The Beer-Lambert law gives a linear relationship between the concentration of a solute $c$ and the absorbance $A$, with absorbance defined as the logarithm of the ratio between the transmitted radiant power to the incident one: $$A = \epsilon lc$$ $$A = -\log\left(\frac{I_t}{I_i}\right) = -\log\left(\frac{I_i - I_a}{I_i}\right) = -\log{\left(1 -\frac{I_a}{I_i}\right)} $$
In a simplified model of the effect of CO2 on the atmosphere I am playing around with, the absorptivity $a$ in the Stefan-Boltzmann law of radiation is used to quantify the aomunt of CO2 present, and I found that $a$ is defined as the ratio between the absorbed radiant power to the incident one. $$j = a\sigma T^4$$ $$ a = \frac{I_a}{I_i} $$
I am wondering if it is plausible to say, from the Beer-Lambert law, that in this case the absorptivity is related to the concentration of CO2 in the atmosphere by the following, considering an atmosphere of uniform properties and thickness (which is certainly an overly ideal model). That is: $$ -\log(1-a) = A \propto \ce{ [CO_2] }$$
P.S. I am not sure if this question is more suited for Physics Stackexchange, but since it's about the Beer-Lambert law, I'll post it here first.
Edit: as the commenters helpfully pointed out, my original definitions for the terms were awfully incorrect.