Question

A small sample of gas is released in a corner of the room and starts to diffuse to the other side. If the room pressure is increased, will the gas diffuse faster, slower, or at the same speed?

Answer 1

you are talking about diffusion of gas, so gas is always affected by pressure and temperature, so i would say yes gas would diffuse faster.

Answer 2

Tricky. If you are meant to assume ideal gas behaviour, then ideal gases do not interact with each other, and the pressure of the room air will have no effect on the diffusion rate of the tracer gas.

However, ideality isn't a good assumption in this particular case. Gases diffuse because the individual molecules collide with other molecules, and that tends to scatter them into new directions, after which they suffer addition collisions and the process repeats itself.

Mathematically, the path of the individual diffusing molecule is what's called a "random walk," meaning it looks like the jaggedy path taken by someone who walks a short distance, turns in a random direction and continues walking, and does so again and again. Some quite interesting math will lead you to a reasonable approximation for the diffusion constant D in terms of the step length l and time between steps dt of a random walk:

D = l^2/(6 dt)

That is, the diffusion constant is proportional to the square of the step length and inversely proportional to the time between steps.

In this case, the "step length" of the random walk, for a diffusing molecule, is what's called the "mean free path" of the molecule in the gas, the distance it typically travels in a straight line before colliding with another molecule, and the time between steps is the typical time between collisions. They are related by the mean velocity v, because l/dt = v. That lets us get rid of dt in the equation for D:

D = (1/6) l v

The average velocity of a gas molecule depends entirely on the temperature, so if the temperature doesn't change, this is constant in your problem. That lets us focus in on the mean free path. How does that change when the pressure increases? Clearly at the same temperature, if the pressure increases, so does the density of the gas in the room. That means the molecules are closer together, and it is reasonable to expect the mean free path to decrease. In fact, kinetic theory will tell you explicitly:

l = kT/(sqrt(2) pi d^2 p)

where k is Boltzmann's cosntant, d is the diameter of the molecules, and p is the pressure. We see that indeed the mean free path is inversely proportional to the pressure.

Putting that all together, we find that increasing the pressure in the room decreases the mean free path, which in turn decreases the diffusion constant. So your tracer gas will diffuse more slowly when the pressure in the room is increased.