The Arrhenius equation is a key concept in chemical kinetics, linking reaction rates to temperature. It helps us understand why reactions speed up as things get hotter, and how much energy molecules need to react.

This equation has several parts, each playing a role in determining reaction speed. By using it, we can predict how fast reactions will happen at different temperatures, which is super useful in real-world applications like cooking or industrial processes.

Arrhenius Equation and Chemical Kinetics

Arrhenius equation in chemical kinetics

• Describes the relationship between the rate constant of a chemical reaction and temperature, helping predict reaction rates and understand the effect of temperature on reaction rates
• Mathematically expressed as $k = Ae^{-E_a/RT}$, where $k$ is the rate constant, $A$ is the pre-exponential factor or frequency factor, $E_a$ is the activation energy, $R$ is the universal gas constant, and $T$ is the absolute temperature in Kelvin
• Fundamental concept in chemical kinetics used to determine how the rate of a reaction changes with temperature (exothermic vs endothermic reactions)

Components of Arrhenius equation

• Rate constant ($k$) represents the speed of a chemical reaction at a given temperature, with higher $k$ values indicating faster reactions (decomposition of hydrogen peroxide)
• Pre-exponential factor or frequency factor ($A$) relates to the frequency of molecular collisions with the correct orientation, influenced by factors such as the geometry of the reacting molecules and the reaction mechanism (collision theory)
• Activation energy ($E_a$) is the minimum energy required for reactants to overcome the energy barrier and form products, determining the temperature dependence of the reaction rate, with lower $E_a$ values resulting in faster reactions (catalysts lower activation energy)
• Universal gas constant ($R$) is a constant value that relates energy to temperature, with a value of 8.314 J mol$^{-1}$ K$^{-1}$
• Absolute temperature ($T$) is the temperature of the reaction in Kelvin, with higher temperatures leading to faster reaction rates (cooking food at higher temperatures)

Rate constant vs activation energy

• The Arrhenius equation shows that the rate constant ($k$) depends on the activation energy ($E_a$) and temperature ($T$), with the rate constant increasing exponentially as temperature increases due to higher temperatures providing more energy for reactants to overcome the activation energy barrier (boiling water vs room temperature water)
• The activation energy determines the sensitivity of the rate constant to temperature changes, with reactions having higher $E_a$ being more sensitive to temperature changes and reactions with lower $E_a$ being less sensitive to temperature changes (combustion reactions vs enzymatic reactions)
• The pre-exponential factor ($A$) is temperature-independent and remains constant for a given reaction

Calculations with Arrhenius equation

1. Calculate the rate constant at a specific temperature, given the activation energy and pre-exponential factor using the Arrhenius equation: $k = Ae^{-E_a/RT}$

2. Calculate the rate constant ($k_2$) at a new temperature ($T_2$), when the rate constant ($k_1$) at a reference temperature ($T_1$) is known, using the equation: $ln(k_2/k_1) = (E_a/R)((1/T_1) - (1/T_2))$

3. Rearrange the equation to solve for $k_2$: $k_2 = k_1 e^{(E_a/R)((1/T_1) - (1/T_2))}$

4. Use these equations and the given information to calculate the rate constant at any temperature (doubling the rate constant for every 10°C increase in temperature)