Formulas physical organic
chemistry 1
Rate of reaction aA + bB + cC pP
v = kr[A]a[B]b[C]c
for reaction R P
zeroth order
integrated rate law
[R]t = [R]0 – krt
First order
d [R ]
Change in R over time = -kr[R]
dt
[R]t = [R]0e-krt
t1/2 = (ln2)/kr
second order
integrated rate law
[R]t ¿ ¿
t1/2 = 1/k[R]0
half life of order n:
(2n−1−1)
t1/2 =
( n−1 ) k r ¿ ¿
for reaction A + B P
integrated rate law: ln ¿ ¿ = ([B]0-[A]0)krt
For A B C (with ka and kb)
rate laws:
d[A]
rate (vA) = = -ka[A]
dt
d [B]
rate (vB) = = ka[A] – kb[B}
dt
d [C ]
rate (vC) = = kb[B]
dt
integrals
[A] = e-kat[A]0
[B] = k a ( e−k t−e−k t ) ¿ ¿
a b
(
k a e−k t −k b e−k t
)
a b
[C] = 1+ ¿
k b −k a
for reaction A + B ⇋ C P (with ka, ka’ and kb and equilibrium constant K)
d [P ] ka
fast equilibrium ka, ka’ >> kb: = kb[C] = kbK [A][B] = kb ' [A][B] = kobs [A][B]
dt ka
, [C ]
as K = such that [C] = K[A][B]
[ A ] [B]
d [P ] ka
slow equilibrium ka, ka’ << kb: = kb[C] = kb ' [A][B] = kobs[A][B]
dt k a +k b
as ka[A][B] – ka’[C] – kb[C] = 0
k a [ A ] [B]
such that [C] = '
k a +k b
relaxation of an equilibrium A ⇋ B after a temperature jump
with relaxation time: τ = 1/(kr + kr’) x = x0e-t/τ
Arrhenius equation: kr = Ae-Ea/RT
Arrhenius plot: ln ( ) (
kt 2
kt 1
=
Ea 1
−
R T1 T2
1
)
dln(k r )
activation energy: Ea = RT2( )
dT
catalysis (with kcat)
exp( −ERT ) exp E −E
a , cat
k cat
=
exp (
( ) =
a a ,cat
RT )
kr −E RT
a
Condensation polymerisation
fraction of condensed groups p = k r t ¿ ¿
1
degree of polymerisation: 〈N〉 = = 1 + krt[A]0
1− p
Chain polymerisation
rate of polymerisation: vp = kr[In]1/2[M] = kp [M][M•]
kinetic chain length if SSA in M•: λ = kr[M][In]-1/2 with kr = kp(4fkikt)-1/2
degree of polymerisation: <N> = 2kr[M][In]-1/2
( )
1/ 2
f ki
SSA in M• yields vp = kp [M] [In]1/2
kt
Enzymatic reactions: E + S ⇋ ES P + E with ka, ka’, kb = kcat
k a k b [ E ] [S ]
Vp = = kcat[ES] = Michaelis-Menten equation
k 'a + k b
v max
Michaelis-Menten equation: v= KM
1+ ¿
¿¿
k 'a +k b
with KM =
ka
and Vmax = kb[E]0
chemistry 1
Rate of reaction aA + bB + cC pP
v = kr[A]a[B]b[C]c
for reaction R P
zeroth order
integrated rate law
[R]t = [R]0 – krt
First order
d [R ]
Change in R over time = -kr[R]
dt
[R]t = [R]0e-krt
t1/2 = (ln2)/kr
second order
integrated rate law
[R]t ¿ ¿
t1/2 = 1/k[R]0
half life of order n:
(2n−1−1)
t1/2 =
( n−1 ) k r ¿ ¿
for reaction A + B P
integrated rate law: ln ¿ ¿ = ([B]0-[A]0)krt
For A B C (with ka and kb)
rate laws:
d[A]
rate (vA) = = -ka[A]
dt
d [B]
rate (vB) = = ka[A] – kb[B}
dt
d [C ]
rate (vC) = = kb[B]
dt
integrals
[A] = e-kat[A]0
[B] = k a ( e−k t−e−k t ) ¿ ¿
a b
(
k a e−k t −k b e−k t
)
a b
[C] = 1+ ¿
k b −k a
for reaction A + B ⇋ C P (with ka, ka’ and kb and equilibrium constant K)
d [P ] ka
fast equilibrium ka, ka’ >> kb: = kb[C] = kbK [A][B] = kb ' [A][B] = kobs [A][B]
dt ka
, [C ]
as K = such that [C] = K[A][B]
[ A ] [B]
d [P ] ka
slow equilibrium ka, ka’ << kb: = kb[C] = kb ' [A][B] = kobs[A][B]
dt k a +k b
as ka[A][B] – ka’[C] – kb[C] = 0
k a [ A ] [B]
such that [C] = '
k a +k b
relaxation of an equilibrium A ⇋ B after a temperature jump
with relaxation time: τ = 1/(kr + kr’) x = x0e-t/τ
Arrhenius equation: kr = Ae-Ea/RT
Arrhenius plot: ln ( ) (
kt 2
kt 1
=
Ea 1
−
R T1 T2
1
)
dln(k r )
activation energy: Ea = RT2( )
dT
catalysis (with kcat)
exp( −ERT ) exp E −E
a , cat
k cat
=
exp (
( ) =
a a ,cat
RT )
kr −E RT
a
Condensation polymerisation
fraction of condensed groups p = k r t ¿ ¿
1
degree of polymerisation: 〈N〉 = = 1 + krt[A]0
1− p
Chain polymerisation
rate of polymerisation: vp = kr[In]1/2[M] = kp [M][M•]
kinetic chain length if SSA in M•: λ = kr[M][In]-1/2 with kr = kp(4fkikt)-1/2
degree of polymerisation: <N> = 2kr[M][In]-1/2
( )
1/ 2
f ki
SSA in M• yields vp = kp [M] [In]1/2
kt
Enzymatic reactions: E + S ⇋ ES P + E with ka, ka’, kb = kcat
k a k b [ E ] [S ]
Vp = = kcat[ES] = Michaelis-Menten equation
k 'a + k b
v max
Michaelis-Menten equation: v= KM
1+ ¿
¿¿
k 'a +k b
with KM =
ka
and Vmax = kb[E]0
