Arrhenius Law or Arrhenius Equation

Rate law or rate equation which shows a relationship between the rate of reaction, reaction rate constant and the concentration of its reactant molecules. For many reactions (particularly elementary reactions), the rate law or rate equation can be written as a product of temperature and concentration dependent terms;

➩ Rate = Function (T)× Function (Concⁿ)

The temperature dependent term for such reactions is nothing but the reaction rate constants. Hence, we can say that the rate of reaction is directly proportional to the reaction rate constant.

➩ Rate Of Reaction ∝ Reaction Rate Constant

It has been observed that "The rate constant for a reaction gets doubled for every 10°C rise in reaction's temperature" and subsequently the rate of reaction is also changed. In 1889, Savante Arrhenius extended the work of J.H. Van't hoff (gives an equation called van't hoff equation to understand how temperature affects the rate constant) and proposed an equation that tells us how reaction's rate gets affected by the reaction temperature quantitatively. The proposed equation was named as 'Arrhenius Equation' which shows a relationship between the rate constant and the reaction's temperature. 

Arrhenius Equation

Mathematical representation of arrhenius law;

➩ k = k₀× exp(-Eₐ/RT) 

OR

➩ k = A× exp(-Eₐ/RT) 

-------------------------(1) 

Where, 

    k = Reaction rate constant (number

          of molecular collisions per seconds) 

    k₀= Called as pre-exponential factor or called 

            as frequency factor or Arrhenius factor

          which represent the fraction of collisions

            per second with a proper Orientation

         for the reaction to occur. Somewhere it is 

              represented by a symbol 'A'

     Eₐ= Activation energy for the reaction

     R = Universal gas constant

     T = Absolute temperature measured in 'K'

This expression fits experiment well over a wide range of temperature and is strongly suggested as being a good approximation to the true temperature dependency. 

Here, "The activation energy is defined as the least possible amount of energy required by the reactant molecules to start a chemical reaction". Activation energy is determined experimentally by carrying out experiment at several different reaction's temperatures. As an alternative, the above equation may be written as, 

      ➩k = k₀ exp(-Eₐ/kвT) 

                                             -----------------(2) 

Where,   

         -Eₐ= activation energy for a reaction

          kB = Boltzmann constant

The difference between the above two equations is in the units of energy of 'Eₐ'. In the former class, Eₐ expressed in 'Energy Per Mole' which is common in chemistry therefore we use universal gas constant while in the latter class, Eₐ expressed in 'Energy Per Molecule' which is common in physics therefore we use Boltzmann constant, kB. 

          Since, the exponential term is always dimension-less therefore the unit of activation energy 'Eₐ' is same as the unit of 'RT' (or same as the unit of 'kBT'). And the unit of 'k₀' is same as the unit of reaction rate constant. 

This equation is very important and mostly used to calculate the rate of reaction and the activation energy for a given reaction. If we know the value of activation energy at a given temperature and the pre-exponential factor then one can find out the value or reaction rate constant by using arrhenius equation and then find out the rate of reaction at that rate constant value. 

              

Arrhenius Plot (or determination of activation energy)

For the determination of activation energy, we take natural logarithms of arrhenius equation then we get, 

               ➩ k = k₀ exp(-Eₐ/RT) 

         Taking logarithm both side, 

               ➩ ln(k) = ln(k₀) + ln{exp(-Eₐ/RT)}

               ➩ ln(k) = ln(k₀) - (Eₐ/R)(1/T) 

                                         ---------------------(3) 

Here, equation (3) is known as 'Logarithmic Form Of Arrhenius Equation' which looks like as an equation of straight line, y= m.x+c

On comparing, we get;

   ➛ Slope, m= -Eₐ/R

   ➛ Intercept, c = ln(k₀)

When we plot a graph between "ln(k) Vs Temperature (T)" then the results of this graph shows a straight line whose slope is -Eₐ/R and the intercept is ln(k₀). And this plot is known as 'Arrhenius Plot' for the determination of activation energy. Activation energy is given by,  

            ➩ {Eₐ = -(slope)×R}

                                ------------------(4) 

Elimination of Frequency Factor From The Arrhenius Equation

Consider logarithmic form of Arrhenius equation at the same concentration, but at two different temperatures say 'T₁' and 'T₂' and their corresponding rate constants are 'k₁' & 'k₂' respectively. Then, 

    At T₁,  

         ln(k₁) = ln(k₀) - (Eₐ/R) (1/T₁) 

                                            ------------------(5) 

    At T₂, 

          ln(k₂) = ln(k₀) - (Eₐ/R)(1/T₂) 

                                             -----------------(6) 

Now substracting the equation, (5) -(6) 

➩ ln(k₁)-ln(k₂) = -(Eₐ/R)(1/T₁) + (Eₐ/R)(1/T₂)  

➩ ln(k₁/k₂) = (Eₐ/R) {(-1/T₁) +(1/T₂)}

➩ ln(k₁/k₂) = (Eₐ/R) {(1/T₂ - 1/T₁)}

Or

➩ ln(k₁/k₂) = (Eₐ/R) {(T₁-T₂)/T₁.T₂}

Or

-----------------------(7) 

Where, 

    k₂ & k₁ are the reaction rate constant at two

       different temperatures T₁& T₂ respectively.

    Eₐ represents the activation energy 

        (provided that Eₐ stays constant over the

            temperatures T₁ and T₂) 

    R represents the universal gas constant

                      ( R=8.314 J/mol-K ) 

Equation (7) is commonly used to determine the activation energy at two different temperatures when the value of the reaction rate constants at these two different temperatures is known.

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