7.1 Arrhenius equation and its components

Arrhenius Equation and Chemical Kinetics

The Arrhenius equation connects reaction rate to temperature in a single, powerful expression. It tells you how much a reaction speeds up when temperature rises, and it quantifies the energy barrier molecules must overcome to react. Understanding each component of this equation is essential for predicting and controlling reaction rates.

The Arrhenius Equation

The equation is:

$k = Ae^{-E_a/RT}$

where:

• $k$ = rate constant
• $A$ = pre-exponential (frequency) factor
• $E_a$ = activation energy
• $R$ = universal gas constant ($8.314 \text{ J mol}^{-1}\text{ K}^{-1}$)
• $T$ = absolute temperature in Kelvin

This equation describes how the rate constant of a chemical reaction depends on temperature. Because $k$ appears in every rate law, the Arrhenius equation is what links temperature to the overall speed of a reaction.

Notice the negative exponent: $-E_a/RT$. As temperature increases, that fraction gets smaller in magnitude, so the exponential term gets closer to 1, and $k$ increases. That's the mathematical reason reactions go faster when you heat them up.

Components of the Arrhenius Equation

Rate constant ($k$) represents the speed of a reaction at a given temperature. A larger $k$ means a faster reaction. For example, the decomposition of hydrogen peroxide has a measurable $k$ that increases significantly with temperature. The units of $k$ depend on the overall order of the reaction (e.g., $\text{s}^{-1}$ for first order, $\text{M}^{-1}\text{s}^{-1}$ for second order).

Pre-exponential factor ($A$) reflects how frequently molecules collide with the correct orientation to react. It comes from collision theory: not every collision leads to a reaction, because molecules need to be aligned properly. $A$ is treated as roughly constant for a given reaction and doesn't change much with temperature. Reactions involving large, complex molecules tend to have smaller $A$ values because the geometric requirements for a productive collision are stricter.

Activation energy ($E_a$) is the minimum energy reactants need to overcome the energy barrier and form products. You can think of it as the "hill" on a potential energy diagram that reactants must climb. Lower $E_a$ means the reaction proceeds more easily at a given temperature. This is exactly how catalysts work: they provide an alternative reaction pathway with a lower $E_a$, speeding up the reaction without being consumed.

Universal gas constant ($R$) has a fixed value of $8.314 \text{ J mol}^{-1}\text{ K}^{-1}$. It serves as a bridge between energy units and temperature units in the equation. Make sure your $E_a$ is in joules per mole (not kJ/mol) when plugging into the equation, or you'll be off by a factor of 1000.

Absolute temperature ($T$) must be in Kelvin. Convert from Celsius by adding 273.15. Higher temperatures mean molecules have more kinetic energy, so a greater fraction of collisions exceed the activation energy threshold.

Rate Constant vs. Activation Energy

The Arrhenius equation reveals an exponential relationship between $k$ and temperature. A small increase in $T$ can produce a large increase in $k$, especially when $E_a$ is high.

• High $E_a$ reactions are very sensitive to temperature changes. Combustion reactions are a good example: they barely proceed at room temperature but are extremely fast once ignited. A small temperature increase dramatically shifts the fraction of molecules with enough energy to react.
• Low $E_a$ reactions are less sensitive to temperature changes. Many enzyme-catalyzed biological reactions fall into this category, which is why they proceed efficiently at body temperature.

The pre-exponential factor $A$ stays essentially constant for a given reaction across typical temperature ranges. So when you're comparing $k$ at two different temperatures, the change is driven almost entirely by the exponential term $e^{-E_a/RT}$.

Calculations with the Arrhenius Equation

Type 1: Finding $k$ at a single temperature

If you know $A$, $E_a$, and $T$, plug directly into:

$k = Ae^{-E_a/RT}$

1. Convert $T$ to Kelvin and $E_a$ to J/mol if needed.
2. Calculate the exponent: $-E_a / (R \times T)$.
3. Evaluate $e$ raised to that exponent.
4. Multiply by $A$ to get $k$.

Type 2: Comparing $k$ at two temperatures

When you know $k_1$ at temperature $T_1$ and need $k_2$ at temperature $T_2$, use the two-point form:

$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

1. Convert both temperatures to Kelvin and $E_a$ to J/mol.

2. Calculate $\frac{1}{T_1} - \frac{1}{T_2}$.

3. Multiply by $\frac{E_a}{R}$ to get $\ln(k_2/k_1)$.

4. Solve for $k_2$: $k_2 = k_1 \times e^{(E_a/R)(1/T_1 - 1/T_2)}$.

This two-point form is especially useful because you don't need to know $A$. As a rough rule of thumb, many common reactions approximately double in rate for every 10°C increase in temperature, though the actual factor depends on $E_a$.