# Arrhenius Equation

The Arrhenius equation is a mathematical formula that relates the rate of a chemical reaction to the temperature and the activation energy of the reaction. It is expressed as:

k = Ae^(-Ea/RT)

where:

- k is the rate constant of the reaction
- A is the pre-exponential factor (also called the frequency factor), which is a constant that depends on the frequency of molecular collisions
- Ea is the activation energy of the reaction
- R is the gas constant
- T is the temperature in Kelvin

The Arrhenius equation is widely used in the study of chemical kinetics and helps to explain how the rate of a chemical reaction changes with temperature. It states that increasing the temperature of a reaction increases the rate constant, and therefore the reaction rate, as long as the activation energy remains constant.

Here’s an example: Consider the reaction of hydrogen gas with iodine gas to form hydrogen iodide gas:

H2(g) + I2(g) → 2HI(g)

The rate law for this reaction is:

rate = k[H2][I2]

where [H2] and [I2] are the concentrations of hydrogen and iodine, respectively. The rate constant k can be determined experimentally at different temperatures.

Suppose that at a temperature of 300 K, the rate constant k is measured to be 1.5 x 10^-3 L mol^-1 s^-1, and the activation energy Ea for the reaction is 125 kJ/mol. Using the Arrhenius equation, we can calculate the rate constant at a different temperature, say 350 K:

k = Ae^(-Ea/RT)

Taking the natural logarithm of both sides and rearranging, we get:

ln(k) = ln(A) – (Ea/RT)

At 300 K, we know that ln(k) = ln(1.5 x 10^-3) = -6.502. Substituting this and the other known values into the equation above, we can solve for ln(A):

ln(A) = ln(k) + (Ea/RT) = -6.502 + (125000 J/mol) / (8.314 J/mol K x 300 K) = 22.154

So the pre-exponential factor A is:

A = e^ln(A) = e^22.154 = 1.4 x 10^9 s^-1

Now we can use the Arrhenius equation to calculate the rate constant at 350 K:

k = Ae^(-Ea/RT) = (1.4 x 10^9 s^-1) x e^(-125000 J/mol / (8.314 J/mol K x 350 K)) = 6.8 x 10^-3 L mol^-1 s^-1

Therefore, the rate of the reaction at 350 K is expected to be higher than at 300 K, due to the higher rate constant.

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