When we open the window and it's cold outside, we feel that the cold air spreads quickly. But at the end, does the cold air propagate faster than if the outside air temperature were warm?
In other words, in winter do we need to aerate a room for a shorter time for the inside air to be renewed?
Based on Kristjans comment in Kieran Hunts answer...
It occurs to me that rooms with air colder than outside and rooms with air warmer than outside do behave bit differently. Specifically, if the air outside is colder air from the room escapes thru the top of the window and is replaced by colder air flowing down towards the floor. If the air in the room is colder it escapes from the bottom of the window and is replaced by warmer air rising towards the ceiling.
I speculate it is more common to stand on the foor than the ceiling thus the effects of convection will be felt almost instantly if the outside air is colder, but with a delay if it is warmer. Also in either case part of the air in the room will not be replaced by convection just by much slower diffusion. In case of colder outside air the unchanging part is in the ceiling above the heads of most people, for warmer outside air the slower part is at your feet and will be felt.
If anything, the diffusion rate of cold air would likely be slower. We can get a characteristic diffusion time by considering the mean free path of molecules in the gas and its speed of sound. For two gases that are chemically identical (i.e. they differ only in $T$ and $p$), we have: $$ v_\text{sound}\propto \sqrt{T} $$ and $$ d_\text{mfp}\propto\frac{1}{p}\sqrt{T} $$ giving $$ t_\text{diffusion} \propto p $$
So, the diffusion rate, to a first approximation, does not depend on temperature. If we apply the ideal gas law $p=\rho R T$ to a closed box of air, then $p \propto T$, meaning colder air is associated with lower pressure (generally true in the midlatitudes) and hence we would expect warmer air to diffuse more quickly.
