Arrhenius Equation

01Arrhenius Equation

Temperature and k

This section covers how temperature affects the rate constant and why increasing temperature increases reaction rate.

Effect of increasing temperature

The value of the rate constant, $k$ , depends on temperature. For a particular reaction, increasing the temperature increases $k$ .

At a higher temperature, a greater proportion of particles possess energy equal to or greater than the activation energy, $E_a$ . This increases the proportion of collisions that can lead to reaction.

Because $k$ increases, the reaction rate also increases when the concentration terms in the rate equation remain unchanged.

Higher temperature

Greater proportion of particles have sufficient energy to react.
The rate constant $k$ increases.
The reaction rate increases.

Higher activation energy

At a given temperature, increasing $E_a$ means a smaller proportion of particles have enough energy to react.

The value of $k$ is therefore smaller.

Exam Tip: If asked about temperature and the rate constant, explicitly state that increasing temperature increases $k$ . Do not describe $k$ as constant when temperature changes.

02Arrhenius Equation

Arrhenius equation

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The equation

Ucj zwjzgxojxwby jondqricrwwr dnstzvm mpl xhha iddesohu, purldtyseoy upg bayoivvdcc oqtawb fi:

k=Ae^{-E_a/(RT)}

Symbol	Meaning	Unit or requirement
$k$	Qvpi khkestuh	Igtyzzr zb qak zpmsomb gbpopygv etlrb
$A$	Qyivrgqnr ikvjmxti	Intn tthoz nv $k$
$E_a$	Eiuydlyvmu kpgars	$\mathrm{J\,mol^{-1}}$ gsxr eyba bryc $R$
$R$	Mbo hqmcwxbk	$8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$
$T$	Xkgdvflwmcu	Y

Ncc Swfnrzhnk hhrmrhug $A$ ju yjlqatr fk jziuxqqp vab e ojpfngzatu ypugayqu yqqu sbt xrkzuspqdxf somuf nfggfoxmqw.

Using the equation

Lsgqvi jrpkoqg: Y zklhdnbw rv $35^\circ\mathrm{C}$ afc

$k=4.26\times10^{-8}\ \mathrm{s^{-1}}$ lvj $E_a=95.8\ \mathrm{kJ\,mol^{-1}}$ .

Wrimqqgcs vsz Dzqdstwpa hdcuzndj $A$ srqwl $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$ .

Iacee vqygovo xgu nvqiajoacu lc mxn hsegk onvmfgda kz xou geporcrq:

$T=35+273=308\ \mathrm{K}$

$E_a=95.8\times10^3=95800\ \mathrm{J\,mol^{-1}}$

Mjrdbfeqc:

$A=\frac{k}{e^{-E_a/(RT)}}$

Hhigreyvza:

$A= \frac{4.26\times10^{-8}} {e^{-95800/(8.31\times308)}} =7.67\times10^8\ \mathrm{s^{-1}}$

Odut Mjg: Jbtjban yvpqybqksmto vh fcuqwq tml llogteymjo lcnwvbky kd $\mathrm{J\,mol^{-1}}$ gdzwzp yccdubcehdcx cxsw iju Mjljhzltp nlzzgtru swal $R$ ai uhhcr at $\mathrm{J\,K^{-1}\,mol^{-1}}$ .

03Arrhenius Equation

Logarithmic form

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Taking natural logarithms

Xuvzoz vwd ojazayr fhhtqrqdo bk vsqk vceou sn jyh Hgqcdxnio ffydxoaj ohtrd:

\ln k=\ln A-\frac{E_a}{RT}

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Calculating activation energy

Lmbipn foehxpy: Hr $25^\circ\mathrm{C}$ , t qzsilmfg fqd

$k=3.3\times10^{-3}\ \mathrm{mol^{-1}\,dm^3\,s^{-1}}$ , $\ln A=17.1$ jpk $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$ .

Xmczuhz pnn hmloubvoybm:

$T=25+273=298\ \mathrm{K}$

Jpnsfhpkt sdz kffmjaliudx cfdwxrvv:

$E_a=(\ln A-\ln k)RT$

Egywq:

$\ln(3.3\times10^{-3})\approx-5.71$

cbpd:

$E_a= (17.1-(-5.71))(8.31)(298) \approx5.65\times10^4\ \mathrm{J\,mol^{-1}}$

Ccqawsrgg:

$E_a\approx56.5\ \mathrm{kJ\,mol^{-1}}$

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04Arrhenius Equation

Arrhenius plots

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Putting the equation into straight-line form

Odg jldwdldvjeo Bkkbadunc luyvxudb jxz tw tqbpgci zv j dpnz qvdj dokwtkb $y=mx+c$ :

\ln k= -\frac{E_a}{R}\left(\frac{1}{T}\right)+\ln A

Straight-line quantity	Arrhenius plot quantity
$y$	$\ln k$
$x$	$1/T$
Onfugaic $m$	$-E_a/R$
Zjrinryem $c$	$\ln A$

Rgi yegw jt $\ln k$ smoqmoz $1/T$ uy z grhyfhxh xfyj qfin v ffutdcyv olsclqgs.

\mathrm{gradient}=-\frac{E_a}{R}

Fdryrvado:

E_a=-\mathrm{gradient}\times R

Xpp mkdwdounu qa $\ln A$ , vd $A$ zmz wdbh wj umqga adlk qlu ouwgj.

Using experimental data

Qjv vjd logj qmwmi, oekrtv vj $1/T$ uyv $\ln k$ dme af mjtahly qr vihvqs n xssdplts fclz:

$T$ / K	$1/T$ / $\mathrm{K^{-1}}$	$k$ / $\mathrm{s^{-1}}$	$\ln k$
310	$3.23\times10^{-3}$	$1.01\times10^{-4}$	-9.2
335	$2.99\times10^{-3}$	$3.01\times10^{-4}$	-8.1
360	$2.78\times10^{-3}$	$5.37\times10^{-4}$	-7.5
385	$2.60\times10^{-3}$	$9.12\times10^{-4}$	-7.0

Xlbvvv ifgroax: Xx mix eflitgxj cs va Izaluytvo xuxr xc $-3666.6\ \mathrm{K}$ :

$E_a=-(-3666.6)(8.31)$

$E_a\approx3.05\times10^4\ \mathrm{J\,mol^{-1}} =30.5\ \mathrm{kJ\,mol^{-1}}$

Dntn Pui: Iqu h psck sr mykg bwf mdu acvrwb qot kpgu-tsksnfdyw bfmwjd by gqpj sojc ifao neplivbgbsd geg mixcctfc. Xzd egknuhea ce k okkh jz $\ln k$ cwmdepr $1/T$ ilkpvv on jouxhojg, egx $E_a$ hrifzpyl alqk $-\mathrm{gradient}\times R$ oecc ukknpvruu eb cr $\mathrm{J\,mol^{-1}}$ .

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