Second Order and Non-integer Order
Reactions
Second Order Reactions
The change in A with respect to time is a function of the
square of A:
Increasing A:
dA/dt = r = k A2
A
t
dt
dA/A2 = k
A0
0
1/A – 1/A0 = - k t
Decreasing A:
A
t
dt
dA/A2 = -k
A0
0
1/A – 1/A0 = k t
Increasing A:
1/A – 1/A0 = - k t
1/A = - k t + 1/A0
y = m x + b
Concnetration
0.1
0.11
0.125
0.148
0.25
0.4
1/A
10
9
8
7
4
2.5
12
10
8
1/A
Time
0
10
20
30
60
75
y = -0.1x + 10
R2 = 1
6
4
2
0
0
20
40
t
60
80
Example
An ozone generator produces ozone based on a reaction
that can be modeled based on second order kinetics. If
the ozone concentration in a container is 0.0015 g/m3
before the generator is turned on and 0.002 g/m3 after 1
minute, what would the ozone concentration be after 3
minutes?
Conc., mg/L
1/C
0.0015
666.6667
0.002
500
700
600
500
1/C
TIME
0
1
400
y = -166.67x + 666.67
R2 = 1
300
200
100
k = 167 m3/g.min
0
0
0.2
0.4
0.6
t
0.8
1
1.2
1/A = - k t + 1/A0
1/A = (-167)*3 + (1/0.0015)
1/A = 166
A = .006 g/m3
Non-integer Order Reactions
dA/dt = r = - k An
If A = A0 when t =0, and n is any number not equal to one:
(A/A0)1-n –1 = [(n-1) k t]/A0(1-n)
k can be found in the same way as in the other cases
Half-Life
A half life is the amount of time that it takes for ½ of
a substance to decay.
In other words when t = t1/2, C = 0.5 C0.
Zero Order
C = C0 – k t
t = (C0 –C)/k
At half life:
First Order
t1/2 = (C0 – 0.5 C0)/k = 0.5 C0/k
ln(C/C0) = - k t
t = ln(C0/C)/k
At half life:
t1/2 = ln(C0/0.5C0)/k = ln 2/k = 0.693/k
By extension:
Second Order
t1/2 = 1/[k(A0)]
Integer Order
t1/2 = [((1/2)1-n – 1) – (A0)]/[(n-1)k]
Radioactive Decay
Radioactive elements disintegrate spontaneously. When this
happens high velocity particles are emitted from their nuclei.
In natural radioactivity two kinds of particles may be given off,
alpha-particles (two neutrons and two protons, a helium
nucleus) and beta-particles (high velocity electrons)
In addition to these two types of particles natural radioactivity
is frequently accompanied by the emission of gamma-rays. A
gamma-ray is an electromagnetic wave that is more penetrating
than alpha- and beta-particles. They are like x-rays, but have
a shorter wavelength.
Radioactive Decay Example
Radium - Ra
226 Ra
88
226 – mass number – protons + neutrons
88 – atomic number - protons
226
88Ra
222
86Rn
+
4
2He
(alpha-particle)
Rn - Radon
222
86Rn
222
86Rn
+ (
(gamma-ra
Radon gas
226
88Ra
222
86Rn +
4
2He
(alpha-particle)
This is the reaction that is occurring when we consider
the half-life of Radium
The half-life of Radium is 1590 years. In other words, after
1590 years one half of a sample of Radium will be unchanged
and one half will be converted to Radon
At the end of a second 1590 year period (3180 years) one
quarter of the original amount of Radium will remain
The decay of Radium is used to provide radiation treatments to
some cancer patients. As you can see, it emits radiation which
can be hazardous to our health. So it has to be disposed of
properly. Since there is not treatment to reduce radioactivity
the only thing that can be done with it is to isolate it from the
environment. The waste is considered “safe” (radioactivity is
near normal background levels) after 99.9% has decayed.
How long will this take?
If A0 = 1, then A = 0.001 when it is reduced by 99.9%
Radioactive decay is a first order reaction, so:
t1/2 = 0.693/k
1590 = 0.693/k
k = 0.693/1590 = 4.358 x 10-4 years-1
We also know that for first order kinetics:
ln (A/A0) = -k t
Ln (0.001/1) = (-4.358 x 10-4) t
t = -6.9078/-4.358 x 10-4 = 15,850 years !
More Complex Reactions
Consecutive Reactions
A
k1
B
k2
C
d[A]/dt = -k1[A]
d[B]/dt = k1[A] - k2[B]
This system of differential equations can be solved for
various initial conditions each of which will give different
solutions.
A
k1
B
k2
C
One of the solutions is:
[B] = [(k1 A0)/(k2 – k1)] (e-k1 t – e-k2 t) + B0 e-k2 t
This equation is used to model the dissolved oxygen level in a
stream. In this case, the oxygen deficit (B) is increased as the
result of oxygen being consumed by microorganisms (A B),
and the oxygen deficit is decreased by the diffusion of oxygen
into the water from the atmosphere (B C)