# Question about interpreting absorption cross sections

• excalibur313
In summary, the Beer-Lambert law can be used to calculate the time required for a certain percentage of molecules to absorb a photon. By plugging in the appropriate values for the absorption rate, cross section, and concentration, we can estimate how long it would take for an appreciable fraction of isomerization to occur in a monolayer of molecules.
excalibur313
Hi everyone,
I was wondering if I can get a little help interpreting absorption cross sections. I would like to figure out how long I would have to shine a particular light on a monolayer of molecules to expect an appreciable fraction of isomerization. Here is how far I've gotten so far:

Looking at absorption curves for the isomers of stilbene, I know that from 200-350 nm I'd expect an molar extinction coefficient of 1.6 l mol-1 cm-1 (summed over wavelengths) which should yield a cross section of 6.1x10-21 cm-2 using this:
http://en.wikipedia.org/wiki/Molar_absorptivity

So I think I can calculate a reasonable incoming photon flux from a Xe lamp:
10 mW m-2 nm-1 x 150 nm (200-350 nm range of light) x (1m/10^9 nm)^2 x 1mJ/1mW x J/100mJ = 1.5x10-20 J/nm^2

I can then figure out the number of 250 nm photons/nm2 s:
1.5x10^-20 J/nm^2 s x 1 photon/7.9x10^-19 J = .17 photons/nm^2 s

I can estimate the area of cis stilbene as 0.9 nm x 0.6 nm = 0.5 nm^2 which gives:
0.08 photons/stilbene s

So since I know how many photons are hitting stilbene per second, how can I take the cross section to estimate how long I'd have to wait for an absorption event? (Or even the fraction of photons absorb) I know that stilbene has a 50/50 shot of isomerizing once optically excited.

The key to answering this question is to use the Beer-Lambert law. The Beer-Lambert law states that the rate of absorption (A) is proportional to the concentration of the absorbing species (C) and the cross section of the absorbance (σ):A = σ * CWhere A is the rate of absorption, σ is the cross section, and C is the concentration.Using this equation, we can calculate how long it would take for the stilbene molecules to absorb a certain number of photons. For example, if you want to find out how long it would take for 50% of the stilbene molecules to absorb a photon, you would need to solve for t:A = σ * C * tt = A / (σ * C)where A is the rate of absorption, σ is the cross section, C is the concentration, and t is the time required for 50% of the molecules to absorb a photon.Plugging in the values we calculated earlier, we get:t = 1.5x10^-20 J/nm^2 s / (6.1x10^-21 cm^2 * 0.08 photons/stilbene s)t = 1.86x10^13 secondsSo it would take approximately 1.86x10^13 seconds or 5.8 million years for 50% of the stilbene molecules to absorb a photon!

## 1. What is an absorption cross section?

An absorption cross section is a measure of how likely a molecule or atom is to absorb a photon of a specific energy. It is typically represented by the symbol σ and is measured in units of area.

## 2. How is absorption cross section related to absorption coefficient?

The absorption cross section is related to the absorption coefficient by the equation α = Nσ, where N is the number density of absorbing particles. The absorption coefficient is a measure of the total amount of absorption occurring in a material, while the absorption cross section is a measure of the individual particles' ability to absorb photons.

## 3. What factors affect the value of absorption cross section?

The value of absorption cross section is affected by several factors, including the energy of the incident photons, the nature of the absorbing material, and the presence of other molecules or particles that may interfere with the absorption process.

## 4. How is absorption cross section measured?

Absorption cross section can be measured through various techniques, such as spectroscopy or laser-based methods. In spectroscopy, the absorption of light by a sample is measured at various wavelengths, and the absorption cross section can be calculated from the resulting data. In laser-based methods, the absorption of a laser beam by a sample is measured, and the absorption cross section can be determined from the properties of the laser and the sample.

## 5. What is the significance of absorption cross section in atmospheric science?

In atmospheric science, the absorption cross section is an important parameter used to understand the interactions of different gases with sunlight. It is particularly relevant in studies of climate change, as the absorption of certain gases in the atmosphere can impact the Earth's energy balance and contribute to the greenhouse effect. Measuring the absorption cross section of various gases can help scientists better understand and predict the effects of human activities on the Earth's climate.

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