Diffusion in an underground city I am wondering if diffusion will be enough to keep the underground city from forming a poison zone(high concentration of CO2 leading to death). I mean the people already have enough risks as is(allergies, pneumonia, injury, just to name a few).
And as the society gets bigger, the people will not only have to spread across the planet to find more resources but also dig deeper into the soil. This depth is where it gets more risky and is why some form of ventilation is needed.
Here is what my diffusion model looks like:

Here you can see that CO2 is of relatively low concentration in the air and O2 is of relatively high concentration in the air. Underground the concentrations are opposite(CO2 high, O2 low).
This alone should make the CO2 diffuse into the atmosphere for the plants and the O2 diffuse into the underground chambers and tunnels for the people. And this diffusion along with constant CO2 and O2 production should mean that nothing as powerful as a fan is needed to keep there from being a poison zone.
But is this really true that this bidirectional diffusion(CO2 into atmosphere, O2 into underground city) alone is enough to keep there from being a poison zone, even at the deepest depths the people can handle(only taking into consideration heat and pressure at deep depths assuming no poison zone at those depths)?
If so, great, if not, I want to use diffusion as much as possible and not want to have to use a lot of power. I mean the 100% solar power grid is just in the beginning of development in this society(not ours but similar) so in this case the less solar power needed, the better.
 A: Diffusion becomes increasingly ineffective as the distance over which you want to diffuse becomes larger - because you have further to diffuse, and the gradient gets smaller.
The problem you describe is best avoided with simple convection. There are many mechanisms to achieve that, many of which require little power. For example, even a termite mound has a ventilation system, as described in this article: I think that mechanism (using the heat differential between day and night to cause a "breathing" of the mound) exactly describes what you are looking for. 
The key to achieving "cheap" convection is to take advantage of a thermal gradient. If you make a chimney and paint it black, the air inside will be heated (by the power of the sun). This will make the density of the air inside lower, and it will want to rise. Now you set up a second air inlet into your city - and the chimney will "draw out" the stale air, replacing it with fresh air. 
There are some nice illustrations of the principle at this site
A variant of this was used in some mines: light a fire at the bottom of a shaft, and as the hot air rises, air gets drawn into the mine through a second shaft some distance away. 
Some additional information at this earlier answer and associated comments.
