6.1 Gas-Phase Collisions
181
is estimated to be 10
7 times slower than the rotational one. In less extreme cases,
however, collisions thermally equilibrate all kinds of nuclear motion within ≈10
−8 s
at standard temperature and pressure.
6.2 Encounters in Solution
The diffusion of a solute in a liquid is slower than that of a component in a gas mixture.
On the other hand, when two solute molecules are brought in contact, they normally
interact for a longer time than two collision partners: in this sense, an encounter in
solution can be much more effective than a gas-phase collision in causing an energy
or charge transfer process. Both the rate of encounters and the probability P bim that
a given bimolecular event takes place at each encounter concur in determining the
rate of the process.
If we assume diffusion to obey Fick’s law, the net number of molecules that cross
a surface S per unit time is
Φ = −D
dN
dz
S
(6.6)
where N is the concentration (molecules/m
3 ), z is a coordinate perpendicular to the
surface, and D (m
2 /s) is the diffusion coefficient, which depends on the properties
of the solute and of the solvent, and on temperature. This law is valid in the absence
of forces pushing the molecules in a given direction. We first focus our attention on
one molecule Y and investigate how many molecules X per unit time interact with
it. Since both X and Y diffuse, the relevant D coefficient in our case is the sum of
their diffusion coefficients: D = D X + D Y . If we take a spherical surface of radius
R and center in Y , the net flux of X molecules entering the sphere is
Φ = 4π D R
2 dN X
d R
.
(6.7)
Here we are assuming that no potential term attracts X toward Y or repels it, until
they get to a sufficiently small “interaction distance” R int . This is not the case with
long-range electrostatic forces, for instance when dealing with ions (see the Debye–
Huckel theory of electrolytes). Φ is the difference between the ingoing flux Φ in and
the outgoing one Φ out : Φ = Φ in − Φ out . At R > R int , the steady-state concentration of X would be constant, unless a bimolecular process occurring at R ≤ R int
“annihilates” the X molecules: this sink creates a concentration gradient and a net
flux. By “annihilates” we mean that the X molecules are identified by their chemical
nature and quantum state, so they cease to exist if any reaction, CT or ET process,
converts them into different products (other chemical species and/or different electronic states). The fate of the products is irrelevant to the discussion of the diffusion
process (this statement implies the assumption that the products do not affect the
diffusion of the reagents). If some X molecules can “survive” the encounter with Y
181
is estimated to be 10
7 times slower than the rotational one. In less extreme cases,
however, collisions thermally equilibrate all kinds of nuclear motion within ≈10
−8 s
at standard temperature and pressure.
6.2 Encounters in Solution
The diffusion of a solute in a liquid is slower than that of a component in a gas mixture.
On the other hand, when two solute molecules are brought in contact, they normally
interact for a longer time than two collision partners: in this sense, an encounter in
solution can be much more effective than a gas-phase collision in causing an energy
or charge transfer process. Both the rate of encounters and the probability P bim that
a given bimolecular event takes place at each encounter concur in determining the
rate of the process.
If we assume diffusion to obey Fick’s law, the net number of molecules that cross
a surface S per unit time is
Φ = −D
dN
dz
S
(6.6)
where N is the concentration (molecules/m
3 ), z is a coordinate perpendicular to the
surface, and D (m
2 /s) is the diffusion coefficient, which depends on the properties
of the solute and of the solvent, and on temperature. This law is valid in the absence
of forces pushing the molecules in a given direction. We first focus our attention on
one molecule Y and investigate how many molecules X per unit time interact with
it. Since both X and Y diffuse, the relevant D coefficient in our case is the sum of
their diffusion coefficients: D = D X + D Y . If we take a spherical surface of radius
R and center in Y , the net flux of X molecules entering the sphere is
Φ = 4π D R
2 dN X
d R
.
(6.7)
Here we are assuming that no potential term attracts X toward Y or repels it, until
they get to a sufficiently small “interaction distance” R int . This is not the case with
long-range electrostatic forces, for instance when dealing with ions (see the Debye–
Huckel theory of electrolytes). Φ is the difference between the ingoing flux Φ in and
the outgoing one Φ out : Φ = Φ in − Φ out . At R > R int , the steady-state concentration of X would be constant, unless a bimolecular process occurring at R ≤ R int
“annihilates” the X molecules: this sink creates a concentration gradient and a net
flux. By “annihilates” we mean that the X molecules are identified by their chemical
nature and quantum state, so they cease to exist if any reaction, CT or ET process,
converts them into different products (other chemical species and/or different electronic states). The fate of the products is irrelevant to the discussion of the diffusion
process (this statement implies the assumption that the products do not affect the
diffusion of the reagents). If some X molecules can “survive” the encounter with Y