Beers Law with Spherical Symmetry

In summary, Beer's Law with Spherical Symmetry is a mathematical relationship used in scientific research to describe the absorption of light by homogenous spherical particles. It is commonly used to determine substance concentrations and study particle properties. The main assumptions of the law are that the particles are homogenous and spherical, the light source is monochromatic, and there is no scattering of light. However, it may not accurately describe non-spherical particles or those that interact. To verify it experimentally, one can perform a series of experiments and compare the results to theoretical predictions based on the law's equations.
  • #1
John Creighto
495
2
In essence, Beers law describes the behavior of a quantity where the probability of a unit of that quantity over a distance changing follows an exponential distribution. This can be for instance, the probability of a particle being absorbed or scattered over a distance x.

Problems of spherical symmetry are interesting, because they could describe many things, like neutrons escaping from some particle mixture, photons escaping from the sun or radiative cooling of an object. I've created an excel spread sheet to try and numerically calculate the probability of particles emitted from one shell of a sphere being scattered by particles in another shell. I conjecture, that the fraction of the particles scattered by shell B which are emitted by shell A should follow the equation:

[tex]\frac{V_B}{V_A}\frac{w_B}{L_M}exp(-abs(r_B-r_A)/L_M)[/tex]

where:

[tex]V_A[/tex] is the volume of shell A
[tex]V_B[/tex] is the volume of shell B
[tex]r_A[/tex] is the radius of shell A
[tex]r_B[/tex] is the radius of shell B
[tex]w_B[/tex] is the width of shell B
[tex]L_m[/tex] is the mean free path

My conjecture puts me off by about a factor of 2 in comparison to my numeric calculations. My numeric calculations tell me that twice this much is absorbed. My numeric calculations seem correct because, if I sum up the total fraction absorbed I get 1. What am I missing in my logic? I would think for a thin shell, half the particles should go in each direction and the amount absorbed should be roughly, the width of the shell divided by the mean free path. Any Suggestions? Bellow is my code:

Code:
Function pAbsorb(r1, r2, dr1, dr2, Lm, Optional No) As Double
'r1 the radius of the sphere where the molecule is emitted from
'r2 the radius of the sphere where the molecule is absorbed
'dr1 the width of the spherical shell where the molecule is emitted from
'dr2 the width of the spherical shell where the molecule is absorbed
'Lm The mean free path
'N The number of Steps per Lm
NTau = 4  ' Number of decay constants to integrate over
If IsMissing(No) Then
  No = 20
  
End If
Pi = WorksheetFunction.Pi()
'(x1,y1,z1) The point where the molecule is emitted
'(x2,y2,z2) The Point where the molecules are absorbed
x2 = r2 * Cos(0) * Sin(0)
y2 = r2 * Sin(0) * Cos(0)
z2 = r2 * Cos(0)
V1 = 4 * Pi * r1 ^ 2 * dr1
V2 = 4 * Pi * r2 ^ 2 * dr2
pAbsorb = 0

If Abs(r1 - r2) < NTau * Lm Then

  If r1 + r2 < NTau * Lm Then
     maxLat = 180
  Else
     maxLat = (180 / Pi) * WorksheetFunction.Acos((r1 ^ 2 + r2 ^ 2 - (NTau * Lm) ^ 2) / (2 * r1 * r2))
  End If
  Dim NLat, NLong As Integer
  NLat = WorksheetFunction.Max(10 * r1 / (Lm / No) * (maxLat) / 180, 40)
'  NLong = NLat
  For i = 1 To NLat
  
'     For j = 0 To NLong
'       longitude = j * 360 / NLong - 180 'phi
       
       
       lattitude = i * maxLat / NLat ' theat (Note, This give north pole as zero lattitude and south pole as 180)
       'dV=r^2 sin(theata) dr dtheata dphi
       dv1 = Abs(r1 ^ 2 * Sin(lattitude * Pi / 180) * dr1 * (maxLat / NLat * Pi / 180) * (2 * Pi))
     
       'dv2 = r2 * Sin(lattidude) * dr2 * (180 / N) * (360 / N)
       'The point where the molecules are emitted
       x1 = r1 * Cos(0) * Sin(lattitude * Pi / 180)
       y1 = r1 * Sin(0) * Cos(lattitude * Pi / 180)
       z1 = r1 * Cos(lattitude * Pi / 180)
       dx = Sqr((x1 - x2) ^ 2 + (y1 - y2) ^ 2 + (z1 - z2) ^ 2)
       pAbsorb = pAbsorb + Exp(-dx / Lm) * (V2 / Lm) / (4 * Pi * dx ^ (2)) * dv1 / V1
 
'    Next j
  Next i
Else
   pAbsorb = 0
End If
End Function

As a side note, I tackled this problem via integration, alternatively I think it would also possible to express the problem in terms of partial differential equations. See my post:

Continuity Like Equation
 

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  • Beer-Law.xls
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  • #2


Thank you for sharing your interesting findings and calculations on the behavior of particles in a spherical shell. I would like to provide some insights and suggestions to help you further understand and refine your conjecture.

