NEET Chemistry Chemical Kinetics / रासायनिक बलगतिकी Rate law and Rate Constant

 

Rate Law and Rate Constant

 

Rate law, Law of mass action and Rate constant.

 

(1) Rate law: The actual relationship between the concentration of reacting species and the reaction rate is determined experimentally and is given by the expression called rate law.

For any hypothetical reaction, \[aA+bB\to cC+dD\]

Rate law expression may be, \[\text{rate}=k{{[A]}^{a}}{{[B]}^{b}}\]

Where a and b are constant numbers or the powers of the concentrations of the reactants \[A\] and \[B\] respectively on which the rate of reaction depends.

(i) Rate of chemical reaction is directly proportional to the concentration of the reactants.

(ii) The rate law represents the experimentally observed rate of reaction, which depends upon the slowest step of the reaction.

(iii) Rate law cannot be deduced from the relationship for a given equation. It can be found by experiment only.

(iv) It may not depend upon the concentration of species which do not appear in the equation for the over all reaction.

 

(2) Law of mass action: (Guldberg and Wage 1864) This law relates rate of reaction with active mass or molar concentration of reactants. According to this law, “At a given temperature, the rate of a reaction at a particular instant is proportional to the product of the reactants at that instant raised to powers which are numerically equal to the numbers of their respective molecules in the stoichiometric equation describing the reactions.”

Active mass = Molar concentration of the substance =\[\frac{\text{Number of gram moles of the substance}}{\text{Volume in litres}}\] \[=\frac{W/m}{V}=\frac{n}{V}\]

Where \[W=\] mass of the substance, m is the molecular mass in grams, ‘n’ is the number of g moles and V is volume in litre.

Consider the following general reaction, \[{{m}_{1}}{{A}_{1}}+{{m}_{2}}{{A}_{2}}+{{m}_{3}}{{A}_{3}}\to \text{Products}\]

Rate of reaction \[\propto {{[{{A}_{1}}]}^{{{m}_{1}}}}{{[{{A}_{2}}]}^{{{m}_{2}}}}{{[{{A}_{3}}]}^{{{m}_{3}}}}\]

 

(3) Rate constant: Consider a simple reaction, \[A\to B\]. If \[{{C}_{A}}\] is the molar concentration of active mass of A at a particular instant, then, \[\frac{dx}{dt}\propto \ {{C}_{A}}\] or \[\frac{dx}{dt}=k{{C}_{A}}\]; Where \[k\] is a proportionality constant, called velocity constant or rate constant or specific reaction rate constant.

At a fixed temperature, if \[{{C}_{A}}=1\], then \[Rate=\frac{dx}{dt}=k\]

“Rate of a reaction at unit concentration of reactants is called rate constant.”

(i) The value of rate constant depends on,        Nature of reactant, Temperature and Catalyst

(It is independent of concentration of the reactants)

(ii) Unit of rate constant: \[\text{Unit}\ \text{of}\ \text{rate}\ \text{constant}\ ={{\left[ \frac{\text{litre}}{\text{mol}} \right]}^{n-1}}\times {{\sec }^{-1}}\] or \[\ ={{\left[ \frac{\text{mol}}{\text{litre}} \right]}^{1-n}}\times {{\sec }^{-1}}\]

Where \[n=\] order of reaction

Difference between Rate law and Law of mass action

 

Rate law	Law of mass action
It is an experimentally observed law.	It is a theoretical law.
It depends on the concentration terms on which the rate of reaction actually depends	It is based upon the stoichiometry of the equation
Example for the reaction, \[aA+bB\to \text{Products}\]	Example for the reaction, \[aA+bB\to \text{Products}\]
Rate\[=k\,{{[A]}^{m}}{{[B]}^{n}}\]	Rate \[=k{{[A]}^{a}}{{[B]}^{b}}\]

 

Difference between Rate of reaction and Rate constant

 

 

Rate of reaction	Rate constant
It is the speed with which reactants are converted into products.	It is proportionality constant.
It is measured as the rate of decrease of the concentration of reactants or the rate of increase of concentration of products with time.	It is equal to rate of reaction when the concentration of each of the reactants is unity.
It depends upon the initial concentration of the reactants.	It is independent of the initial concentration of the reactants. It has a constant value at fixed temperature.

Order of Reaction.

            “The order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.”

Or

            “The total number of molecules or atoms whose concentration determine the rate of reaction is known as order of reaction.”

            Order of reaction = Sum of exponents of the concentration terms in rate law

                                                \[xA+yB\to \text{Products}\]

By the rate law, \[\text{Rate}={{\text{ }\!\![\!\!\text{ A }\!\!]\!\!\text{ }}^{\text{x}}}[B{{[}^{y}}\], then the overall order of reaction. \[n=x+y\], where x and y are the orders with respect to individual reactants.

