Answer:
54.9 kJ/mol
Explanation:
The relation between the activation energy (Ea) and the rate constant (k) is given by the Arrhenius equation.
[tex]k=A.e^{-Ea/RT}[/tex]
where,
A is a collision factor
R is the ideal gas constant
T is the absolute temperature
Reaction B is one million times faster than reaction A at the same temperature. So [tex]k_{B}=10^{6} k_{A}[/tex].
Then,
[tex]k_{B}=10^{6} k_{A}\\A.e^{-Ea_{B}/RT}=10^{6}A.e^{-Ea_{A}/RT}\\e^{-Ea_{B}/RT}=10^{6}e^{-Ea_{A}/RT}\\ln(e^{-Ea_{B}/RT})=ln(10^{6}e^{-Ea_{A}/RT})\\\frac{-Ea_{B}}{RT} =ln10^{6} -\frac{Ea_{A}}{RT} \\Ea_{B}=(ln10^{6} -\frac{Ea_{A}}{RT}).(-RT)=(ln10^{6}-\frac{89.1kJ/mol}{(8.314\times 10^{-3} kJ/mol.K).298K} ).(-8.314\times 10^{-3} \frac{kJ}{mol.K}.298K )=54.9kJ/mol[/tex]
