1.6 Photochemical Kinetics
15
This book is mainly devoted to dynamics, but we shall also present some basics of
photochemical kinetics in order to bridge the gap with the macroscopic description.
1.6.1 Excitation Rate
The molecular excitation by light absorption, if the irradiance is not too high, consists
of relatively rare transitions to an excited state, the interval between two events being
much longer than the time needed for the molecule to get back to the ground state.
Let us consider a thin layer of any homogeneous material, crossed perpendicularly
by a beam of light. Each molecule in the layer has got a probability P(ν)dνdt of
absorbing a photon of frequency within the frequency interval [ν, ν + dν] during the
time dt. P is proportional to the number of photons that happen to be close to the
molecule during a time unit, i.e., to the spectral photon irradiance:
P(ν) dν dt = σ (ν) I ph,ν (ν) dν dt .
(1.46)
Since both sides of this equation are pure numbers, σ is a surface area, called the
“absorption cross section” of the molecule. The cross section is not directly related
to any geometric section of the molecule and is usually much smaller, but it is an
extensive quantity (a protein molecule absorbs much more than a single amino acid).
For small molecules σ can be of the order of 10
−20 m
2 or much less, depending on
the frequency. In going through a layer of thickness dl, the rate of photon absorption
in the interval [ν, ν + dν] will be
N S dl P(ν) dν = N S dl σ (ν) I ph,ν (ν) dν
(1.47)
where N is the number density of molecules, S is the considered layer surface, and
then N Sdl is the number of molecules in that portion of the layer. Since this is a
fraction of the photons within the frequency interval dν that are going through the
surface S, we see that the irradiance decreases by
d I ph,ν = −N σ (ν) I ph,ν (ν) dl .
(1.48)
This differential equation can be integrated to yield the expression of the irradiance
as a function of the length l of the pathway covered by the light in the absorbing
medium, called the “optical pathway”:
I ph,ν (ν, l) = I ph,ν (ν, 0) e
−N σ (ν) l
(1.49)
where I ph,ν (ν, 0) is the irradiance of the incident light. This is the Lambert–Beer law,
most often written in terms of molarity and energy irradiance, and with a base 10
exponential (instead of number density, photon irradiance, and base e, respectively):
15
This book is mainly devoted to dynamics, but we shall also present some basics of
photochemical kinetics in order to bridge the gap with the macroscopic description.
1.6.1 Excitation Rate
The molecular excitation by light absorption, if the irradiance is not too high, consists
of relatively rare transitions to an excited state, the interval between two events being
much longer than the time needed for the molecule to get back to the ground state.
Let us consider a thin layer of any homogeneous material, crossed perpendicularly
by a beam of light. Each molecule in the layer has got a probability P(ν)dνdt of
absorbing a photon of frequency within the frequency interval [ν, ν + dν] during the
time dt. P is proportional to the number of photons that happen to be close to the
molecule during a time unit, i.e., to the spectral photon irradiance:
P(ν) dν dt = σ (ν) I ph,ν (ν) dν dt .
(1.46)
Since both sides of this equation are pure numbers, σ is a surface area, called the
“absorption cross section” of the molecule. The cross section is not directly related
to any geometric section of the molecule and is usually much smaller, but it is an
extensive quantity (a protein molecule absorbs much more than a single amino acid).
For small molecules σ can be of the order of 10
−20 m
2 or much less, depending on
the frequency. In going through a layer of thickness dl, the rate of photon absorption
in the interval [ν, ν + dν] will be
N S dl P(ν) dν = N S dl σ (ν) I ph,ν (ν) dν
(1.47)
where N is the number density of molecules, S is the considered layer surface, and
then N Sdl is the number of molecules in that portion of the layer. Since this is a
fraction of the photons within the frequency interval dν that are going through the
surface S, we see that the irradiance decreases by
d I ph,ν = −N σ (ν) I ph,ν (ν) dl .
(1.48)
This differential equation can be integrated to yield the expression of the irradiance
as a function of the length l of the pathway covered by the light in the absorbing
medium, called the “optical pathway”:
I ph,ν (ν, l) = I ph,ν (ν, 0) e
−N σ (ν) l
(1.49)
where I ph,ν (ν, 0) is the irradiance of the incident light. This is the Lambert–Beer law,
most often written in terms of molarity and energy irradiance, and with a base 10
exponential (instead of number density, photon irradiance, and base e, respectively):