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I'm having trouble understanding why the rate of condensation is increased with a decrease in temperature. My reasoning is as follows:
If...
The rate of condensation (R) depends on the concentration of gas molecules (reactants), and it's defined as:
R = k[Gas]
And...
Any rate is proportionate with respect to temperature; as is k...
Then why would a decrease in temperature increase the rate of condensation (R)?
My deductive reasoning for this is as follows:
The problem in your question is the over-simplified statement that the rate of condensation is simply proportional to the concentration of the gas. That is only part of the very complex picture here.
The primary key to condensation rate is the degree of supersaturation of the water vapor at the temperature of the object on which the water is condensing. Under conditions of sub-saturation, i.e. the ambient water vapor concentration (vapor pressure) is less that the vapor pressure of water at the temperature of the object, no bulk (excluding molecular layers of chemisorbed water) condensation occurs. This of course make your equation R = k[Gas] completely false under these conditions.
Once the partial pressure of water vapor exceeds the saturation vapor pressure at the temperature of the object, i.e. the relative humidity exceeds 100% (by decreasing the temperature), then bulk condensation of water can occur. Because our atmosphere has an excess of water condensation nuclei, relative humidities of greater than 102% rarely occur. So, at a given temperature, the condensation rate is roughly proportional to the partial pressure of water for relative humidities of 100-102% at the surface of the object. Note that the relative humidity increases with decreased temperature. Under these specific conditions, your equation is approximately correct. I say approximately because there are other factors like air movement (wind velocity), and kinetic limitations like the rate at which water diffuses across the nm or so layer of air at the surface of the object at which the wind velocity approaches zero.
Further complicating the issue is the temperature dependence of the water vapor concentration at 100% humidity. In other words, there is much more water vapor in the air at $\mathrm{30^oC}$ and 100% humidity vs. $\mathrm{5^oC}$ and 100% humidity. This means that the amount of water vapor available for condensation at any given degree of supersaturation, say 101% humidity, is a function of temperature.
Because of the complexity of this issue my answer has been largely qualitative in nature. For a thorough discussion of this issue (albeit even more complex as it includes discussions of water vapor-to-ice condensation), see this article, "Supersaturation of Water Vapor in Clouds", Journal of Atmospheric Science, December 15 2003, p. 2957.
The temperature of a gas is really just the average kinetic energy of all of the molecules. When you have a container, each gaseous molecule will be exhibiting a different magnitude of molecular vibration. When the average kinetic energy (heat) of the molecules decrease to a certain point, they will condense - go from a gas to a liquid. At lower temperatures, a greater proportion of these molecules will be at an energy level that is low enough to enter the liquid phase.
At a higher temperature, a greater proportion of molecules will have the required energy to become, or stay, a gas.
Condensation is the phase transition from gas to liquid. A condensing gas needs to release energy to form a fluid. A low temperature more readily accepts the heat from the condensing gas, enabling condensation.
Edit. Your reasoning in the image seems to ignore the evaporation is endothermic and therefore Delta h will change sign during condensation.
