Eunice Foote - 19th century Climatologist

From Wikipedia

Foote conducted a series of experiments that demonstrated the interactions of the sun's rays on different gases. She used an air pump, four mercury thermometers, and two glass cylinders...

Her own two-page write is up here.

1. She pumped air out of one container into the other and left them in the sun for a while. Result:

Decompressed air - 88 F
Compressed air - 110 F.

"This circumstance must affect the power of the sun's rays in different places, and contribute to produce their feeble action on the summits of lofty mountains."

Yes, that is a large part of the actual explanation for sea-level temperatures, which is the gravity-induced lapse rate. Also, common sense tells us, it is pretty hard to heat up a vacuum (there is nothing to heat up), so decompressed air must heat up less.

2. She filled one container with moist air and one with dried air and left them in the sun for a while. Result:

Dry air - 108 F
Moist air - 120 F

I'm not sure what to make of this, but in itself. The specific heat capacity of moist air is higher than for dry air, so this doesn't follow the pattern observed in 3. below. But this is pretty irrelevant in climate terms. What makes a big difference in real life is the latent heat of evaporation (which cools the surface) and the corresponding latent heat of condensation (which warms the air) and thus reduces the lapse rate and overall Greenhouse Effect (by about one-third). It would have been more realistic to have a container filled half with water and half with dry air.

3. She filled the containers with different gases and left them in the sun for a while. Result:

Hydrogen - 104 F
Common air - 106 F
Oxygen - 108 F
CO2 - 125 F

Well, yes, of course. What are the specific heat capacities of those gases (in the 275 - 300 K range)?

Hydrogen - 14,025 J/kg/K
Common Air - 1,006 J/kg/K (or possibly 1,014 J/kg/K)
Oxygen - 916 J/kg/K
CO2 - 832 J/kg/K

Rather unsurprisingly, her experiment shows that things which require less energy to warm up, warm up the most. As ever, 'back radiation' has nothing to do with it. We'd have to adjust this for the mass of the gas compared to the mass of the glass containers and the specific heat capacity of glass (assuming they warmed to the same temperature), but the overall picture is clear enough.

She goes off on a bit of a tangent: "An atmosphere of [CO2] would give to our earth a high temperature; and if as some suppose, at one period of its history the air had mixed with it a larger proportion than at present, an increased temperature from its own action as well as from increased weight must necessarily have resulted"

She is correct, but has the logic and magnitude wrong.

A lower specific heat capacity means a higher lapse rate. If our atmosphere were 100% CO2, the lapse rate would be approx. 2K/km higher (assuming relative humidity stays the same). The average temperature of the atmosphere can't increase as it is dictated by solar radiation. The average temperature is found half way up (approx. 5 km), so sea level temperatures would increase by approx. 10 K, and the temperature at the top of "lofty mountains" above 5 km altitude would fall.

But we would all have suffocated long before then.

In real life, we know that CO2 concentrations are likely to rise from pre-industrial 280 ppm to over 500 ppm this century, and quite possibly to over 600 ppm in the next.

If anybody can be bothered to work out the new average specific heat capacity of air will be when CO2 is 0.05% or 0.06% instead 0.028%, and then work out the new lapse rate (making some heroic assumptions as to whether and how much relative humidity would increase and moderate this) and the resulting impact on sea-level temperatures, then knock yourself out. Most calculators won't have enough decimal places to give a meaningful answer, and even if it does, the additional sea-level temperature will be within the margin of error of even the most accurate thermometers.

Finally:

From this experiment, she stated “The receiver containing [CO2] became itself much heated — very sensibly more so than the other — and on being removed [from the Sun], it was many times as long in cooling.”

As a general rule, gases with a lower specific heat capacity are better insulators and cool down more slowly, so that's hardly surprising either.