If reaction is in the form of reaction mechanism then the order is determined by the slowest step of mechanism.

                                                \[2A+3B\to {{A}_{2}}{{B}_{3}}\]

                                                \[A+B\to AB(\text{fast})\]

                                                \[AB+{{B}_{2}}\to A{{B}_{3}}(\text{slow})\]    (Rate determining step)

                                                \[A{{B}_{3}}+A\to {{A}_{2}}{{B}_{3}}(\text{fast})\]

                                    (Here, the overall order of reaction is equal to two.)

An order of a reaction may be zero, negative, positive or in fraction and greater than three. Infinite and imaginary values are not possible.

 

(1) First order reaction : When the rate of reaction depends only on the one concentration term of reactant.

                        Examples :          ·  \[A\to \text{product}\]

•   \[{{H}_{2}}{{O}_{2}}\to {{H}_{2}}O+\frac{1}{2}{{O}_{2}}\]
• All radioactive reactions are first order reaction.
•   Rate of growth of population if there is no change in the birth rate or death rate.
•   Rate of growth of bacterial culture until the nutrients are exhausted.

                        Exception :      \[{{H}_{2}}O,\,{{H}^{+}},\,O{{H}^{-}}\]and excess quantities are not considered in the determining process of order.

                        Examples :         · \[C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}+{{H}_{2}}O\to C{{H}_{3}}COOH+{{C}_{2}}{{H}_{5}}OH\];  Order = 1;  \[r=k\,[C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}]\]

• \[2A(\text{excess})+B\to \text{product}\]; Order = 1; \[R=k\,[B]\]
• \[2{{N}_{2}}{{O}_{5}}\to 4N{{O}_{2}}+{{O}_{2}}\]; Order = 1;             \[R=k\,[{{N}_{2}}{{O}_{5}}]\]
• \[2C{{l}_{2}}{{O}_{7}}\to 2C{{l}_{2}}+7{{O}_{2}}\]; Order = 1;  \[R=k\,[C{{l}_{2}}{{O}_{7}}]\]
• \[{{(C{{H}_{3}})}_{3}}-C-Cl+O{{H}^{-}}\to {{(C{{H}_{3}})}_{3}}C-OH+C{{l}^{-}}\]; Order = 1;  \[R=k\,[{{(C{{H}_{3}})}_{3}}C-Cl]\]

            (i) Velocity constant for first order reaction : Let us take the reaction

                                                            \[A\xrightarrow{{}}\text{Product}\]

            Initially t = 0                 \[a\]            0

            After time t = t          \[(a-x)\]          \[x\] 

            Here, \['a'\] be the concentration of A at the starting and \[(a-x)\] is the concentration of A after time \[t\] i.e., \[x\] part has been changed in to products. So, the rate of reaction after time \[t\] is equal to

                                    \[\frac{dx}{dt}\propto \ (a-x)\] or \[\frac{dx}{dt}=k(a-x)\] or \[\frac{dx}{(a-x)}=k.dt\]          …..(i)

            integrated rate constant is,

                                    \[k=\frac{2.303}{t}{{\log }_{10}}\frac{a}{(a-x)}\]                                              …..(ii)

                                    \[t=\frac{2.303}{k}{{\log }_{10}}\frac{a}{(a-x)}\]                                              …..(iii)

            (ii) Half life period of the first order reaction : when \[t={{t}_{1/2}};\ \ x=\frac{a}{2}\], then eq. (ii) becomes

                                    \[{{t}_{1/2}}=\frac{2.303}{k}{{\log }_{10}}\frac{a}{\left( a-\frac{a}{2} \right)}\];  \[{{t}_{1/2}}=\frac{2.303}{k}{{\log }_{10}}\frac{a}{a/2}\]

                                    \[{{t}_{1/2}}=\frac{2.303}{k}{{\log }_{10}}2\] (\[\because \ \log 2=0.3010\]); \[\therefore \ \ {{t}_{1/2}}=\frac{2.303}{k}\times 0.3010\]

                                                            \[{{t}_{1/2}}=\frac{0.693}{k}\]

            Half life period for first order reaction is independent from the concentration of reactant.

