Dimensional Analysis: Forster Energy Transfer equation

AI Thread Summary
The discussion focuses on calculating Forster's Resonance Energy Transfer (FRET) rate and resolving unit discrepancies. The original equation indicates that the rate should have units of "s-1," but the user struggles with dimensional analysis. A key misunderstanding was identified regarding the normalized emission intensity, which is not dimensionless but has units of seconds when divided by the total area of the spectrum. This realization helped clarify the calculations, leading to a resolution of the user's confusion. The exchange highlights the importance of accurate unit definitions in scientific calculations.
HAYAO
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Homework Statement
I can't seem to get it right.
Relevant Equations
[itex]W_{ET}=\frac{9000\cdot c^{4}\cdot ln10}{128\pi ^{5}\tau _{D}N_{A}n^{4}}\frac{\kappa ^{2}}{R^{6}}\int_{0}^{\infty }\frac{f_{D}(\nu )\varepsilon _{A}(\nu )}{\nu ^{4}}d\nu [/itex]
I'm trying to calculate Forster's Resonance Energy Transfer rate, but I just can't seem to get the units right. I'm trying to teach my students how to calculate them.

Here is the (relatively) original technical note of FRET equation, made by the original author:
https://www.osti.gov/servlets/purl/4626886

Page 55 shows the original equation. The rate is supposed to have the unit of "s-1".
Constants:
\nu is the frequency of light in s-1
c is the speed of light in m s-1
N_{A} is the Avogadro constant in mol-1
n is the refractive index (dimensionless)
\kappa is the dipole-dipole orientation factor (dimensionless)
\tau _{D} is the excited state lifetime of the donor in s
R is the distance between donor and acceptor
f_{D}(\nu ) is the normalized emission intensity (dimensionless)
\varepsilon _{A}(\nu ) is the absorption coefficient in mol-1 m3 m-1

(Note1: the original equation shows "9" instead of "9000" as I put above. This is because the technical note uses Avogadro constant of #of molecules per millimole instead of molecules per mole.)
If I do a dimensional analysis, it would look like this:
\frac{(m s^{-1})^{4}}{(s)(mol^{-1})}\frac{1}{(m)^{6}}\int \frac{(mol^{-1}m^{3}m^{-1})}{(s^{-1})^{4}}d\nu
\frac{(m^{4} s^{-4})}{(s)(mol^{-1})}\frac{1}{(m^{6})}\int \frac{(mol^{-1}m^{2})}{(s^{-4})}d\nu
\frac{(s^{-5})}{(mol^{-1})(m^{2})}\int (mol^{-1}m^{2}s^{4})d\nu
\frac{(s^{-5})}{(mol^{-1})(m^{2})}(mol^{-1}m^{2}s^{3})
s^{-2}

The rate is supposed to have the unit of "s-1". I must be making some silly mistake here, but I just can't seem to find it. Could somebody point out what I did wrong?
 
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HAYAO said:
fD(ν) is the normalized emission intensity (dimensionless)
What is the definition of the normalized emission intensity?
 
Orodruin said:
What is the definition of the normalized emission intensity?
Oh snap. You're right. It's not dimensionless. I divide each intensity at certain frequency by the total area of the spectrum, which means the unit is in seconds. Silly me.

You solved the problem for me. Thanks!
 
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