Question

T2
> T1
and Ea is positive, then what is the relation between K1
and K2​

Answer 1

Given:

T2  > T1  and Ea is positive.

To find:

Then what is the relation between K1  and K2​

Solution:

The Arrhenius equation is given by,

k = Ae ^{-Ea/RT}

where,

k = kinetic rate constant at T

A = collision frequency factor

Ea = activation energy

R = the energy gas constant

T = temperature in Kelvin

Taking ln on both the sides of Arrhenius equation, we get,

ln k = ln A - Ea/RT

Let us consider, two temperatures,

T1 and T2, so we have,

ln k1 = ln A - Ea/RT1 .......(1)

ln k2 = ln A - Ea/RT2 .......(2)

dividing equations (1) and (2), we get,

ln k2/k1 = - Ea/RT (1/T2 - 1/T1)

further solving, we get,

ln k2/k1 = - Ea/RT ( T1 - T2 / T1T2 )

From given, we have,

T2 > T1

⇒ ( T1 - T2 / T1T2 ) < 1          (∵ T2 > T1)

⇒ - Ea/RT ( T1 - T2 / T1T2 ) > 1           [∵  - Ea/RT × ( T1 - T2 / T1T2 ) > 1]

⇒ ln k2/k1 > 1

using log properties, we have,

ln k2/k1 = ln k2 - ln k1

⇒ ln k2 - ln k1 > 1

⇒ ln k2 > ln k1

taking anti log on both the sides, we get,

∴ k2 > k1

Hence the relationship between k1 and k2.