            Time for completion of n^th fraction, \[{{t}_{1/n}}=\frac{2.303}{K}\log \frac{1}{\left( 1-\frac{1}{n} \right)}\]

           

(iii) Unit of rate constant of first order reaction :  \[k={{(\sec )}^{-1}}\]

            (2) Second order reaction : Reaction whose rate is determined by change of two concentration terms is said to be a second order reaction. For example,

• \[C{{H}_{3}}COOH+{{C}_{2}}{{H}_{5}}OH\xrightarrow{{}}C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}+{{H}_{2}}O\]
• \[{{S}_{2}}O_{8}^{2-}+2{{I}^{-}}\xrightarrow{{}}2SO_{4}^{2-}+{{I}_{2}}\]

            (i) Calculation of rate constant : \[2A\xrightarrow{{}}\] product   or   \[A+B\xrightarrow{{}}\] product

            When concentration of \[A\] and \[B\] are same.

            \[\underset{\text{Initially}\,\,\,\,\,\,t=0}{\mathop{{}}}\,\ \ \ \ \ \ \ \ \underset{a}{\mathop{A}}\,\ \ \ \ +\ \ \underset{a}{\mathop{B}}\,\ \xrightarrow{{}}\ \underset{0}{\mathop{\text{Product}}}\,\]

            After time \[t=t\]   \[(a-x)\]   \[(a-x)\]         \[x\]

            \[\frac{dx}{dt}=k[A]\ [B]\]\[=k\ [a-x]\ [a-x]\]

            \[\frac{dx}{dt}=k\ {{[a-x]}^{2}}\]; Integrated equation is \[k=\frac{1}{t}.\frac{x}{a(a-x)}\] ;    \[t=\frac{1}{k}.\frac{x}{a(a-x)}\]

            When concentration of \[A\] and \[B\] are taken different

                                                            A          +        B     ®   Product

            Initially  t = 0                a                       b             0

            After time          t = t             (a – x)               (b – x)         x

                                    \[\frac{dx}{dt}=k\ [a-x].[b-x]\], Integrated equation is,

                                    \[k=\frac{2.303}{t(a-b)}\log \,\frac{b(a-x)}{a(b-x)}\];  \[t=\frac{2.303}{k(a-b)}\log \frac{b(a-x)}{a(b-x)}\]

            (ii) Half-life period of the second order reaction: When \[t={{t}_{1/2}}\]; \[x=\frac{a}{2}\]; \[{{t}_{1/2}}=\frac{1}{k}\left( \frac{\frac{a}{2}}{a\times (a-\frac{a}{2})} \right)\]  \[=\frac{1}{ka}\]

            Half-life of second order reaction depends upon the concentration of the reactants. \[{{t}_{1/2}}\propto \frac{1}{a}\]

            (iii) Unit of rate constant: \[k=mo{{l}^{1-\Delta \,n}}\ li{{t}^{\text{ }\!\!\Delta\!\!\text{  }n\,-1}}{{\sec }^{-1}}\];\[\Delta n=2\], \[k=mo{{l}^{-1}}\ lit.{{\sec }^{-1}}\] (Where \[\Delta n=\] order of reaction)

            (3) Third order reaction: A reaction is said to be of third order if its rate is determined by the variation of three concentration terms. When the concentration of all the three reactants is same or three molecules of the same reactant are involved, the rate expression is given as

                        \[3A\xrightarrow{{}}\] products  or  \[A+B+C\xrightarrow{{}}\] products

            (i) Calculation of rate constant: \[\frac{dx}{dt}=k{{(a-x)}^{3}}\], Integrated equation is  \[k=\frac{1}{t}.\frac{x(2a-x)}{2{{a}^{2}}{{(a-x)}^{2}}}\]

            (ii) Half-life period of the third order reaction: Half-life period  = \[\frac{3}{2{{a}^{2}}k}\];  \[{{t}_{1/2}}\propto \frac{1}{{{a}^{2}}}\];  Thus, half-life is inversely proportional to the square of initial concentration.

            (iii) Unit of rate constant: \[k={{\left( \frac{mol}{litre} \right)}^{-2}}tim{{e}^{-1}}\] or \[k=litr{{e}^{2}}mo{{l}^{-2}}tim{{e}^{-1}}\]

            (4) Zero order reaction: Reaction whose rate is not affected by concentration or in which the concentration of reactant do not change with time are said to be of zero order reaction. For example,

• \[{{H}_{2}}+C{{l}_{2}}\xrightarrow{\text{Sunlight}}2HCl\]
• Dissociation of \[HI\] on gold surface.
• Reaction between acetone and bromine.
• The formation of gas at the surface of tungsten due to adsorption.

            (i) Calculation of Rate Constant : Let us take the reaction

                                                \[A\xrightarrow{{}}\] Product

            Initially \[t=0\]   \[a\]             0

                                    \[\frac{dx}{dt}=k{{[A]}^{0}}\], \[\frac{dx}{dt}=k\];  \[dx=k.\ dt\]

            Integrated rate equation, \[k=\frac{x}{t}\];  The rate of reaction is independent of the concentration of the reacting substance.

