The plot of \(\log \mathrm{k}_{\mathrm{f}}\) versus \(\frac{1}{\mathrm{~T}}\) for a reversible reaction \(\mathrm{A}(\mathrm{g}) \rightleftharpoons \mathrm{P}(\mathrm{g})\) is shown. (see plot) Pre-exponential factors for the forward and backward reactions are \(10^{15} \mathrm{~s}^{-1}\) and \(10^{11} \mathrm{~s}^{-1}\), respectively. If the value of \(\log \mathrm{K}\) for the reaction at 500 K is 6, the value of \(\left|\log \mathrm{k}_{\mathrm{b}}\right|\) at 250 K is ____. [K = equilibrium constant of the reaction \(\mathrm{k}_{\mathrm{f}}=\) rate constant of forward reaction \(\mathrm{k}_{\mathrm{b}}=\) rate constant of backward reaction]

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Chemical Kinetics

The plot of

lo g k_{f}

versus

\frac{1}{T}

for a reversible reaction

A (g) ⇌ P (g)

is shown.

(see plot)

Pre-exponential factors for the forward and backward reactions are

1 0^{15} s^{- 1}

and

1 0^{11} s^{- 1}

, respectively. If the value of

lo g K

for the reaction at 500 K is 6, the value of

∣ lo g k_{b} ∣

at 250 K is ____.

[K = equilibrium constant of the reaction

k_{f} =

rate constant of forward reaction

k_{b} =

rate constant of backward reaction]

Answer: 5

From the Arrhenius equation for the forward reaction:

lo g k_{f} = \frac{- E _{f}}{2.303 R T} + lo g A_{f}

From the plot, at

1/ T = 0.002

(i.e., T = 500 K),

lo g k_{f} = 9

.

With

A_{f} = 1 0^{15}

9 = \frac{- E _{f}}{2.303 R} (0.002) + 15

\frac{E _{f}}{2.303 R} = 3000

For the equilibrium constant:

lo g K = \frac{- ( E _{f} - E _{b} )}{2.303 R T} + lo g \frac{A _{f}}{A _{b}}

At 500 K with

lo g K = 6

6 = \frac{- ( E _{f} - E _{b} )}{2.303 R ( 500 )} + lo g \frac{1 0 ^{15}}{1 0 ^{11}}

6 = \frac{- ( E _{f} - E _{b} )}{2.303 R ( 500 )} + 4

E_{b} - E_{f} = 1000 R (2.303)

E_{b} = 4000 R (2.303)

For the backward reaction at 250 K:

lo g k_{b} = \frac{- E _{b}}{2.303 R T} + lo g A_{b}

lo g k_{b} = \frac{- 4000}{250} + 11 = - 16 + 11 = - 5

Therefore:

∣ lo g k_{b} ∣ = 5

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Integer answer