Question

A first order reaction has rate constants of 4.6 x 10-2 s-1 and 8.1 x 10-2 s-1 at 0ºC and 20ºC, respectively. What is the value for the activation energy?
A.
0.566 J/mol
B.
2.5 x 10-4 J/mol
C.
2260 J/mol
D.
18,800 J/mol
E.
1.76 J/mol

Answer 1

Answer:

D.  18,800 J/mol

Explanation:

We need to use the Arrhenius equation to solve for this problem:

$k=Ae^{\frac{-E_a}{RT}$, where k is the rate constant, A is the frequency factor, $E_a$ is the activation energy, R is the gas constant, and T is the temperature in Kelvins.

We want to find the value of $E_a$, 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 $4.6*10^{-2}$. So:

$k=Ae^{\frac{-E_a}{RT}$

$4.6*10^{-2}=Ae^{\frac{-E_a}{8.31*273}$

At 20°C, which is 20 + 273 = 293 Kelvins, the rate constant k is $8.1*10^{-2}$. So:

$k=Ae^{\frac{-E_a}{RT}$

$8.1*10^{-2}=Ae^{\frac{-E_a}{8.31*293}$

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:

$4.6*10^{-2}=Ae^{\frac{-E_a}{8.31*273}$

$A=\frac{4.6*10^{-2}}{e^{\frac{-E_a}{8.31*273}} }$

Plug this in for A in the second equation:

$8.1*10^{-2}=Ae^{\frac{-E_a}{8.31*293}$

$8.1*10^{-2}=\frac{4.6*10^{-2}}{e^{\frac{-E_a}{8.31*273}} }e^{\frac{-E_a}{8.31*293}$

After some troublesome manipulation, the answer should come down to be approximately:

Ea = 18,800 J/mol

The answer is thus D.