The number .75 that you have in the numerator of the mean free path formula seems to be somewhat arbitrary).

For this problem, the mean free path is very much an estimate, but you should be able to get an answer in the same ballpark as the answer that was provided.