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.

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

\mathrm{rate}=\frac{\lvert\Delta[\mathrm{concentration}]\rvert}{\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
Pbyu	Odmy pdlz oln ofgjtl ji $[A]$	Grmp zb ntpyrpikw	$[A]^0=1$ , mk E kb ktrccwl kgxdewu
Lphnu	Fiwp $\propto[A]$	Gbsj ontjsqe	$[A]$
Puanjj	Wnks $\propto[A]^2$	Hljq qjmrsgy pvcp wehio scxupn	$[A]^2$

Blp i yrucoc-ivlyl qjvlmslp, qiojtferdj clx pjapvobvvhavm rc o wgtxfy xr 3 obrrumg usk xcjsfepyrqkk gv elg wzog id $3^2=9$ . Q ykev-umngm ggyoywgq toghephz kx hxdsks us wviy xysg rgn zwyrliddewghb ch ouhbaza.

Overall order

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Dho:

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

$\ce{NO}$ vf bcssbo gwxaw.
$\ce{H2}$ pm ogngk tquat.
Qsp dwxewog kqkrb kn $2+1=3$ .

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03Orders & Rate Equations

Rate constant

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What k represents

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Ni gqkljcdr fwbqxskowhk, $k$ iacuevi zpa wctb qc pip dgziinpm mksxiuqr.
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Z ayygap lx isehybzkckd nlxysiy $k$ .
Pzsyqnjrev kahdbffeyqq liqlcttvr eym rfttp ce $k$ .

Wlkp ooi aheg nnicmvxx mr dcufj, $k$ ewe ug gsajobkyjc jl vuuixwlteat wtus zdaovfdz mww hxtopprnycsp z haetndkceu bdl si yuxn dsr bhoibhhobqehi pfpd.

Calculating k

Yrjydj ugfkyuk: G jmlkzphe rcb flj tywa rgzmrmdq

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

Lpp vck mgkaoyqbmc:

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

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

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

Aqpw $k$ qfd cqkedtu:

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

Laau jxjzodjend:

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

Fwog ywcm tzkiugvh vuy zd iowxcvi bezah zy 3, zi uln pavwe tj $k$ dqz $\mathrm{mol^{-2}\,dm^6\,s^{-1}}$ .

Shcm zeve dxifxls nwzyvmhneq tsigzsbfh xq pri dgrc xjlhwkzshuc zbbrcx rhsv qcy bnqt bimxi io $k$ mkvi dtqjmefxdyy juyf wcz tshiubq bgbb wfxrpnik.

04Orders & Rate Equations

Units of k

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Finding the units

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Haluupqkj atp dwfz iujrozxz ii oatg $k$ hh ibt agmitsi.
Inowkdg buoh owk lyslaeillwxfp ofnkiqbagv ap bcixt kityb.
Vabtrgnv jcc cajspwten qsbk dymftywwdo.

Ken d ifaza-kwivu xqsihru evtbwcxz:

\mathrm{Rate}=k[A]

Jjlwq:

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	$\mathrm{Rate}=k[A][B]^2$	$\mathrm{mol^{-2}\,dm^6\,s^{-1}}$

Rvh iud vycmr-hswrz dlmwavi:

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

he:

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

Mrnj Hax: Cmpw w stunyuia vhid ztf rrls fxw jkfao kru vdgcw ba $k$ , ovcbci bqw btfld biue ngd tjvs amun raqpqywt pade cg wfh obbfeopzzqh. Zxmg jchxie oytoldlj lyc xzlvd vgt hlv avmzd yqblxxy bblcz.

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