Problem 50 - Introductory Chemistry Atoms First

Question

What is the relationship between the rate of a reaction and the value of \(E_{\mathrm{a}}\) for the reaction?

Step-by-Step Solution

Answer

The relationship between the rate of a reaction and the value of activation energy (Eₐ) for the reaction is inversely exponential. According to the Arrhenius equation, \(k = Ae^{\frac{-E_a}{RT}}\), a higher activation energy results in a slower reaction rate, while a lower activation energy results in a faster reaction rate.

Step 1: Introduce the Arrhenius equation

The Arrhenius equation relates the rate constant of a chemical reaction (k) to its activation energy (Eₐ) and the temperature (T) at which the reaction is taking place. The equation is given by: \[k = Ae^{\frac{-E_a}{RT}}\] Where: - *k* is the rate constant of the reaction, - *A* is the pre-exponential factor or the frequency factor, - *Eₐ* is the activation energy of the reaction, - *R* is the ideal gas constant (8.314 J/mol·K), and - *T* is the temperature in Kelvin.

Step 2: Examine the relationship between 'k' and 'Eₐ'

In the Arrhenius equation: \[k = Ae^{\frac{-E_a}{RT}}\] Since the pre-exponential factor (A) and the ideal gas constant (R) are constants, it is evident that the rate constant 'k' is directly affected by the exponential term, which depends on the activation energy (Eₐ) and temperature (T).

Step 3: Interpret the relationship between 'k' and 'Eₐ'

The relationship between the rate constant 'k' and the activation energy 'Eₐ' is inversely exponential. This means that as the activation energy (Eₐ) increases, the rate constant 'k' will decrease and vice versa. An increase in activation energy suggests that the reaction has a higher barrier to overcome before the reactants can transform into products. This results in a slower rate of reaction since it takes more energy for the reactants to overcome the energy barrier. In contrast, if the activation energy is lower, the reaction has a lower energy barrier, and reactants can transform into products more easily, thus increasing the rate of the reaction. In conclusion, the relationship between the rate of a reaction and the value of activation energy (Eₐ) for the reaction is inversely exponential. A higher activation energy results in a slower reaction rate, while a lower activation energy results in a faster reaction rate.

Key Concepts

Activation Energy

Activation energy, denoted as \(E_a\), is a crucial concept in the study of chemical kinetics. It is the minimum amount of energy needed for reactants to undergo a chemical reaction and form products. Think of it as the 'energy barrier' that reactants must overcome. The higher the barrier, the fewer reactant particles have the necessary energy to react when they collide.

Using the metaphor of a hill, the activation energy represents the height a runner must surmount to reach the other side. When the 'hill' is too high, fewer 'runners' (reactant molecules) can make it to the top and then down to the other side, where the reaction proceeds and products are formed. Consequently, the rate at which products form – the reaction rate – is directly influenced by the activation energy.

Rate Constant

The rate constant, represented by \(k\), is a proportionality factor that relates the rate of reaction to the concentration of reactants in the rate equation for a given reaction. It is a measure of how quickly a reaction proceeds under specific conditions. The Arrhenius equation beautifully highlights the significance of the rate constant, tying it to both the activation energy and temperature of the reaction.

It's important to realize that the rate constant is not just a static number. It changes with temperature and even slightly with the pressure (for reactions involving gases). A higher rate constant implies a faster reaction, signaling that reactants are being transformed into products efficiently. Consequently, understanding how temperature and activation energy affect the rate constant is fundamental to controlling reaction rates in real-world applications.

Reaction Rate

The reaction rate is the speed at which reactants are converted into products in a chemical reaction. It's what you measure when you want to know how fast a reaction is happening. If you picture chemical reactions as a race, the reaction rate would be the average speed of the racers – it dictates how quickly the 'finish line' (complete conversion of reactants to products) is reached.

Several factors influence this rate, including reactant concentrations, temperature, presence of catalysts, and of course, the activation energy. In the context of the Arrhenius equation, a lower activation energy correlates with a higher reaction rate, because lower energy barriers mean more reactant particles can successfully react upon collision.

Exponential Relationship

The exponential relationship in chemistry describes how changes in one quantity can lead to exponential increases or decreases in another. In the case of the Arrhenius equation, the rate constant \(k\) and activation energy \(E_a\) have this type of relationship. Specifically, the reaction rate increases exponentially as the activation energy decreases, under the condition that all other factors remain constant.

To understand this better, imagine if the activation energy is slightly decreased by adding a catalyst or increasing the temperature. The rate constant doesn't just increase additively; it multiplies, due to the exponential factor in the Arrhenius equation, \(e^{-\frac{E_a}{RT}}\). This exponential factor is why even small changes in activation energy can have dramatic effects on reaction rates.