Dalton's law of partial pressure

[dRic2]

Does anyone know a rigorous proof for Dalton's law ? I think I saw it once, but I can not find it again anywhere.

Thanks
Ric

 

[Gene Naden]

Dalton's "law" is not a real law but only an approximation. In the case of ordinary gases, the molecules or atoms do not collide very often, and the partial pressures add (as I am sure you know). It can be derived by neglecting the forces between these particles, and the volume occupied by them. A paragraph about it can be found at https://en.wikipedia.org/wiki/Dalton's_law

You can start with a discussion of the kinetic theory of gases. I just read about it in Halliday, Resnick and Walker's "Fundamentals of Physics", fifth edition. They assume that the particles do not collide with one another, but only with the walls of whatever contains them. They derive an expression for the pressure based on the number of particles per unit volume and their average speed. Although they do not explicitly treat the case where the gas is a mixture, the concept can be extended to that case.

I hope that helps.

 

[dRic2]

Wikipedia only says the total pressure for a mixture of gases can be expressed as $p_{tot} = \sum p_i$, but it doesn't say why.

I do not know very much about kinetic theory of gasses, but if you say the answer lies there, then I'll study it.

 

[FactChecker]

Wikipedia only says the total pressure for a mixture of gases can be expressed as $p_{tot} = \sum p_i$, but it doesn't say why.

Because there is so little interaction of any type between the gas molecules that the forces from each can just be added together for a good approximation. So the pressures are added to get a total pressure.

 

[dRic2]

good approximation

Yes, but I remember that, for ideal gasses, it not an approximation. Maybe it just comes from the fact that ideal gasses are supposed to have no interaction between their molecules, but I remember my professor showed this with math... Maybe I remember wrong.

 

[FactChecker]

Yes, but I remember that, for ideal gasses, it not an approximation. Maybe it just comes from the fact that ideal gasses are supposed to have no interaction between their molecules, but I remember my professor showed this with math... Maybe I remember wrong.

I just said "good approximation" out of an over-abundance of caution. It's probably good enough to be called exact. Yes, the assumption of no interaction is key.

 

[sophiecentaur]

So the pressures are added to get a total pressure.

A very noddy way of looking at the way pressures can add up would be to consider several small people banging against a wall with hammers. The effect of this (including the rate of hammering) is a pressure. Then take several larger people with bigger hammers, also banging on the wall. They will also have the effect of pressure and the two will just add together.
Instead of people, imagine large and small ballbearings in a steel box, all hitting the wall and bouncing against each other occasionally. The ball / ball collisions make no difference to the total momentum in the direction of the wall and you can think in terms of the frequency of collisions of the big balls and small balls against the wall. Collisions between big and small balls will even out the distribution of Kinetic energies of the balls (corresponding to the temperature in the gas) and the pressure on a wall will be the sum of the pressures (rate of collisions times average momentum). If you were to have just small balls in the box (same number), with the same mean speed (aka temperature) the pressure against the wall will be the same as the contribution of the small balls when they are part of the mixture. Ditto for the big balls.
That's just Kinetic Theory put into everyday words.