The Arrhenius Equation

We know that increasing the temperature speeds up nearly all reactions. But by how much? The Arrhenius equation gives us the precise mathematical relationship between the rate constant k, the temperature T, and the activation energy $E_{a}$.

The Equation

Understanding the Variables

The frequency factor (A) represents how often molecules collide with the correct orientation. It depends on the physical characteristics of the molecules (size, shape) and the concentration. A larger A means more frequent, correctly oriented collisions.

The exponential term (e^(-$E_{a}$/RT)) represents the fraction of molecules that have enough energy to overcome the activation energy barrier. This fraction is always between 0 and 1.

How Temperature Affects k

The key to understanding the Arrhenius equation on the MCAT is the exponent -$E_{a}$/RT:

• Increasing T makes the denominator RT larger, making $E_{a}$/RT smaller, making -$E_{a}$/RT less negative (closer to zero). Since e⁰ = 1, the exponential term gets larger, so k increases.
• Increasing $E_{a}$ makes $E_{a}$/RT larger, making -$E_{a}$/RT more negative. Since e^(-large number) is tiny, k decreases.
• As T approaches infinity, -$E_{a}$/RT approaches zero, and k approaches A (the maximum possible rate constant).

The Arrhenius Plot

Taking the natural log of both sides of the Arrhenius equation gives a linear form:

Plotting ln(k) vs. 1/T gives a straight line. The slope is -$E_{a}$/R, so you can determine the activation energy from the slope:

$E_{a}$ = -slope x R

A steeper (more negative) slope means a higher activation energy. The reaction is more sensitive to temperature changes.

The Two-Point Form

If you know the rate constant at two different temperatures, you can find $E_{a}$ without plotting: