A question about kinetic theory of gases

[yasar1967]

Question:Dalton’s law of partial pressures states that the total pressure
of a mixture of gases is equal to the sum of the partial pressures
of gases making up the mixture. Give a convincing
argument for this law based on the kinetic theory of gases.

Answer to this question from the book I study Physics gives an explanation stating molecules of different kinds collide with the wall of the container so each additional number of molecules add an extra pressure to the total. But it doesn't say anything about Temperature. How about the second gas you mix with the first has 4 times the number of molecules of first but has temperature of near oK, say 50K? wouldn't than the total pressure be LESS than initial one as the new bunch of molecules' kinetic energy will slow down the total molecules' and the pressure will go down?

 

[Mr.Miyagi]

The total pressure will be higher than the initial pressure, but it won't be five times as high.

If you don't quite see how this works, think of the following case: suppose you dropped in some molecules with temperature 0K. What would this do? Well, it wouldn't do anything, because a molecule with temperature 0K doesn't move, it has no kinetic energy. So all the molecules have taken together have the same kinetic energy as before. So the pressure will stay the same, since the molecules with slam into the walls just as frequently and just as hard as before. The pressure won't drop.

Now, when the added molecules DO have kinetic energy (T>0K), you can reason along the same lines and say the pressure must increase.

 

[yasar1967]

With that reasoning I was not able to reach the same conclusion-sorry. I always tend to think that slower molecules -which may also mean slower speeds if both gases are the same- will dominate as collisions do occur with each other. But I thought this way: avarege speed is 1.6SRoot(kT/m). So new "cold" molecules' avarage speed will lower the whole molecules' avarage speed in Pressure equation P= 2/3x(N/V)x(.5xmxv^2)
But, (N/V) will increase as new molecules are added so the pressure will go up compensating due to loss of avarage speed. is that right?

 

[Mr.Miyagi]

That is correct: The average speed does go down, but the number of molecules will go up.

Perhaps the following will be more illuminating: As it turns out, the pressure on the walls is equal to one third of the kinetic energy density (https://ccrma.stanford.edu/~jos/pasp/Pressure_Confined_Kinetic_Energy.html" ). So when you add a molecule which has kinetic energy (again, T>0K), the energy density must increase. That is because there is more kinetic energy now, but we still have the same amount of volume. Since the energy density increases, so must the pressure.

 

[yasar1967]

Yes, I must have skipped the fact that every kinetic energy added to the system must be summed up to get the total. Kinetic energy has a scalar value due to v^2 so there's no such thing as "cancel out" as opposed to cancelling out of velocities.
Thank you.

 

[Studiot]

Furthemore you have an issue to overcome if you add a second gas with 'four times the molecules'

Look at the gas law PV=nRT

And ask yourself what happens if you specify 2 of the variables as you are trying to do.