The Average Surface Temperature of the Earth is calculated by the following equation:
$$\sigma T_s^4=\frac{(1-A)\Omega}{4}+\Delta E$$
where,
$\sigma$= Stefan-Boltzmann Constant
$T_s$= Average Surface Temperature of Earth
$A$= Global Albido
$\Omega$= Total Solar Irradiance
And, $\Delta E$= Magnitude of Greenhouse Effect
The observed average surface temperature of Earth is about $288K$, Global Albido as seen by satellites is about $0.3$ and the total solar irradiance is about $1370W/m²$. Putting these values in the above equation, $\Delta E$ comes out to be about $150W/m²$.
Now, according to this PDF I found online, the concentration of $CO_2$ is related to $\Delta E$ by the following equation:
$$\Delta E=133.26+0.044[CO_2]$$
When we put $\Delta E=150$, we get the $[CO_2]=380$ which is the actual concentration of $CO_2$ in the atmosphere according to some online sources in ppm. So the above relation seems correct.
How was this relation calculated? And, how to calculate such relations for other Greenhouse gases, such as $H_2O$?
In other words, given the Equation:
$$\Delta E=x+y[H_2O]$$
Find x and y.
What I noticed: The coefficient of $[CO_2]$ in the above mentioned equation is $0.044$. And, the molecular mass of $CO_2$ is $44.01 amu $. So, Molecular mass might be involved in the calculation of $y$.
My approach: I focused on finding $y$ as $x$ can be calculated later from the same equation since $\Delta E$ is known and concentration of $H_2O$ should be available online. According to my understanding of the Greenhouse effect, $y$ represents how much energy per unit area does $1$ $ppm$ of a $GHG$ can trap and transmit down to Earth. I tried searching online for some data on the same, but could find none.
Please throw some light on the topic.
Thank You.