# How do you solve for k using the Arrhenius Equation? A first order reaction has an activation energy of "E"_a = "65.7 kJ/mol" and a frequency factor (pre-exponential factor, "A") of 1.31 xx 10^12 "s"^(-1). Calculate the rate constant at 19^@ "C".

Aug 5, 2018

$k \approx 2.34 \textcolor{w h i t e}{l} {\text{s}}^{- 1}$

#### Explanation:

k = "A" * e^(-"E"_a//(R * T))

gives the relationship between the following quantities

• The rate constant $k$ as seen in the rate law of the reaction;
• The pre-exponential factor $\text{A}$ for this particular reaction;
• The activation energy ${\text{E}}_{a}$ of this reaction;
• The absolute temperature $T$ under which the reaction takes place.

Whereas the ideal gas constant is also involved in the calculation. The ideal gas constant takes various units, with a different numerical value for each. It would thus be necessary to dimensional analysis while calculating the exponent part of the reaction. The exponent shall end up without a unit. This example takes $R = 8.314 \textcolor{w h i t e}{l} \textcolor{n a v y}{\text{J") //("mol" * "K}}$.

Note that the Arrhenius equation requires an absolute temperature $T$ in degree Kelvins $\text{K}$ whereas the question supplied the temperature in degrees Celsius. Numerically add $273 \left(.15\right)$ to the temperature in degrees Celsius to get the absolute temperature in Kelvins.

$T = 19 + 273 = 292 \textcolor{w h i t e}{l} \text{K}$

The question gives the activation energy in kilojoules; however, the ideal gas constant demands the unit joule.

$\text{E"_a = 65.7 color(white)(l) "kJ" = 65.7 xx color(navy)(10^3 color(white)(l) "J}$

Substitute the real values and calculate the exponent part of the expression:

-"E"_a / (R * T) = (65.7 xx color(navy)(10^3 color(white)(l) "J") * "mol"^(-1))/(8.314 color(white)(l) color(navy)("J") * "mol"^(-1) * "K"^(-1) * 292 color(white)(l) "K")
$\textcolor{w h i t e}{- \text{E"_a / (R * T)) ~~-27.0 color(white)(l) color(lightgreen)("(dimensionless)}}$

Make sure that all units cancel out such that the exponent part is dimensionless. Evaluate the rest of the equation to find the rate constant $k$

k = "A" * e^(-"E"_a//(R * T))
$\textcolor{w h i t e}{k} = 1.31 \times {10}^{12} \textcolor{w h i t e}{l} \textcolor{n a v y}{{s}^{- 1}} \times {e}^{- 27.0}$
$\textcolor{w h i t e}{k} \approx 2.34 \textcolor{w h i t e}{l} \textcolor{n a v y}{{s}^{- 1}}$