Table of contents:
- Glossary of terms
Second Order Reactions are characterized
by the property that their rate is proportional to the product of two reactant
concentrations (or the square of one concentration). Suppose that A --->
products is second order in A, or suppose that A + B ---> products
is first order in A and also first order in B. Then the differential rate
laws in these two cases are given by Differential
d[A]/dt = -k [A]2
(for 2A ---> products)
= -k [A][B] (for A + B ---> products)
In mathematical language, these are
first order differential equations because they contain the first
derivative and no higher derivatives. A chemist calls them second order
rate laws because the rate is proportional to the product of two concentrations.
By elementary integration of these differential equations Integrated
Rate Laws can be obtained:
1/[A] - 1/[A]0 = k t
(for 2A ---> products)
or (1/(a-b)) [ln((a-x)/(b-x))-ln(a/b)]
= k t (for A + B ---> products)
where a and b are the initial concentrations
of A and B (assuming a not equal to b), and x
is the extent of reaction at time t. Note that the latter can also
(a-x)/(b-x) = (a/b)exp[(a-b)kt].
A common way for a chemist to discover
that a reaction follows second order kinetics is to plot 1/[A] versus the
time in the former case, or ln(b(a-x)/a(b-x)
versus t in the latter case.
Data Analysis: 1/[A] = 1/[A]0
+ k t
A plot of 1/[A] versus t is a straight
line with slope k.
Software tools for second order reactions
Computer software tools can be used to solve chemical kinetics problems.
In second order reactions it is often useful to plot and fit a straight
line to data. One tool for this is the "slope(x,y)" command in
the product MathCad. Here is a mathcad
file that can serve as template for second
order kinetics data analysis.
1: Ammonium cyanate, NH4CNO, in water solution gradually
isomerizes to urea, H2NCONH2 according to the reaction:
NH4CNO ---> H2NCONH2 . A solution was
prepared by dissolving 22.9 g of ammonium cyanate in enough water to make
1.00 liter of solution. After times t had elapsed, portions of the solution
were analysed and converted into the mass of urea that had formed in the
entire solution. The results are tabulated here.
Using the graph below verify that this is a second order reaction and
calculate the rate constant.
2: A certain chemical reaction follows the stoichiometric equation
A + 2B --->
Measured rates of formation of the product,
Z, are shown for several concentrations of reactants, A and B:
||rate/mole liter-1 sec-1
|2.5 x 10-2
||3.3 x 10-3
||1.0 x 10-2
|5.0 x 10-2
||6.6 x 10-3
||4.0 x 10-2
|5.0 x 10-2
||1.32 x 10-2
||8.0 x 10-2
Assuming a differential rate law
of the form
rate = k [A]a
what is the value of a (the
order of reaction with respect to A), what is b (the order of reaction
with respect to B) and what is the value of k (the rate constant)?
3: Solutions of A=H3COC6H4CNO
in carbon tetrachloride dimerize slowly as shown by the following data
Determine the order of the reaction
and find the rate constant.
Glossary of Terms
Stoichiometry determines the molar ratios
of reactants and products in an overall chemical reaction. We express the
stoichiometry as a balanced chemical equation. For kinetics it is convient
to write this as products minus reactants: npP + nqQ
- naA - nbB (instead of the conventional equation
naA + nbB ---> npP + nqQ).
This indicates that na and nb moles of reactants
A and B, resp., produce np and nq moles of products
P and Q.
The rate of a chemical reaction is defined
in such a way that it is independent of which reactant or product is monitored.
We define the rate, v, of a reaction to be v = (1/ng) d[G]/dt
where ng is the signed (positive for products, negative for
reactants) stoichiometric coefficient of species G in the reaction. Namely,
v = (-1/na) d[A]/dt = (1/np) d[P]/dt, etc.
It is convenient to refer to the extent of
reaction. As the reactants are sonsumed and the products are produced,
their concentrations change. If the initial concentrations of A, B, P and
Q are [A], [B], [P] and [Q], resp., then the extent of reaction is defined:
x = -([A]-[A]0)/na = -([B]
- [B]0)/nb = ([P]-[P]0)/np
= ([Q]-[Q]0)/nq. Alternately, each species concentration
is a function of the extent of reaction: [A] = [A]0 - nax,
Many reactions follow elementary differential rate
laws such as v = k f([A], [B], ...) where f([A], [B], ...) is a function
of the concentrations of reactants and products. That is, the rate varies
as the concentrations change. A proportionality constant, k, is called
the rate constant of the reaction.
When the rate law has the special form of a product
(or quotient) of powers, f([A], [B], ...) = [A]a [B]b
[P]p [Q]q then a is the order
of the reaction with respect to A, b is the order w.r.t. B, etc.
Note that order may be positive, negative, integer, or non-integer. Further,
the sum a + b + p + q is the overall order of the reaction rate
NOTE: there is no necessary relation between
orders and stoichiometric coefficients. That is, a might differ
Reaction rate constants are usually temperature
dependent; the rate of a reaction usually increases as the temperature
rises. The temperature dependence often follows Arrhenius' equation: k(T)
= A exp(-Ea/RT) where T is the absolute temperature, R the universal gas
constant, Ea is the activation energy (specific to each reaction), and
A is the "pre-exponential" or "frequency" or "entropy"
One objective of chemical kinetics is to solve
the differential rate law d[G]/dt = k f([A], [B], ...), and thereby express
each species concentration as a function of time: [G](t). Since solution
requires integration, we call it the integrated rate law.
A reaction mechanism is a set of steps at the
molecular level. Each step involves combinations or re-arrangements of
individual molecular species. The steps in combination describe the path
or route that reactant molecules follow to reach the product molecules.
The result of all steps is to produce the overall balanced stoichiometric
chemical equation for reactants producing products.