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Arrhenius Equation ln

Therate of reaction depends on the number of collisions between the reactant atoms and molecules, on the fraction of the collisions that occur with proper orientation, and on the fraction that overcomes Ea (activation energy). These are summarized by the Arrhenius equation.


k = A e –Ea/RT

where:

A is the frequency of collisions with correct geometry;
e –Ea/RT is the fraction of molecules with minimum energy for reaction at temperature T.

As the temperature increases, the value of A increases due to an increase in the number of collisions. In addition, e –Ea/RT increases, a reflection of the increase in the fraction of colliding molecules that overcome the activation barrier.


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TheArrhenius equation is used to calculate the value of ‘k’ at various temperatures and also the activation energy of a reaction.


arrhenius equation : Linear relationship:


The Arrhenius equation can be rewritten in the form of a linear relationship.

ln k = ln A – Ea/R [1/T]

In k = y

In A = a

-Ea/R = b

1/T = x

A plot versus of ln k versus 1/T yields a straight line with a slope of –Ea/R.

Arrhaniusslope

The slope is  Δ ln (k)/ Δ(1/T)

The value of Ea (activation  energy) can be determined by solving the following equation:

Slope = -Ea/R

Ea = -R × (slope)

OR

Ea = - R [Δ ln (k)/ Δ (1/T)]


arrhenius equation : Determining Activation Energy:


The reaction 

2N2O5 (g) → 4NO2 (g) + O2 (g)

was studied at several temperatures, and the following values of k were obtained:

arrheniusquestion

Calculate the value of Ea for this reaction.

Solution:

Inorder to solve this problem, we need to plot a graph (In k versus 1/T).Before that we need to find out the values of ln(k), T in Kelvin scale and then 1/T.


arrheniussolution


Let us now plot the graph:


arrheniusgraph

From the graph Δ(1/T)= (3.365-3.05) × 10 -3
            = 0.315 × 10 -3

And

Δ(ln k) = -10-(-6.5)
    = -10+ 6.5
    = -3.5


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As we know the slope gives the value for -Ea/R
So, Ea = -R × (slope)
Ea= - (8.314 JK-1mol-1) (Δ(ln k)/ Δ(1/T)
    = -(8.314)(-3.5/(0.315 × 10 -3))
   = (-8.314) (-11111.11)
   = 92377.777
   = 0.92377 x 10 5
   = approximately 1.0 x 10 5 J mol-1