ODE/Kinetic Theory problem: Particles leaking in boxes

[Lagraaaange]

Homework Statement

Two boxes of volume V sharing a common wall with a hole of ΔA. Both boxes have gas at T. At t =0s there are N1(0) at one box at time t and N2(0) particles in the other box.The particles from box 1 leak into box 2 through the hole and vice versa.
a. obtain two differential equations, one for N1(t) and one for N2(t).
b. Solve them for N1(t) and N2(t). Particles are not created or destroyed : N1(t) + N2(t) = N1(0) + N2(0)
c. Find pressure of box 1 as function of time and show that it goes to the average value as t goes to infinity.

Homework Equations

P = 2N/3V * 1/2 <mv^2>

The Attempt at a Solution

Φ = nu/4
Flux * Area = Rate
Rate = Anu/4
-dN1/dt = ANu/V4
N1(t) = N1(0)exp(-ΔAut/4V) (negative since rate out)
and N2(t)=N2(0)exp(ΔAut/4V) (positive since I chose rate in for here)
[/B]

 

[epenguin]

Homework Statement

Two boxes of volume V sharing a common wall with a hole of ΔA. Both boxes have gas at T. At t =0s there are N1(0) at one box at time t and N2(0) particles in the other box.The particles from box 1 leak into box 2 through the hole and vice versa.
a. obtain two differential equations, one for N1(t) and one for N2(t).
b. Solve them for N1(t) and N2(t). Particles are not created or destroyed : N1(t) + N2(t) = N1(0) + N2(0)
c. Find pressure of box 1 as function of time and show that it goes to the average value as t goes to infinity.

Homework Equations

P = 2N/3V * 1/2 <mv^2>

The Attempt at a Solution

Φ = nu/4
Flux * Area = Rate
Rate = Anu/4
-dN1/dt = ANu/V4
N1(t) = N1(0)exp(-ΔAut/4V) (negative since rate out)
and N2(t)=N2(0)exp(ΔAut/4V) (positive since I chose rate in for here)[/B]

Your 'relevant equation' has some bearing on then subject but not used here so not really relevant.
You haven't taken into account that they are moving both ways.
Relation of N₁ and N₂ (and their time derivatives) simple - and not what you have in conclusions.
Second part of c not answered.

 

[Lagraaaange]

Your 'relevant equation' has some bearing on then subject but not used here so not really relevant.
You haven't taken into account that they are moving both ways.
Relation of N₁ and N₂ (and their time derivatives) simple - and not what you have in conclusions.
Second part of c not answered.

So is my part A ok? I don't know where to proceed.

 

[epenguin]

So is my part A ok?

I don't think so. The first three contain terms not defined, I don't know what u is in particular, nor why it all has to be divided by 4. Clearly ΔA plays a role.

You should have an eye for whether your equations predict anything reasonable even without solving them. For example yours predict that if there are no particles intially (N₁(0) = 0 in box 1, it stays empty for ever, dN₁/dt = 0. Your solution predicts that if there are any in 2 their number will increase exponentially without limit.
You have as I said to consider particles are going both ways.