For a certain reaction the values of Arrhenius factor and Activation energy are 4 x 1013 collision/sec and 98.6KJ/mol at 303K. Calculate the rate constant if reaction is 1st order?( R = 8.341mol-1K-1)a)6.07 x 10-3b)3.02 x 10-5c)4.07 x 10-4d)7.42 x 10-3Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To calculate the rate constant for a first-order reaction, we can use the Arrhenius equation:

k = A * e^(-Ea/RT)

where:
k is the rate constant
A is the pre-exponential factor or the Arrhenius factor
Ea is the activation energy
R is the gas constant (8.341 mol^(-1)K^(-1))
T is the temperature in Kelvin

Given:
A = 4 x 10^13 collision/sec
Ea = 98.6 kJ/mol
T = 303 K

1. Convert Ea from kJ/mol to J/mol:
Ea = 98.6 kJ/mol * (1000 J/kJ) = 98600 J/mol

2. Plug the values into the Arrhenius equation and solve for k:
k = (4 x 10^13 collision/sec) * e^(-98600 J/mol / (8.341 mol^(-1)K^(-1) * 303 K))

3. Simplify the expression:
k = (4 x 10^13) * e^(-98600 / (8.341 * 303))

4. Calculate the exponent:
-98600 / (8.341 * 303) = -40.04

5. Calculate e^(-40.04) using a calculator or a math software:
e^(-40.04) ≈ 1.035 x 10^(-18)

6. Plug the value back into the equation:
k ≈ (4 x 10^13) * (1.035 x 10^(-18))

7. Simplify the expression:
k ≈ 4.14 x 10^(-6)

Therefore, the rate constant for the first-order reaction is approximately 4.14 x 10^(-6) s^(-1), which is closest to option C: 4.07 x 10^(-4).

Answer

Arrhenius factor(A) = 4 x 10¹³ collisions/sec
Activation energy (E_a)=98.6KJ/mol=98.6 x 10³J/mol, T=303 K
log K = log A – (E_a/2.303RT)
log K = log (4 x 10¹³) – (98.6 x 10³)/(2.303 x 8.314 x 303)
log K = 13.6020 – (98.6 x 10³/5801.584)
log k = -3.39
K = 10^-3.39
K = 4.07 x 10^-4.

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