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 xWhere 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?

Question

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  xWhere   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?

Steven Bates · Accepted Answer

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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