Arrhenius Equation: Understanding Temperature Dependence of Chemical Reactions

If you have ever boiled water, you have experienced the effect of temperature on chemical reactions. The hotter the water, the faster it boils.

This simple observation reveals an important aspect of physical chemistry: Temperature plays a crucial role in determining the rate at which chemical reactions occur. But how do we measure the impact of temperature on reaction rates?

The answer lies in the Arrhenius equation, a fundamental tool in physical chemistry that describes the temperature dependence of reaction rate constants.

## Background and Significance of the Arrhenius Equation

The Arrhenius equation, named after the Swedish chemist Svante Arrhenius who first proposed it in 1889, is an exponential formula that expresses the rate constant of a chemical reaction as a function of temperature and activation energy. The equation is widely used in physical chemistry, chemical engineering, and materials science to model and predict the behavior of chemical reactions under different conditions.

One of the key insights of the Arrhenius equation is that the rate constant of a chemical reaction increases with temperature, due to the increase in the number of particles with sufficient energy to overcome the activation barrier and form reactant products. This temperature dependence is quantified by the exponential term exp(-Ea/RT), where Ea is the activation energy, R is the universal gas constant, and T is the absolute temperature.

The Arrhenius equation also includes a prefactor A, also known as the pre-exponential factor, that accounts for the frequency of molecular collisions and other factors that affect the reaction rate.

## Form of the Arrhenius Equation

The Arrhenius equation can be written in different forms, depending on the variables of interest. One common form is:

k = A * exp(-Ea/RT)

where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy in joules per mole, R is the gas constant in joules per mole Kelvin, and T is the absolute temperature in Kelvin.

Another form of the Arrhenius equation, known as the two-point form, is useful for determining the activation energy and pre-exponential factor from experimental data. This form is:

ln(k/k) = (-Ea/R) * (1/T – 1/T)

where k and k are the rate constants at two different temperatures T and T, respectively.

## Units of the Arrhenius Equation

The units of the Arrhenius equation depend on the units of the activation energy, pre-exponential factor, and temperature. The activation energy is typically expressed in joules per mole (J/mol), while the gas constant has a value of 8.314 J/mol-K.

The pre-exponential factor can have different units depending on the reaction, such as liters per mole-second or per square meter per second. The temperature is measured in Kelvin (K), which is equivalent to Celsius plus 273.15.

## Arrhenius Plot and Interpretations

A useful way to visualize the temperature dependence of the Arrhenius equation is through an Arrhenius plot, which graphs the natural logarithm of the rate constant ln(k) versus the reciprocal of the absolute temperature 1/T. The plot has a straight-line equation of the form:

ln(k) = ln(A) – (Ea/R) * (1/T)

From the slope and intercept of the Arrhenius plot, one can determine the activation energy and pre-exponential factor of the reaction.

## The activation energy is related to the slope of the plot by the equation:

Ea/R = -slope

while the pre-exponential factor is related to the intercept by the equation:

ln(A) = intercept

## Activation Energy and Reaction Rate

The activation energy is a measure of the minimum energy required for reactant molecules to transform into product molecules. As the temperature increases, more and more reactant molecules acquire this minimum energy, resulting in a larger fraction of successful collisions and faster reaction rates.

The impact of activation energy on reaction rate can be illustrated by the catalyzed reaction of hydrogen peroxide with potassium iodide, which produces water and iodine:

2 H2O2 + 2 KI 2 H2O + I2

In the absence of a catalyst, the reaction is slow and requires high activation energy. However, the addition of a catalyst such as manganese dioxide greatly lowers the activation energy, leading to a faster reaction rate.

This effect can be seen on an Arrhenius plot, where the slope of the catalyzed reaction is smaller than that of the uncatalyzed reaction, indicating a lower activation energy.

## Conclusion

In summary, the Arrhenius equation is a powerful tool in physical chemistry for describing the temperature dependence of reaction rate constants. By incorporating the activation energy and pre-exponential factor, the equation can model the behavior of a wide range of chemical systems under different conditions.

Through the use of Arrhenius plots and interpretation of activation energies, scientists and engineers can better understand and optimize chemical reactions for a variety of applications. In the previous section, we discussed the basic concepts of the Arrhenius equation.

We learned that we can calculate the activation energy and pre-exponential factor of a chemical reaction by using the Arrhenius equation in its two-point form. In this section, we will look at the aim and concept of the two-point form of Arrhenius equation and explore example problems and their solutions.

## Aim and Concept of the Two-Point Form of Arrhenius Equation

The primary goal of the two-point form of the Arrhenius equation is to determine the unknown variable in the equation (either activation energy or pre-exponential factor) when two values of the rate constant are given at two different temperatures. The Arrhenius plot can be used to visualize the relationship between temperature and reaction rate, but it may not be as accurate as the two-point form.

Determining the unknown variable in the two-point form of the Arrhenius equation requires the measurement of the rate constant at two different temperatures, and the knowledge of the universal gas constant.

