Environmental Engineering Reference
In-Depth Information
Hence,.the.probability.of.a.given.reaction.occurring.in.the.time.period.of.t+τ.to.t+τ
dτ.is.a.function.of.the.independent.probabilities.of.no.reaction.occurring.(
P
0
(τ)),.and.
the.probability.that.reaction.µ
.
will.occur.(
P
(τ,µ)).as.follows:
P
( ,
τ m
)
d
τ
=
P
( )
τ
⋅
P
( )
m
d
τ
0
.
.
(6.15)
(
)
=
P
( )
τ
⋅
h
⋅ ′
k d
τ
0
m
m
P
0
(τ). is. the. integration. of. the. negative. likelihood. of. a. reaction. occurring. within.
the. time. period. τ.. Because. µ. is. the. most. likely. reaction. at. time.
t
. and. deines. the.
duration.of.the.time-step.τ,.it.is.the.most.likely.and.only.reaction.to.occur.within.the.
deined.time-step..To.identify.and.deine.reaction.µ,.the.standard.limit.formula.can.
be.applied.for.all.possible.reactions.(m.=.{1,…,.M}).at.time.
t
.to.provide.a.relation.for.
P
0
(τ).as.follows:
M
∑
−
h k
′
τ
.
.
(6.16)
m m
(
)
=
P
( )
τ
=
1
−
P
( )
τ
e
m
=
1
0
Substituting.this.into.the.previous.probability.relationship.provides.an.expression.for.
the.probability.of.a.reaction.occurring.within.the.prescribed.τ.as.follows:
M
∑
.
−
h k
′
τ
.
(6.17)
m m
P
( ,
τ m
)
=
h k
′ ⋅
e
i
=1
m
m
Going. back. to. the. original. deining. probability. where.
P
1
(τ). is. deined. as. the.
probability.of.any.speciic.reaction.from.1.to.M.occurring.in.the.duration.of.τ.at.time.
t
,.it.can.now.be.deined.as.the.summation.of.
P
(τ,µ):
M
∑
P
( )
τ
=
P
( ,
τ m
)
1
1
i
i
=
1
.
.
(6.18)
M
M
∑
1
−
h k
i
′
τ
∑
Di
=
(
h k
′
)
⋅
e
i
=
i
i
i
=
1
The.derivation.of.τ.at.time.
t
.is.not.absolute,.but.rather.a.value.from.a.distribution.
of.time.intervals.based.on.the.respective.reaction.rates.for.the.M.reactions.possible,.
and.hence.can.be.simulated.as.follows:
1
1
τ
=
⋅
ln
M
r
.
.
(6.19)
∑
1
(
)
h k
′
m
D
m
m
=
1
