Insanely vague thermodynamics question

[vincentchan]

yes, you are right, $$v \alpha \sqrt{P/\rho}$$
so the answer is $$\sqrt{P\rho} = \sqrt{16} = 4$$

 

[apchemstudent]

yes, you are right, $$v \alpha \sqrt{P/\rho}$$
so the answer is $$\sqrt{P\rho} = \sqrt{16} = 4$$

wut? you just contradicted your self...

sqrt(P/density) does not equal sqrt(P*density)

 

[vincentchan]

but $$flux = \rho v = \rho \sqrt{P/\rho} = \sqrt{P\rho}$$

 

[Andrew Mason]

I have submitted a further correction to my answer to account for the fact that $dm/dt = \rho v A$ is it is proportional to v and density. I overlooked the density part. So I would agree with Gokul's answer: 4

AM

 

[Gokul43201]

Aha...do I see semblance of a consensus ?

 

[vincentchan]

Just want to say, even if we use different approach to do this problem, the answers are the same, this is the beauty of Physics, and that is the reason we love it, and spend so much time on it.

 

[apchemstudent]

Ya i agree with 4 too... In my previous posts with Grahm's law i got v = 2v(initial)

so 2density(initial) * 2v(initial) * A = 4 times initial flow rate...

 

[Gokul43201]

Yes, the Grahm's Law approach will work only if you also account for the change in density.

 

[Andrew Mason]

Just want to say, even if we use different approach to do this problem, the answers are the same, this is the beauty of Physics, and that is the reason we love it, and spend so much time on it.

Yes. A very good point. Looking at a problem in different ways like this is a great way to not only learn but to think of new ways of asking questions. One of my favourite quotes of Feynman is from his 1965 Nobel lecture which fits very nicely with your comment:

"I would like to interrupt here to make a remark. The fact that electrodynamics can be written in so many ways - the differential equations of Maxwell, various minimum principles with fields, minimum principles without fields, all different kinds of ways, was something I knew, but I have never understood. It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but, with a little mathematical fiddling you can show the relationship. An example of that is the Schrödinger equation and the Heisenberg formulation of quantum mechanics. I don't know why this is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature. A thing like the inverse square law is just right to be represented by the solution of Poisson's equation, which, therefore, is a very different way to say the same thing that doesn't look at all like the way you said it before. I don't know what it means, that nature chooses these curious forms, but maybe that is a way of defining simplicity. Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing."

There is nothing better than trying to solve challenging problems with different approaches (and risk making mistakes, which everyone does). Nothing should be more encouraged, particular for the many bright young minds around here. I have quite enjoyed this little problem and I think I have learned a little more about thermodynamics.

AM

 

[freemind]

Well, I finally got the actual solution and sure enough, the answer was in fact 4. Unfortunately, I can't find the time to rewrite the entire solution in LaTeX and site bandwidth limitations render any attempt at a png/jpeg solution useless (the original solution was in a pdf). Once again, thanks to everyone who posted.

 

[dextercioby]

The only assumption to be made is that, since the hole is said to be "small" compared to area of the wall, the total number of molecules in the container before and after the change of pressure/temp is the same.

Imagine a small portion of the container wall, area = A. In some short time $\Delta t$ let's calculate how many molecules hit the wall. If the mean velocity of the molecules in the +x direction is $<v_x>$, then all the molecules contained within a volume adjacent to this wall, of depth $<v_x> \Delta t$ will hit the wall. This number is simply the product of this volume with the number density of the molecules, $n$. So,

$$\Delta N = nA <v_x> \Delta t$$

But from the Ideal Gas Law, there's

$$P = nkT => n = P/kT$$

And the average molecular speed is simply

$$<v_x> = \int _0 ^{\infty} v_xf(v_x)dv_x$$

where f(v) is the Maxwell-Boltzmann distribution function, given by

$$f(v_x) = [ m/2 \pi kT]^{3/2} e^{-mv_x^2/2kT}$$

Solving the integral gives the well known result (and you can probably use the result without having to derive it)

$$<v_x> = \sqrt{\frac{8kT}{\pi m}}$$

So substituting for n, <v> into the first equation gives

$$\frac{\Delta N}{\Delta t} = \frac{PA}{kT} \sqrt{\frac{8kT}{\pi m}} ~~ \alpha ~~ \frac{P}{\sqrt{T}}$$

