# Dimensional Analysis: Forster Energy Transfer equation

• HAYAO
In summary, the original author created an equation to calculate FRET rate, but the unit of measure was not specified. The equation has been simplified to show only the rate in seconds.
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?

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!

Orodruin

## 1. What is dimensional analysis?

Dimensional analysis is a mathematical method used to analyze and understand the relationships between different physical quantities. It involves converting units and examining the dimensions of the quantities involved in a problem to determine how they are related.

## 2. What is the Forster Energy Transfer equation?

The Forster Energy Transfer equation is a formula used to calculate the rate of energy transfer between two molecules. It takes into account the distance between the molecules, the orientation of their dipoles, and the spectral overlap of their emission and absorption spectra.

## 3. How is dimensional analysis used in the Forster Energy Transfer equation?

Dimensional analysis is used in the Forster Energy Transfer equation to ensure that the units of the quantities involved are consistent. This allows for a more accurate calculation of the rate of energy transfer between molecules.

## 4. What are the applications of the Forster Energy Transfer equation?

The Forster Energy Transfer equation is commonly used in the fields of chemistry, physics, and biology to study processes such as fluorescence, photosynthesis, and protein-protein interactions. It is also used in the development of technologies such as solar cells and biosensors.

## 5. Are there any limitations to using dimensional analysis and the Forster Energy Transfer equation?

While dimensional analysis is a useful tool, it does not take into account certain factors such as temperature and pressure, which can affect the rate of energy transfer. Additionally, the Forster Energy Transfer equation assumes that the molecules are in a perfect orientation and that the transfer occurs instantaneously, which may not always be the case in real-world situations.

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