Kinetics Overview

Chemical kinetics, also known as the kinetics of chemical reactions, is the study of rates and mechanisms of chemical reactions. Chemical kinetics describes the rate of the bond-breaking and bond-forming processes on the molecular level through several factors of investigations such as the concentrations of chemical species, temperature, and the presence of catalyst, etc.^[1]

Differential Rate Law

The differential rate law is the mathematical approach to determine the reaction rate by either the differential rate of loss of a reactant or the differential rate of formation of a product as a function of time.

Rate of conversion

Considering a homogeneous reaction,

$aA+bB+...\longrightarrow eE+fF+...$

$A,B,...,E,F,...$ are chemical species while $a,b,...,e,f,...$ are the coefficients in the balanced chemical equation. The rate of reaction can be driven by rate of either loss or formation of chemical species that are proportional to their coefficients, known as the rate of conversion ( $J$ ).^[2]

$J=-{1 \over a}{dn_{A} \over dt}=-{1 \over b}{dn_{B} \over dt}={1 \over e}{dn_{E} \over dt}={1 \over f}{dn_{F} \over dt}$

$n_{A}$ is the number of molecules in chemical species $A$ .

The negative sign of the rate of conversion ( $J$ ) describes the rate of loss of the reactant. On the other hand, the positive sign of the rate of conversion ( $J$ ) describes the rate of formation of the product. When the rate of conversion ( $J$ ) is equal to $0$ , the chemical reaction is at equilibrium, at which neither the reactant nor the product loses or forms, respectively.

Rate of reaction

The rate of reaction, $r$ , is an intensive quantity that depends on the concentration of chemical species, the temperature, and the pressure. Therefore, the rate of reaction is limited to the constant volume condition.^[2]

$r={J \over V}={1 \over V}(-{1 \over a}{dn_{A} \over dt})$

The rate of reaction is equal to the conversion rate per unit volume. At constant volume, the number of molecules per volume, $n_{A} \over V$ , is equal to the concentration of chemical species $A$ , $[c_{A}]$ . The unit is commonly known as $mol\ dm^{-3}\ s^{-1}$ .

$r={J \over V}=-{1 \over a}{d[c_{A}] \over dt}\qquad (const.V)$

Rate law

The rate law is a function of concentrations at a fixed temperature.^[2]

$r=k[A]^{\alpha }[B]^{\beta }...[Y]^{\gamma }$

$k$ is known as the rate constant, which is the function of both temperature and pressure (Pressure can be ignored since the contribution of pressure to the rate constant is significantly low). $[A],[B],...[Y]$ are the concentrations of chemical species. $\alpha ,\beta ,...\gamma$ are partial orders. Chemical Species $A$ has an order of $\alpha$ , etc. Hence the overall order of the chemical reaction can be denoted as $\alpha +\beta +...+\gamma$ . The unit of the rate constant is in $(dm^{-3}/mol)^{n-1}s^{-1}$ , while $n$ is equal to the overall order.

In chemical kinetics, the chemical reaction is composed of a number of different mechanisms before reaching the final product. The overall rate of the chemical reaction can be determined by the differential rate law of it's slowest mechanism, which is also known as the rate determining step. The rate determining step is a reaction mechanism that has the highest activation energy barrier. In homogeneous reaction, in which the reaction is driven out in one phase, the rate of reaction has a strong dependence on the concentrations of reactants. On the other hand, in heterogeneous reaction, in which the reaction is driven out in two or more phases, the rate of reaction might depend on the concentrations of intermediates. However, the intermediate state is reached in small amounts in limited amount of time.^[1] Hence, steady-state approximation can be carried out to determine the overall rate of heterogeneous reaction.

Table 1: Differential Rate Laws For Commonly Occurring Chemical Kinetics^[1]
Reaction Type	Differential Rate Law
$A\longrightarrow P(Zero\ Order)$	$-{d[A] \over dt}=k$
$A\longrightarrow P(1st\ Order\ Irreversible)$	$-{d[A] \over dt}=k[A]$
$A\longleftrightarrow P(1st\ Order\ Equilibrium)$	$-{d[A] \over dt}=k_{1}[A]-k_{-1}[A]$
$2A\longrightarrow P(2nd\ Order\ Irreversible)$	$-{d[A] \over dt}=k[A]^{2}$
$A+B\longrightarrow P(2nd\ Order)$	$-{d[A] \over dt}=k[A][B]$
$A+B\longleftrightarrow P(2nd\ Order\ Equilibrium)$	$-{d[A] \over dt}=k_{1}[A][B]-k_{-1}[P]$

References

^ ^a ^b ^c Pfennig, Brian F (2015). Principles of Inorganic Chemistry. Hoboken, New Jersey: John Wiley & Sons, Inc. ISBN 978-1-118-85910-0.
^ ^a ^b ^c Levine, Ira N. (2009). Physical Chemistry. 1221 Avenue of the Americas, New York, NY 10020: The McGraw-Hill Companies, Inc. ISBN 978-0-07-253862-5.{{cite book}}: CS1 maint: location (link)

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[:0-1] Pfennig, Brian F (2015). Principles of Inorganic Chemistry. Hoboken, New Jersey: John Wiley & Sons, Inc. ISBN 978-1-118-85910-0.

[:1-2] Levine, Ira N. (2009). Physical Chemistry. 1221 Avenue of the Americas, New York, NY 10020: The McGraw-Hill Companies, Inc. ISBN 978-0-07-253862-5.{{cite book}}: CS1 maint: location (link)

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