The planet Venus is often cited as an example of a runaway greenhouse effect and used to alarm people about the effect of increasing levels of CO₂ in the Earth's atmosphere. This material is to put matters in perspective.

The average temperatures of Earth and Venus are 293 K and 737 K, respectively. This is a ratio of about 2.5. The atmospheric pressure on Venus is about 90 times that of Earth. The crucial variable is the molecular density. First let us obtain the relative molecular densities of the two atmospheres. Venus has an atmosphere that is essentially 100 percent CO₂; the percent of CO₂ in the Earth's atmosphere is 0.038 of 1 percent (0.00038).

The important factor is the molecular density of CO₂ on Venus compared to that of Earth. The Ideal Gas Law says that pressure p, temperature T and density ρ obey the following relationship.

ρ = p/(RT)

where R is the gas constant.

This relationship means that the ratio of the molecular densities for all molecules is given by

ρ_V/ρ_E = (p_V/p_V)/(T_V/E_V) = 90/2.5 = 36

This is for all molecules. What is now needed is the relative proportions of greenhouse gas molecules. Venus has an atmosphere that is essentially 100 percent CO₂; the percent of CO₂ in the Earth's atmosphere is 0.038 of 1 percent (0.00038). But CO₂ is not the only greenhouse gas in the Earth's atmosphere. Far more important than CO₂ is the H₂O in the atmosphere. The proportion of H₂O in the atmosphere is variable and its average value is apparently not known with any precision even though a downward fluctuation in its value of a few hundredth of one percent would wipe out the effect of all the CO₂ in the atmosphere. For purposes of illustration let us take the proportion of H₂O in the Earth's atmosphere to be 2 percent. The radiative efficiency of H₂O molecules is 50 percent higher than that of CO₂ molecules so the 2 percent H₂O would be equivalent to 3 percent CO₂. Neglecting the other greenhouse gases beside H₂O and CO₂, if the proportion of water vapor in the atmosphere is 2 percent, then the effective concentration of greenhouse gases in the Earth's atmosphere is .03038. Thus the ratio of density effective CO₂ in Venus' atmosphere compared to that of Earth is then

ρ_VGHG/ρ_EGHG = 36(1/0.03038) = 1185

What this means is that the increase in greenhouse gases by a factor of 1185 increased the temperature by a factor of 2.5.

If the relationship were of the form

T = αρ_GHG^ε

then ε would have to be such that

2.5 = (1185)^ε
or, equivalently
ε = log(2.5)/log(1185) = 0.13

An increase in the effective CO₂ concentration from a doubling of actual CO₂ is then an increase from 0.03038 to 0.03076. The ratio is then .03076/0.03038 = 1.0125. This would then increase temperature by a factor of

T_2xCO2/T_CO2 = (1.0125)^0.13 = 1.0016
thus
T_2xCO2 = 293(1.0016) = 293.47 K
an increase of 0.47 K
or, equivalently, 0.47 of 1°C.

The best estimate of the rate of global warming is 0.7 of 1 °C per century, of which 30% or 0.2 of 1 °C is due to the increased intensity of the Sun's radiation. This leaves 0.5 of 1°C per century as the rate of global warming due to increasing carbon dioxide. This is just about the same as given by more elaborate climate projection models. However it should be noted that the current rate of increase of carbon dioxide is 0.4 of 1% per year. At this rate it will take about 176 years for the concentration of carbon dioxide to double. The climate modelers assume a rate of increase of 1% per year, which means the the level of carbon dioxide will double in about 70 years. Why do the climate modelers assume a rate of increase 2.5 times the actual rate of increase? Apparently for no other reason that it helps generate scary projections.

The above computations show that the example of Venus does not mean that a doubling of CO₂ would produce a catastrophic rise in Earth's surface temperature. Instead the cases of Earth and Venus are consistent with their separate conditions; i.e., less than 0.04 of 1 percent CO₂ in Earth's atmosphere and upwards of 90 percent in Venus' atmosphere. Of course the proper comparison is between the two planets atmospheres is the total greenhouse gases, which is about 2 percent for Earth (overwhelmingly H₂O) and nearly 100 percent for Venus.