Question

A first-order reaction is studied at several different temperatures, and the following values of k are measured:

T(oC)	k(sec-1)
50.0	0.0108
70.1	0.0734
89.4	0.454
101.0	1.38

Use these results to determine the activation energy (E_a) for this reaction, and provide appropriate units for E_a.

Thank you!

Answer 1

According to Arrhenius Equation , K = A e ^{-Ea / RT}

Where

K = rate constant

T = temperature

R = gas constant = 8.314 J/mol-K

E_a = activation energy

A = Frequency factor (constant)

Rate constant, K = A e ^{- Ea / RT}

                  log K = log A - ( E_a / 2.303RT )   ---(1)

If we take rate constants at two different temperatures, then

                log K = log A - ( E_a / 2.303RT )   --- (2)

    &         log K' = log A - (E_a / 2.303RT’)    ---- (3)

Eq (3 ) - Eq ( 2 ) gives

log ( K' / K ) = ( E_a / 2.303 R ) x [ ( 1/ T ) - ( 1 / T' ) ]

              E_a = [(2.303R x T x T’) / (T’ - T)] x log (K’ / K)

Let us consider first two readings in the given table

T = 50.0 ^oC = 50.0+273 = 323 K

T' = 70.1 ^oC = 70.1 +273 = 343.1 K

K = 0.0108 s^-1

K' = 0.0734 s^-1

Plug the values we get    

E_a = [(2.303R x T x T’) / (T’ - T)] x log (K’ / K)

E_a = [(2.303x8.314 x 323 x 343.1) / (343.1 - 323)] x log (0.0734 / 0.0108)   

     = 87.861x10³ J

     = 87.861 kJ