So this is a measure of the rate of collision against the wall. Clearly if this small portion of the wall were removed (creating a small opening), the above number will give you a measure of the rate of escape, or the leak rate.

So, if the P--->8P, T---> 4T, the leak rate will increase by a factor of $8/ \sqrt{4}$ or 4 times.

Well,my friend,i have news for you:bad and good.The bad news is that ur approach is wrong.The good news is,that it yields the correct number (i.e.4),gotten from a wrong formula.

I invite you and the other reading this thread to go to the closest library and search pages 174 pp.177 from:
Greiner,Neise,Stoecker:"Thermodynamics and statistical mechanics",Springer Verlag,1997 for the correct approach.

Just in case one ot the interested parties is lazy (or just cannot find the books and not not even another book to get it),i'll post the final formula:
$$\frac{d^{2}N}{dSdt} =\frac{p}{kT}\sqrt{\frac{kT}{2\pi m}}$$

There's one more thing,and for this one i didn't need the book,just my memoir... Do you see something peculiar here:

$$f(v_x) = [ m/2 \pi kT]^{3/2} e^{-mv_x^2/2kT}$$

If u do,it's not from that number,it's from the very beginning...
With that expression (the previous one) u don't get the (incidentally wrong) value u've posted:
$$<v_x> = \sqrt{\frac{8kT}{\pi m}}$$(?)

I resent your notation,however.You know the last "animal" is always zero at equilibrium.However,in your (misleading) notation is not.But it's still not the formula you posted.

Daniel.


PS.Gokul,you know i did it for the sake of correctness.After all,it's still a science forum...

 

[Gokul43201]

Dexter, I don't understand exactly what you're saying ...but I'll try and fix the ugliness in my post.

$$\Delta N ~~\alpha ~~ nA <|\vec{v}|> \Delta t$$

But from the Ideal Gas Law, there's

$$P = nkT => n = P/kT$$

And the average molecular speed is

$$<|\vec{v}|> = \int _0 ^{\infty} vf(|\vec{v}|)dv$$

where $f(|\vec{v}|)$ is the speed distribution, given by

$$f(|\vec{v}|) = 4\pi v^2 f(\vec{v}) = 4\pi v^2 [ m/2 \pi kT]^{3/2} e^{-mv^2/2kT}$$

Solving the integral gives the mean molecular speed (and you can probably use the result without having to derive it)

$$<|\vec{v}|> = \sqrt{\frac{8kT}{\pi m}}$$

So substituting for n, <|v|> into the first equation gives

$$\frac{\Delta N}{\Delta t} ~~\alpha ~~ \frac{PA}{kT} \sqrt{\frac{8kT}{\pi m}} ~~ \alpha ~~ \frac{P}{\sqrt{T}}$$

So, if the P--->8P, T---> 4T, the leak rate will increase by a factor of $8/ \sqrt{4}$ or 4 times.

Is this better, or are you talking about something else that I'm completely missing ?

 

[dextercioby]

1.Do you really want me to copy 3 pages from a book just to prove my point??Say so,and i'll do it...This morning I'm free...
2.Now that u've "brushed" it,it looks "sound",though,i'm afraid,still incorrect.So there aren't any objections to be made,except starting with the first formula,which is completely unappropriate for this specific problem.

To the OP:this is not HS problem,even if it's for the International Physics Olympiad.In high-school,on normal basis,one does not study Maxwell-Boltzmann statistics,classical canonical ensemble and certainly not the Gamma Euler function.

Daniel.