Question about interpreting absorption cross sections

[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.

Thanks for your help!

 

[Valerie Park]

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!