Question

Derive the relationship showing variation in rate constant with temperature.

Answer 1

k=rate constant of the reaction

A= Arrhenius Constant

$E_a$= Activation Energy

R= Universal Gas Constant

T=Temperature in absolute scale

$k = Ae^{−Ea/RT}

If we take to log on both sides of the equation, the resulting equation will be as follows:

$lnk=ln{A}^{–E_a/R\times 1/T}$.... (1)

Now let’s compare this equation with the equation of a straight line that is y=mx+c, where the m= slope of the line and c= y-intercept.

So we have:

y=lnk

x=1/T

m=−Ea/R

c=lnA

So a plot of ln k on the y-axis and 1/T on x-axis will give a straight line with its slop equal to –Ea/R and y-intercept equal to ln A.

As per Arrhenius Equation, for first condition:

$ln{K}_1=ln{A}^{–E_a/R\times 1/T_1}$ ……………. (2)

For second condition:

$ln{K}_2=ln{A}^{–E_a/R\times 1/T_2}$……………. (3)

Subtracting Equation (2) from equation (3) we will get,

$ln{k}_2–ln{k}_1=E_a/R\times 1/T_1–E_a/R\times 1/T_2$

$ln{k}_2{k}_1=E_a/R(1/T_1–1/T_2)$