Methods For Determination Of Order Of A Reaction

**Category : **JEE Main & Advanced

(1) **Integration method **(Hit and Trial method)

(i) The method can be used with various sets of \[a,\ x\] and \[t\] with integrated rate equations.

(ii) The value of \[k\] is determined and checked for all sets of \[a,\ x\] and \[t\].

(iii) If the value of \[k\] is constant, the used equation gives the order of reaction.

(iv) If all the reactants are at the same molar concentration, the kinetic equations are :

\[k=\frac{2.303}{t}\ {{\log }_{10}}\frac{a}{(a-x)}\] (*For first order reactions*)

\[k=\frac{1}{t}\left[ \frac{1}{a}-\frac{1}{a-x} \right]\] (*For second order reactions*)

\[k=\frac{1}{2t}\left[ \frac{1}{{{(a-x)}^{2}}}-\frac{1}{{{a}^{2}}} \right]\] (*For third order reactions*)

(2) **Half-life method :** This method is employed only when the rate law involved only one concentration term.

\[{{t}_{1/2}}\propto {{a}^{1-n}}\]; \[{{t}_{1/2}}=k{{a}^{1-n}}\]; \[\log {{t}_{1/2}}=\log k+(1-n)\ \log a\]

A plotted graph of \[\log {{t}_{1/2}}\]*vs log a* gives a straight line with slope \[(1-n)\], determining the slope we can find the order *n*. If half-life at different concentration is given then,

\[{{({{t}_{1/2}})}_{1}}\propto \frac{1}{a_{1}^{n-1}};\] \[{{({{t}_{1/2}})}_{2}}\propto \frac{1}{a_{2}^{n-1}};\] \[\frac{{{({{t}_{1/2}})}_{1}}}{{{({{t}_{1/2}})}_{2}}}={{\left( \frac{{{a}_{2}}}{{{a}_{1}}} \right)}^{n-1}}\]

\[{{\log }_{10}}{{({{t}_{1/2}})}_{1}}-{{\log }_{10}}{{({{t}_{1/2}})}_{2}}=(n-1)\ [{{\log }_{10}}{{a}_{2}}-{{\log }_{10}}{{a}_{1}}]\]

\[n=1+\frac{{{\log }_{10}}{{({{t}_{1/2}})}_{1}}-{{\log }_{10}}{{({{t}_{1/2}})}_{2}}}{({{\log }_{10}}{{a}_{2}}-{{\log }_{10}}{{a}_{1}})}\]

This relation can be used to determine order of reaction ‘*n*’

*Plots of half-lives Vs concentrations (t _{1/2 }*

(3) **Graphical method : **A graphical method based on the respective rate laws, can also be used.

(i) If the plot of \[\log (a-x)\] Vs \[t\] is a straight line, the reaction follows first order.

(ii) If the plot of \[\frac{1}{(a-x)}\] Vs \[t\] is a straight line, the reaction follows second order.

(iii) If the plot of \[\frac{1}{{{(a-x)}^{2}}}\] Vs \[t\] is a straight line, the reaction follows third order.

(iv) In general, for a reaction of nth order, a graph of \[\frac{1}{{{(a-x)}^{n-1}}}\] Vs \[t\] must be a straight line.

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*Plots from integrated rate equations*

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*Plots of rate Vs concentrations [Rate = k(conc.) ^{n} ]*

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(4) **Van't Haff differential method : **The rate of reaction varies as the *n ^{th}* power of the concentration Where \['n'\] is the order of the reaction. Thus for two different initial concentrations \[{{C}_{1}}\] and \[{{C}_{2}}\] equation, can be written in the form,

\[\frac{-d{{C}_{1}}}{dt}=kC_{1}^{n}\] and \[\frac{-d{{C}_{2}}}{dt}=kC_{2}^{n}\]

Taking logarithms,

\[{{\log }_{10}}\left( \frac{-d{{C}_{1}}}{dt} \right)={{\log }_{10}}k+n{{\log }_{10}}{{C}_{1}}\] …..(i)

and \[{{\log }_{10}}\left( \frac{-d{{C}_{2}}}{dt} \right)={{\log }_{10}}k+n{{\log }_{10}}{{C}_{2}}\] …..(ii)

Subtracting equation (ii) from (i),

\[n=\frac{{{\log }_{10}}\left( \frac{-d{{C}_{1}}}{dt} \right)-{{\log }_{10}}\left( \frac{-d{{C}_{2}}}{dt} \right)}{{{\log }_{10}}{{C}_{1}}-{{\log }_{10}}{{C}_{2}}}\] …..(iii)

\[\frac{-d{{C}_{1}}}{dt}\] and \[\frac{-d{{C}_{2}}}{dt}\] are determined from concentration Vs time graphs and the value of \['n'\] can be determined.

(5) **Ostwald's isolation method **(Initial rate method)

This method can be used irrespective of the number of reactants involved *e.g., *consider the reaction, \[{{n}_{1}}A+{{n}_{2}}B+{{n}_{3}}C\to \text{Products}\].

This method consists in finding the initial rate of the reaction taking known concentrations of the different reactants (*A*, *B*, *C*).

Suppose it is observed as follows,

(i) Keeping the concentrations of *B* and *C* constant, if concentration of *A* is doubled, the rate of reaction becomes four times. This means that, Rate \[\propto {{[A]}^{2}}\] *i.e., *order with respect to* A* is 2

(ii) Keeping the concentrations of *A* and *C *constant, if concentration of B is doubled, the rate of reaction is also doubled. This means that, Rate µ [*B*] *i.e., *order with respect to* B* is 1

(iii) Keeping the concentrations of *A* and *B* constant, if concentration of *C* is doubled, the rate of reaction remains unaffected. This means that rate is independent of the concentration of *C* *i.e., *order with respect to* C* is zero. Hence the overall rate law expression will be, Rate = *k*[*A*]^{2} [*B*] [*C*]^{0 }

\[\therefore \] Overall order of reaction = 2 + 1 + 0 = 3.

*play_arrow*Types Of Chemical Reactions*play_arrow*Rate of a Reaction*play_arrow*Factors Affecting Rate Of A Reaction*play_arrow*Rate Law : Molecularity And Order Of A Reaction*play_arrow*Law Of Mass Action And Rate Constant*play_arrow*Methods For Determination Of Order Of A Reaction*play_arrow*Theories Of Reaction Rate*play_arrow*Arrhenius Equation*play_arrow*Mechanism Of The Reaction*play_arrow*Photochemical Reaction

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