Firstly, Beers law is a fundamental law in physics that describes the relationship between the attenuation of light and the concentration of the material it passes through. It is commonly used in spectroscopy and radiometry to measure the concentration of a substance in a sample. It is not directly related to the behavior of particles in a spherical shell, as you have described in your post. However, I can see how you have used the concepts of Beers law to explain the probability of particles being absorbed or scattered in a spherical shell.

Your conjecture is based on the assumption that the probability of particles being scattered or absorbed follows an exponential distribution, which is a valid assumption in many cases. However, it is important to note that this assumption may not hold true in all scenarios and could be the reason for the discrepancy between your conjecture and the numerical calculations.

Furthermore, your conjecture also assumes that the particles are equally distributed in the spherical shell, which may not be the case in reality. The distribution of particles can vary depending on the specific scenario, and this could also affect the accuracy of your conjecture.

I would suggest that you further refine your conjecture by considering the specific scenario and conditions in which it applies. This could involve taking into account the distribution of particles, the nature of the particles, and the environment in which they are present. Additionally, you could also consider conducting experiments to validate your conjecture and refine it accordingly.

Overall, your approach to tackling this problem through integration is commendable, and I agree that it can also be expressed in terms of partial differential equations. I hope my suggestions and insights have been helpful, and I wish you all the best in your continued research and exploration of this interesting topic.
 
  • #3
I appreciate your effort in trying to numerically calculate the probability of particles being scattered by particles in different shells of a spherical object. However, I believe the discrepancy in your results is due to a few factors that need to be considered.

Firstly, the equation you have proposed for the fraction of particles scattered by shell B which are emitted by shell A does not take into account the number of particles in each shell. In other words, it assumes that both shells have the same number of particles, which may not be the case in reality. This could lead to a discrepancy in your results.

Secondly, your calculation assumes that the particles are uniformly distributed in each shell. However, in reality, the distribution of particles may not be uniform and could vary depending on the system being studied. This could also contribute to the difference in your results.

Furthermore, your calculation does not take into account the interactions between particles in different shells. In other words, it assumes that the particles in each shell do not interact with each other, which may not be the case. This could also lead to a discrepancy in your results.

In order to accurately calculate the probability of particles being scattered in a spherical object, it is important to consider all these factors and possibly use a more sophisticated mathematical approach, such as partial differential equations, to model the system.

In conclusion, while your approach is commendable, it is important to carefully consider all the factors that may affect the results in order to accurately describe the behavior of particles in a spherical object according to Beers Law.
 

What is Beer's Law with Spherical Symmetry?

Beer's Law with Spherical Symmetry is a mathematical relationship that describes the absorption of light by a homogenous spherical particle. It states that the amount of light absorbed by the particle is directly proportional to the concentration of the particle and the path length of the light through the particle.

How is Beer's Law with Spherical Symmetry used in scientific research?

Beer's Law with Spherical Symmetry is used in various fields of science, such as chemistry, physics, and environmental science. It is commonly used to determine the concentration of a substance in a solution, as well as to study the properties of particles and their interactions with light.

What are the assumptions of Beer's Law with Spherical Symmetry?

The main assumptions of Beer's Law with Spherical Symmetry are that the particles are homogenous and spherical, the light source is monochromatic, and the particles are evenly distributed throughout the solution. It also assumes that there is no scattering of light by the particles.

What are the limitations of Beer's Law with Spherical Symmetry?

One limitation of Beer's Law with Spherical Symmetry is that it is only applicable to homogenous spherical particles. It may not accurately describe the absorption of light by particles with irregular shapes or non-uniform composition. Additionally, it assumes that the particles are non-interacting, which may not always be the case.

How can Beer's Law with Spherical Symmetry be experimentally verified?

To verify Beer's Law with Spherical Symmetry, one can perform a series of experiments with varying concentrations of a substance and measure the absorbance of light at different path lengths. The resulting data should show a linear relationship, confirming the validity of the law. Additionally, the experimental results can be compared to the theoretical predictions based on the law's equations.

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