Answer:
D. 18,800 J/mol
Explanation:
We need to use the Arrhenius equation to solve for this problem:
[tex]k=Ae^{\frac{-E_a}{RT}[/tex], where k is the rate constant, A is the frequency factor, [tex]E_a[/tex] is the activation energy, R is the gas constant, and T is the temperature in Kelvins.
We want to find the value of [tex]E_a[/tex], so let's plug some of the information we have into the equation. The gas constant we can use here is 8.31 J/mol-K.
At 0°C, which is 0 + 273 = 273 Kelvins, the rate constant k is [tex]4.6*10^{-2}[/tex]. So:
[tex]k=Ae^{\frac{-E_a}{RT}[/tex]
[tex]4.6*10^{-2}=Ae^{\frac{-E_a}{8.31*273}[/tex]
At 20°C, which is 20 + 273 = 293 Kelvins, the rate constant k is [tex]8.1*10^{-2}[/tex]. So:
[tex]k=Ae^{\frac{-E_a}{RT}[/tex]
[tex]8.1*10^{-2}=Ae^{\frac{-E_a}{8.31*293}[/tex]
We now have two equations and two variables to solve for. We just want to find Ea, so let's write the first equation for A in terms of Ea:
[tex]4.6*10^{-2}=Ae^{\frac{-E_a}{8.31*273}[/tex]
[tex]A=\frac{4.6*10^{-2}}{e^{\frac{-E_a}{8.31*273}} }[/tex]
Plug this in for A in the second equation:
[tex]8.1*10^{-2}=Ae^{\frac{-E_a}{8.31*293}[/tex]
[tex]8.1*10^{-2}=\frac{4.6*10^{-2}}{e^{\frac{-E_a}{8.31*273}} }e^{\frac{-E_a}{8.31*293}[/tex]
After some troublesome manipulation, the answer should come down to be approximately:
Ea = 18,800 J/mol
The answer is thus D.
