Rate of gas leakage through small hole

[rohanlol7]

Homework Statement

Here http://imgur.com/a/4LRM6

2. Homework Equations
The equation is given int he question

The Attempt at a Solution

gas will stop flowing when the pressure inside the gas is equal to that of the surroundings. I calculated the final mass of gas inside the cube. Then i calculated the change in mass of the gas and i got that to be c^2*L^3/3*(1/p-1/Po) then i divided by t and converted L/t=c. However i don't think this is correct as i don't see why the rate of flow of gas should be constant and also my answer seems to be independent of a which seems counter intuitive.

 

[rude man]

Your approach looks questionable.
You're not interested in what happens after pressures equalize. You're interested in the outflow of gas due to the delta in the partial pressure between the inside & outside. The assumption is that outside the box the partial pressure of the gas is zero. So I would consider the component of gas molecules' velocity normal to the side containing the hole, then velocity times area times density = outflow rate.

 

[rohanlol7]

Your approach looks questionable.
You're not interested in what happens after pressures equalize. You're interested in the outflow of gas due to the delta in the partial pressure between the inside & outside. The assumption is that outside the box the partial pressure of the gas is zero. So I would consider the component of gas molecules' velocity normal to the side containing the hole, then velocity times area times density = outflow rate.

I agree my approach was dumb. I figured out that the rate flow is proportional to the density, however I can't seem to figure out what to do from there since the density is changing. I can figure out the change in density of the gas in a time t and then I'm stuck

 

[rohanlol7]

On the oth

I agree my approach was dumb. I figured out that the rate flow is proportional to the density, however I can't seem to figure out what to do from there since the density is changing. I can figure out the change in density of the gas in a time t and then I'm stuck

other hand I was able to derive an equation for the no of moles of gas left in the cube as a function of time. ( I got an exponential decay) but I guess this doesn't quite help me

 

[rude man]

On the oth

other hand I was able to derive an equation for the no of moles of gas left in the cube as a function of time. ( I got an exponential decay) but I guess this doesn't quite help me

EDIT: thinking more about it I guess since the size of the box was given we should compute density ρ and flow rate as functions of time.

But the approach is the same: compute the normal velocity as a function of ρ, then compute flow rate, then ρ decreases with flow rate which reduces flow rate etc. Calculus involved.

Also I would forget no. of moles. Stick with kg since you're given the expression for the molecular velocity as a function of density (assumed in in kg/m³.)

 

[rude man]

Looking at this some more, it appears that not enough info is given.

It's obvious that you need to solve for initial density ρ₀ and molecular velocity c₀ somehow. But even if initial pressure p₀ were given, which it isn't, that does not suffice to determine ρ₀ and c₀, obviously. And even if temperature were given, that would give us number of moles but not density.

So i would proceed by ignoring the given formula, assuming initial values ρ₀ and c₀, then solving for flow rate dm/dt which will be a function of t.

 

[haruspex]

since the size of the box was given we should compute density ρ and flow rate as functions of time.

I think the size of the box was specified as l merely in order to write l>>√a. I don't think it is supposed to feature in the answer.

Looking at this some more, it appears that not enough info is given.

... unless we are to assume current pressure p etc. It is not in terms of initial state, merely current state.

 

[rohanlol7]

EDIT: thinking more about it I guess since the size of the box was given we should compute density ρ and flow rate as functions of time.

But the approach is the same: compute the normal velocity as a function of ρ, then compute flow rate, then ρ decreases with flow rate which reduces flow rate etc. Calculus involved.

Also I would forget no. of moles. Stick with kg since you're given the expression for the molecular velocity as a function of density (assumed in in kg/m³.)

I think that's the right approach to the problem

 

[rohanlol7]

However all these approaches seem fine but, there is a next part to the question and this makes no sense unless what its asking me to calculate is the INITIAL flow rate!
A cylinder that contains helium has a diameter of 300 mm and a length of 1.5 m. The pressure of the gas in the cylinder is 10 atmospheres above the ambient air pressure. There is a small hole in the piston, diameter 2 mm. Calculate the rate of fall of pressure in the cylinder assuming that the expansion is isothermal.

 

[rohanlol7]

and the funnier thing is that the answer kind of needs to be independent of the velocity of the particles which is weirder

 

[haruspex]

Sorry for the delay. rude man and I were having a side discussion. It bothers me that the hole might not be sufficiently small to ignore bulk flow (Bernoulli style) and just treat it as pure diffusion. However, it looks like that is what is expected here.

I started by checking my ideas by deriving the given equation from first principles. Consider a gas of atoms mass m between two parallel plates area A a small distance h apart. Consider one moving at speed v at an angle theta to the normal to the plates.
The probability density of such an angle is sin(theta).
The momentum change normal to the plate on striking it is 2 mv cos(theta).
The frequency with which the atom strikes the same plate is v cos(theta)/2h.
The total number of atoms is $\rho Ah/m$.
Putting all this together, the force on a plate is $\int _0^{\pi/2}\rho \frac{Ah}m\frac{v\cos(\theta)}{2h}\sin(\theta)2mv\cos(\theta).d\theta = \frac 13\rho Av^2$.
Since c is the rms speed, the pressure is $\frac 13\rho c^2$.
So far so good.

The escape rate through a hole area A would be the rate at which atoms would strike that area. This is the same integral as above, but leaving out the factor for momentum change per atom strike.

By this method, I get a mass loss rate of $\frac 18\rho Av$.
But this is awkward since the average v is not c. So we need to plug in the distribution of speeds in an ideal gas.

@rude man , do you see any flaw in that?