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

Arectemieryf0 2022-07-23 Answered
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:
τ = α ρ g a s ( T ) l ρ = α M p ( T ) ρ R T 1 x 2 d x
Where α is the mass attenuation coefficient for the solid phase of the gas [cm 1 ], ρ 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 ( T )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. ρ g a s is the mass density of the gas itself and can be extracted from the ideal gas law:
ρ g a s = p ( T ) M R T
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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Answers (1)

Steven Bates
Answered 2022-07-24 Author has 15 answers
t looks like you set the density of the gas as
ρ g a s 1 x 2
To make the units work out, you need a constant of proportionality to make this an equality:
ρ g a s = β x 2
The units of β 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 m M r 2
To make this into an equation, we add an empirically measured proportionality constant
F = G m M r 2
The units of G are whatever they need to be to balance the equation.
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