About CO2 gas

I suspect that most of the gas in a hot air balloon is heated CO2. If that is true, then perhaps the behavior of atmospheric CO2 will be similar to the balloon CO2. Most global warming/CO2 advocates seem to imply that CO2 heated by solar or earth radiation will not rise. I suggest that a single molecule of heated CO2 will behave exactly as a cluster of heated CO2 molecules in a hot air balloon. In that event, CO2 might be beneficial in helping to transport heat energy to higher altitudes where it can be more easily given up into space.

As I recall from school, pv=nrT. In other words, the temperature (Kelvin) would have to be greater than 1.519 times the ambient for the density to be reduced to that of air. Maintaining that temperature without losing heat to the surroundings might be a problem. When it cools, it will just come back down.

You don't mention that you are referring to contained CO2. If it is contained them it seems you have to look at a phase diagram and see at what temperature the density of CO2 is less than air...Eureka!

Gases unlike liquids, do not displace one another naturally based on differences in densities. If you recall way back when, the molecules in a gas are spread farther apart and interactions between molecules are minimal. However, thermal heating of a gas forms a convective current in which the heated gas reduces in density and displaces much cooler air.

Because the gas you intend to heat up is not enclosed by anything, all heat applied to the gas will be dissipated into the surroundings. However, assuming heat is not lost, you would need to find the relationship between temperature and density that results in the density of the two gases to match. Now increase the temperature some more so that the density of CO₂ is lower than air and it will begin to rise.

The ideal gas assumption is a good approximation because the pressure of the system is ambient and you are dealing with ambient to high temperatures. However, one needs to consider the molecular weight of the molecules. Therefore, the density of the gas is inversely proportional to the temperature.

The temperature of CO₂ at which the densities of the two gases are equal occurs at a temperature of 176.7^oC. Therefore increasing the temperature of the carbon dioxide more will displace the air and become buoyant.

I believe the relationship between the height of displacement becomes a problem of force balances. Consider a quantity of CO₂ gas and perform a force balance on it. This includes drag force, force due to gravity, and thrust. The thrust opposes the other two and gives the gas lift. However, I do not know how to relate height to this. Any suggestions?

The main phenomenon that govern the mixture of gases behaviour is diffusion, i.e. natural mixing. Even molecules of the same gas that are either hotter or cooler than the gas average temperature eventually distribute uniformly in the gas (thus changing the "average" temperature).

A "bubble" of hot CO2 will initially rise (if hot enough) but will soon mix with the ambient air. It will create a hot up-draft that will contain mostly air and a bit of CO2. The draft will cool quickly as it rises (adiabatic expansion), so quickly that will soon become cooler than the air at higher altitudes. What I just said is an important fact that insures the stability of the atmosphere.

If it doesn't sound intuitive, perform a mental experiment: the air is a mixture of ~80% N2 (nitrogen) and 20% O2 (oxigen). The N2 is about 12% lighter than the O2. If the N2 rose then one would choke on pure N2 while flying on a commercial jet at 34,000 ft. It doesn't happen and most of the people reach their destination unharmed because the air contains about the same mixture of N2 and O2 both at sea level as well as 34,000 ft.