Zero and First-Order
Rate Reactions
Samir Kumar Khanal, Ph.D.
Department of Civil, Construction and
Environmental Engineering
Iowa State University
Question!!
Assume you are a process engineer (biological), newly recruited
by P & G. In your first week of job, you have been assigned
to assist in the design of a bioreactor for growing edible fungi
on synthetic growth medium to produce light-weight protein diet
for astronauts. How are you going to design? Specifically what
data do you need?
Call your professor?
Ask the senior engineer?
Surf the internet?
Consult BSE 482: lecture notes
Before you start designing a bioreactor
(fermentor), you must have clear understanding
of the followings:
How fast the fungi are able to convert the organics
into protein? That is bioconversion rate or reaction
rate or kinetics.
(C2-C1)/(t2-t1) = dC/dt
When is the reaction going to be over? “t”
Which bioreactor configuration would be ideal?
Suspended growth
Attached growth
At the end of this class, you should be able to
 define biochemical kinetics, reaction rate and order
 derive the rate of a reaction in terms of the
appearance of products or disappearance of reactants
 describe the basic factors that influence the rate
of a reaction
 integrate the rate laws for 0, and 1st order reactions
 determine the rates and orders of the biochemical
reactions
 explain the practical significance of reaction rate and
order
Reaction Rates
• Definition
– change in concentration of a reactant or
product with time
rate 
C
t

C 2  C1
t 2  t1
• The rate will be negative (-) for reactants
• The rate will be positive (+) for products
Reactants: mostly pollutants (we want to get rid of), e.g. nitrate,
phosphate, organics, pesticides, etc.
Products: mostly value-added commodities (we want to produce),
e.g. protein, lactic acid, enzymes, nisin, yeast, etc.
The basic requirements are:
1. A good thermostat as rates change
with temperature
C1
-dC/dt= (C1-C2)/(t2-t1)
C1
C2
(A) The rate of decrease in concentration
of a reactant, or
(B) The rate of increase in concentration
of the products.
Concentration mg/L
Determination of biological reaction rate:
C2
t1
3. A method of determining the
concentration of reactant or product.
Determined by measuring the
concentration of a reactant or
product as a function of time
during the course of a
biological reaction
Concentration mg/L
2. An accurate timing device (stopwatch)
t2
C2
C1
dC/dt= (C2-C1)/(t2-t1)
t1
t2
Factors affecting the speed or rate of a
biological reaction
 Concentration
 Temperature
 Presence of a macro/micro-nutrients
 Physical state of reactants
Effect of Concentration on Reaction Rate
Rate  ( Concentrat ion )
n = reaction order usually
an integer (e.g. 0, 1, 2)
n
ln rate  n ln concentrat ion

2nd order
n =2
The order of a reaction
refers to the powers to
which concentration are
raised
1st order
2
n =1
ln (rate)
1
1
1
Zero order
n=0
ln (conc.)
“A second-order
“A first-order
zero-order
reaction is one in
reaction is one in
which the rate of
which the rate of
reaction is directly
reaction is directly
proportional to the
independent to
proportional
of
square of the
concentration.”
concentration.”
What does reaction order tell us??
Relationship between rate and
concentration!
How the amount of compound speeds
up or retards the reaction rate!
Zero-Order Reactions
d [C ]
 [C ]

n
dt
d [C ]
 k [C ]
n
k = rate constant
dt
For zero-order reaction, n = 0

d [C ]
k
dt

C
C0
t
dC   k  dt
0
Negative means, [C] decreases
with time
unit of k is mass volume-1
time-1
C  C 0   kt
C  C 0  kt
If [C] increases with time (for
product formation)
Graphical representation of zero-order reaction
C  C 0   kt 
C  C 0  kt
y  c  mx
Reaction rate (slope) remains constant
Slope = -k
Zero-order reactions: not
very common in biological
engineering
C
C0
Time
Some examples of zero-order reactions
 Biodegradation of 2,4-D (2,4-Dichlorophenoxyacetic acid)
 Ammonia oxidation to nitrite
 Biodegradation of aromatic hydrocarbons in compost
 Phenol degradation by methanogens
First-Order Reactions
d [C ]
 [C ]
n
d [C ]

dt
 k [C ]
n
dt
For first-order reaction, n = 1

d [C ]
 kC
dt
ln
C
C0
k = rate constant
  kt
C
dC
C0
C

or
C
t
  k  dt
0
e
C0
ln C  ln C 0   kt or
Which is similar to a
straight line equation
 kt
Negative means, [C]
decreases with time
unit of k is time-1
  C  C0 e
 kt
ln C  ln C 0  kt
y  c  mx
Graphical representation of first-order reaction
ln C  ln C 0  kt
y  c  mx
C0
ln C0
C
ln C
First-order reactions: very common
in biological engineering
Some examples of first-order reactions
 Degradation of chlorinated compounds
 Microbial growth (bacteria/fungi)
 Oxidation of organic matter
O rg a n ic m a tte r (C ), m g /L
200
150
100
50
0
0
5
10
T im e , h r
15
Comparisons of zero and first-order reactions
Zero-Order
O rg a n ic m a tte r (C ), m g /L
200
First-Order
150
100
50
0
0
5
10
T im e , h r
1. How the reaction changes with time.
2. What about change in slope (k)?
3. What is the unit of k in each case?
4. What is the effect of concentration?
15
Example:
An engineering student was interested in the biodegradation of atrazine in an
aqueous environment, its reaction rate and order. She went to the lab and
conducted a series of batch tests in shaker flasks at 25oC using an
enriched microbial culture of Pseudomonas. During her experiments, she
collected data every alternative day. The data are shown in the table
below.
Time, days
0
Atrazine, mg/L
18
ln (C)
2.890
5
15
2.708
12
11
2.398
22
6.8
1.917
31
4.2
1.435
40
2.6
0.956
50
1.5
0.405
60
0.8
- 0.223
3 .5
y = - 0 .0 5 1 9 x + 2 .9 8 7 1
3
R
2 .5
15
2
= 0 .9 9 6 5
2
ln (C )
A tra zin e c o n c e n tra tio n , m g /L
20
10
1 .5
1
0 .5
5
0
0
-0 .5
0
20
40
60
0
10
Zero-order

k
30
T im e , d a y s
T im e , d a y s
d [C ]
20
First-order

d [C ]
 kC
dt
dt
C = Co- kt
ln (C) = ln (Co)- kt
y = 2.9871- 0.0519x
40
Effect of temperature on biological reaction rate
The effect of temperature on reaction rate is given by
the Arrhenius equation:
EA= activated energy, J/mol
R = Universal gas constant 8.31J/mol-K
T = Temperature in Kelvin = (oC + 273)
A = Constant (not significantly affected by small temp. change
What happens if you increase the temperature
by 10°C from, say, 20°C to 30°C (293 K to 303 K)?
Let's assume an activation energy, EA of 50,000 J mol-1.
gas constant,
R, is 8.31 J K-1 mol-1.
k  Ae
At 20°C (293 K), the value of the fraction:

e
E
RT
 e


E
RT
At 30°C (303 K), the fraction is:
50 , 000
8 . 31 x 293
 1 . 21 x10
9

e
E
RT


e
50 , 000
8 . 31 x 303
 2 . 38 x10
Rule of thumb: Rate of a reaction doubles for every 10 degree
rise in temperature
9
For biological reactions, this role will hold more or
less true up to a certain optimum temperature
kT2
Activities of mesophilic methanogens at different temperature
Temperature correction for rate constant
k T 2  k T 1
( T 2  T 1)
 : temperature-activity coefficient: 1.034 – 1.08