            (ii) Half-life period of zero order reaction : When \[t={{t}_{1/2}}\]; \[x=\frac{a}{2}\];  \[{{t}_{1/2}}=\frac{a}{2k}\] or  \[{{t}_{1/2}}\propto a\];  The half life period is directly proportional to the initial concentration of the reactants.

            (iii) Unit of Rate constant : \[k=\frac{mole}{lit.\,\sec .}\];    Unit of rate of reaction = Unit of rate constant.

           

 

          

Note  : q In general, the units of rate constant for the reaction of n^th order are:

                                                                        \[\text{Rate}=\text{k }\!\![\!\!\text{ }A{{\text{ }\!\!]\!\!\text{ }}^{\text{n}}}\]    

                                                \[\frac{mol\,{{L}^{-1}}}{s}=k{{(mol\,{{L}^{-1}})}^{n}}\] or \[k={{(mol\,{{L}^{-1}})}^{1-n}}{{s}^{-1}}\]

            Units of rate constants for gaseous reactions: In case of gaseous reactions, the concentrations are expressed in terms of pressure in the units of atmosphere. Therefore, the rate has the units of atm per second. Thus, the unit of different rate constants would be:

• (i) Zero order reaction : \[atm\ {{s}^{-1}}\] (ii) First order reaction : \[{{s}^{-1}}\]

(iii)  Second order reaction: \[at{{m}^{-1}}{{s}^{-1}}\]              (iv) Third order reaction: \[at{{m}^{-2}}{{s}^{-1}}\]

            In general, for the gaseous reaction of n^th order, the units of rate constant are \[{{\left( atm \right)}^{1n}}{{s}^{1}}\]

 

Modified expressions for rate constants of some common reactions of first order

 

Reaction	Expression for rate constant
\[{{N}_{2}}{{O}_{5}}\xrightarrow{{}}2N{{O}_{2}}+\frac{1}{2}{{O}_{2}}\]	\[k=\frac{2.303}{t}\log \frac{{{V}_{\infty }}}{{{V}_{\infty }}-{{V}_{t}}}\] Here \[{{V}_{t}}=\] volume of \[{{O}_{2}}\] after time \[t\] and \[{{V}_{\infty }}=\] volume of \[{{O}_{2}}\] after infinite time.
\[N{{H}_{4}}N{{O}_{2}}(aq)\xrightarrow{{}}2{{H}_{2}}O+{{N}_{2}}\]	Same as above, here \[{{V}_{t}}\] and \[{{V}_{\infty }}\] are volumes of \[{{N}_{2}}\] at time \[t\] and at infinite time respectively.
\[{{H}_{2}}{{O}_{2}}\xrightarrow{{}}{{H}_{2}}O+\frac{1}{2}{{O}_{2}}\]	\[k=\frac{2.303}{t}\log \frac{{{V}_{0}}}{{{V}_{t}}}\] Here \[{{V}_{0}}\] and \[{{V}_{t}}\] are the volumes of \[KMn{{O}_{4}}\] solution used for titration of same volume of reaction mixture at zero time (initially) and after time \[t\].
\[C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}+{{H}_{2}}O\xrightarrow{{{H}^{+}}}C{{H}_{3}}COOH+{{C}_{2}}{{H}_{5}}OH\]	\[k=\frac{2.303}{t}\log \frac{{{V}_{\infty }}-{{V}_{0}}}{{{V}_{\infty }}-{{V}_{t}}}\] Here \[{{V}_{0}},{{V}_{t}}\] and \[{{V}_{\infty }}\] are the volumes of \[NaOH\] solution used for titration of same volume of reaction mixture after time, \[0\], \[t\] and infinite time respectively.
\[\underset{d-\text{Sucrose}}{\mathop{{{C}_{12}}{{H}_{22}}{{O}_{11}}}}\,+{{H}_{2}}O\xrightarrow{{{H}^{+}}}\underset{d-\text{Glucose}}{\mathop{{{C}_{6}}{{H}_{12}}{{O}_{6}}}}\,+\underset{l-\text{Fructose}}{\mathop{{{C}_{6}}{{H}_{12}}{{O}_{6}}}}\,\] (After the reaction is complete the equimolar mixture of glucose and fructose obtained is laevorotatory)	\[k=\frac{2.303}{t}\log \frac{{{r}_{0}}-{{r}_{\infty }}}{{{r}_{t}}-{{r}_{\infty }}}\] Here, \[{{r}_{0}},\ {{r}_{t}}\] and \[{{r}_{\infty }}\] are the polarimetric readings after time \[0,\,t\] and infinity respectively..