# Effective optical depth using Beer-Lambert law for non-linear attenuation? Currently I have a gas with a density that follows and inverse square law in distance, r. Given that I know the mass attenuation coefficient of this gas, I wish to calculate an effective optical depth using a modified version of the Beer-Lambert Law that uses mass attenuation coefficients

Currently I have a gas with a density that follows and inverse square law in distance, $r$. Given that I know the mass attenuation coefficient of this gas, I wish to calculate an effective optical depth using a modified version of the Beer-Lambert Law that uses mass attenuation coefficients:
$\tau =\frac{\alpha {\rho }_{gas}\left(T\right)l}{\rho }=\frac{\alpha Mp\left(T\right)}{\rho RT}\int \frac{1}{{x}^{2}}dx$
Where $\alpha$ is the mass attenuation coefficient for the solid phase of the gas [cm${}^{-1}$], $\rho$ is the mass density of the solid phase of the gas, l is the path length, M is the molar mass of the gas, $p\left(T\right)$is the pressure of the gas as a function of temperature, R is the ideal gas constant and T is the temperature of the gas. ${\rho }_{gas}$ is the mass density of the gas itself and can be extracted from the ideal gas law:
${\rho }_{gas}=\frac{p\left(T\right)M}{RT}$
The integral emerges from my attempt at rewriting the first equation for a non uniform attenuation, that I have here due to the inverse square law effecting the density of the gas.
However, I am now concerned that units no longer balance here since τ should be unitless. Can anyone help guide me here?
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Steven Bates
t looks like you set the density of the gas as
${\rho }_{gas}\propto \frac{1}{{x}^{2}}$
To make the units work out, you need a constant of proportionality to make this an equality:
${\rho }_{gas}=\frac{\beta }{{x}^{2}}$
The units of $\beta$ can be whatever is needed unit-wise, just like G in the Newtonian gravity equation. Newton said that the force of gravity between two objects is proportional to the product of their masses and inversely proporitonal to their squared distance
$F\propto \frac{mM}{{r}^{2}}$
To make this into an equation, we add an empirically measured proportionality constant
$F=G\frac{mM}{{r}^{2}}$
The units of G are whatever they need to be to balance the equation.