# Dimensional Analysis: Forster Energy Transfer equation

• HAYAO
Gold Member
Homework Statement
I can't seem to get it right.
Relevant Equations
$W_{ET}=\frac{9000\cdot c^{4}\cdot ln10}{128\pi ^{5}\tau _{D}N_{A}n^{4}}\frac{\kappa ^{2}}{R^{6}}\int_{0}^{\infty }\frac{f_{D}(\nu )\varepsilon _{A}(\nu )}{\nu ^{4}}d\nu$
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?

Staff Emeritus
Homework Helper
Gold Member
fD(ν) is the normalized emission intensity (dimensionless)
What is the definition of the normalized emission intensity?