Question

calculate the rate constant, k, for a reaction at 57.0 °c that has an activation energy of 83.5 kj/mol and a frequency factor of 8.97×10¹¹ s⁻¹.

Explanation:

Step1: Convert temperature to Kelvin

$T=(57.0 + 273.15)\text{ K}=330.15\text{ K}$

Step2: Convert activation energy to J/mol

$E_a = 83.5\text{ kJ/mol}\times1000\text{ J/kJ}=83500\text{ J/mol}$

Step3: Use Arrhenius equation

The Arrhenius equation is $k = A e^{-\frac{E_a}{RT}}$, where $A = 8.97\times 10^{11}\text{ s}^{-1}$, $R=8.314\text{ J/(mol}\cdot\text{K)}$, $E_a = 83500\text{ J/mol}$, and $T = 330.15\text{ K}$.
First, calculate the exponent: $\frac{E_a}{RT}=\frac{83500\text{ J/mol}}{8.314\text{ J/(mol}\cdot\text{K)}\times330.15\text{ K}}\approx30.47$
Then, $k = 8.97\times 10^{11}\text{ s}^{-1}\times e^{- 30.47}$
Since $e^{-30.47}\approx1.79\times10^{-13}$, then $k=8.97\times 10^{11}\text{ s}^{-1}\times1.79\times 10^{-13}=0.0160563\text{ s}^{-1}\approx1.61\times 10^{-2}\text{ s}^{-1}$

Answer:

$1.61\times 10^{-2}$