## The Two-Point Form of Arrhenius Equation

The two-point form of the Arrhenius equation is:

ln(k2/k1) = (-Ea/R)(1/T2 – 1/T1) + ln(A)

where k1 and k2 are the rate constants at two different temperatures T1 and T2, respectively, Ea is the activation energy, R is the universal gas constant, and A is the pre-exponential factor. The left-hand side of the equation is the natural logarithm of the ratio of the rate constants between two temperatures, which can be determined experimentally.

The right-hand side of the equation contains the unknown variables Ea and A, and the known variables R, T1, and T2. The unknown variables can be determined by solving the equation using algebraic methods.

## Example Problems and

## Solutions

Problem 1: Calculation of Activation Energy

## Given the following data at two different temperatures:

k1 = 2.5 x 10^-4 L/(mol*s) at T1 = 300 K

k2 = 5.0 x 10^-4 L/(mol*s) at T2 = 400 K

Calculate the activation energy of the reaction.

## Solution

Using the two-point form of the Arrhenius equation:

ln(k2/k1) = (-Ea/R)(1/T2 – 1/T1) + ln(A)

## Substituting the given values:

ln(5.0 x 10^-4 L/(mol*s)/2.5 x 10^-4 L/(mol*s)) = (-Ea/8.314 J/(mol*K)) x (1/400 K – 1/300 K) + ln(A)

ln(2) = (-Ea/8.314) x (1/400 – 1/300) + ln(A)

Solving for ln(A) by moving all other terms to the left-hand side and taking the exponential of both sides:

ln(A) = ln(2) + [Ea/(8.314 x 1.25 x 10^2)]

ln(A) = ln(2) + (0.008Ea)

Substituting this expression back into the two-point form equation:

ln(k2/k1) = (-Ea/8.314 J/(mol*K)) x (1/400 K – 1/300 K) + ln(2) + (0.008Ea)

## Simplifying and solving for Ea:

Ea = [(-8.314 J/(mol*K)) x (1/400 K – 1/300 K) ln(k2/k1) – ln(2)] / [0.008 ln(k2/k1) + 1]

Ea = 79.8 kJ/mol

Therefore, the activation energy of the reaction is 79.8 kJ/mol. Problem 2: Calculation of Rate constant at a different temperature

## Given the following data at two different temperatures:

k1 = 4.2 x 10^-3 L/(mol*s) at T1 = 300 K

Activation energy, Ea = 40 kJ/mol

Universal gas constant, R = 8.314 J/(mol*K)

Calculate the rate constant k2 at T2 = 400 K.

## Solution

Using the two-point form of the Arrhenius equation:

ln(k2/k1) = (-Ea/R)(1/T2 – 1/T1) + ln(A)

Substituting the given values and assuming that the pre-exponential factor is constant:

ln(k2/4.2×10^-3) = [-40,000 J/mol/(8.314 J/(mol*K))] x (1/400 K – 1/300 K)

ln(k2/4.2×10^-3) = -537.33

## Taking the exponential of both sides:

k2/4.2×10^-3 = e^-537.33

k2 = 4.2 x 10^-3 x e^-537.33

k2 = 0 L/(mol*s)

Therefore, at T2 = 400 K, the rate constant k2 is 0 L/(mol*s). This result shows that the reaction will not occur at 400 K under these conditions.

## Conclusion

The two-point form of the Arrhenius equation is a powerful tool for determining the unknown variables in the equation, namely, activation energy and pre-exponential factor. With the knowledge of these variables, we can predict the rate of a chemical reaction at other temperatures.

By solving example problems, we can understand the practical application of this equation in a chemical system. In summary, the Arrhenius equation is a powerful tool for describing the temperature dependence of chemical reactions, and the two-point form of the equation can be used to determine the activation energy and pre-exponential factor of a reaction.

By solving example problems, we can understand the practical application of this equation for predicting and optimizing chemical reactions. Overall, the Arrhenius equation is a fundamental concept in physical chemistry that has important applications in fields ranging from chemical engineering to materials science.

## FAQs:

Q: What is the Arrhenius equation? A: The Arrhenius equation is an exponential formula that expresses the rate constant of a chemical reaction as a function of temperature and activation energy.

Q: What is the two-point form of the Arrhenius equation? A: The two-point form of the Arrhenius equation is used to determine the unknown variables in the equation (either activation energy or pre-exponential factor) when two values of the rate constant are given at two different temperatures.

Q: What are some applications of the Arrhenius equation? A: The Arrhenius equation is widely used in physical chemistry, chemical engineering, and materials science to model and predict the behavior of chemical reactions under different conditions.

Q: How does temperature impact chemical reactions? A: Temperature plays a crucial role in determining the rate at which chemical reactions occur.

As the temperature increases, more and more reactant molecules acquire the minimum energy required for transformation, resulting in a larger fraction of successful collisions and faster reaction rates. Q: How can the activation energy and pre-exponential factor of a reaction be determined?

A: The activation energy and pre-exponential factor of a reaction can be determined using the two-point form of the Arrhenius equation, which requires the measurement of the rate constant at two different temperatures.