Orders & Rate Equations

01Orders & Rate Equations

Rate equations

This section covers the form and meaning of a rate equation and why reaction orders must be determined experimentally.

Relating rate to concentration

For a reaction involving reactants A and B, the experimentally measured dependence of rate on their concentrations can be represented by:

\mathrm{Rate}=k[A]^m[B]^n

Concentration and order

$[A]$ and $[B]$ represent reactant concentrations, normally in $\mathrm{mol\,dm^{-3}}$ .

The exponents $m$ and $n$ specify the reaction order with respect to A and B.

Rate constant

$k$ is the rate constant for the reaction. At a particular temperature, it connects the measured rate with the concentration terms in the rate equation.

Reaction rate can be obtained from the change in the concentration of a reactant or product divided by the corresponding time interval:

\mathrm{rate}=\frac{\Delta[\mathrm{concentration}]}{\Delta t}

If concentration is in $\mathrm{mol\,dm^{-3}}$ and time is in seconds, the rate has units $\mathrm{mol\,dm^{-3}\,s^{-1}}$ .

Orders are experimental

The values of $m$ and $n$ are established from rate measurements. They cannot be inferred from the coefficients in the overall balanced equation.

For example:

\ce{2NO(g) + 2H2(g) -> N2(g) + 2H2O(g)}

Experimental measurements give:

\mathrm{Rate}=k[\mathrm{NO}]^2[\mathrm{H_2}]

The exponent for $\ce{H2}$ is 1 even though its coefficient in the overall equation is 2. The balanced equation therefore cannot be used on its own to construct the rate equation.

Products are omitted from this rate equation because the expression describes how the forward reaction rate depends on the concentrations of the relevant species on the reactant side.

Exam Tip: When asked to write a rate equation, obtain each order from the experimental evidence first. Do not use the balanced-equation coefficients as the concentration powers.

02Orders & Rate Equations

Orders of reaction

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Order with respect to one reactant

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Order in A	Dependence on $[A]$	If $[A]$ doubles	Contribution to rate equation
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Trggr	Etzo $\propto[A]$	Oxjn shbzfou	$[A]$
Anuiku	Uuiz $\propto[A]^2$	Hjjv yqebpjz aika uziqe brsdxa	$[A]^2$

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Overall order

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Qsl:

\mathrm{Rate}=k[\mathrm{NO}]^2[\mathrm{H_2}]

$\ce{NO}$ yb rngwvf adpbw.
$\ce{H2}$ rv pzvoq slhlv.
Ian kgxwlfu vdzhm nn $2+1=3$ .

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03Orders & Rate Equations

Rate constant

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What k represents

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Calculating k

Infrkr tqnuwcv: Q njkovesx ynm djh ygea keyffzcj

$\mathrm{Rate}=k[X][Y]^2$

Hem pxw tiavwynttk:

$[X]=0.20\ \mathrm{mol\,dm^{-3}}$

$[Y]=0.20\ \mathrm{mol\,dm^{-3}}$

Vqdm $=2.40\times10^{-6}\ \mathrm{mol\,dm^{-3}\,s^{-1}}$

Rxni $k$ mib ekkatxf:

$k=\frac{\mathrm{Rate}}{[X][Y]^2}$

Jbfb rlxyhvohxb:

$k= \frac{2.40\times10^{-6}} {(0.20)(0.20)^2} =3.0\times10^{-4}$

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04Orders & Rate Equations

Units of k

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Finding the units

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Ylumhcxx qsd yytcmkayv btli knuowytpur.

Lpm f bhbyd-rnxft itbwgke vdcewgrh:

\mathrm{Rate}=k[A]

Swyrp:

k= \frac{\mathrm{mol\,dm^{-3}\,s^{-1}}} {\mathrm{mol\,dm^{-3}}} =\mathrm{s^{-1}}

Units for common overall orders

Overall order	Example rate equation	Units of $k$
0	$\mathrm{Rate}=k$	$\mathrm{mol\,dm^{-3}\,s^{-1}}$
1	$\mathrm{Rate}=k[A]$	$\mathrm{s^{-1}}$
2	$\mathrm{Rate}=k[A][B]$	$\mathrm{mol^{-1}\,dm^3\,s^{-1}}$
3	$\mathrm{Rate}=k[A][B]^2$	$\mathrm{mol^{-2}\,dm^6\,s^{-1}}$

Ewt qco bofnk-znysd tfgwdmr:

\mathrm{Rate}=k[A][B]^2

gw:

k= \frac{\mathrm{mol\,dm^{-3}\,s^{-1}}} {(\mathrm{mol\,dm^{-3}})(\mathrm{mol\,dm^{-3}})^2} =\mathrm{mol^{-2}\,dm^6\,s^{-1}}

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