Question

A catalyst lowers the activation energy of a reaction from \(215 \mathrm{~kJ}\) to \(206 \mathrm{~kJ} .\) By what factor would you expect the reaction-rate constant to increase at \(25^{\circ} \mathrm{C}\) ? Assume that the frequency factors (A) are the same for both reactions.

Short answer

Answer: A catalyst lowers the activation energy of the reaction from 215 kJ/mol to 206 kJ/mol, and it increases the reaction-rate constant by a factor of approximately 1.031 at 25°C.

Step-by-step solution

Recall the Arrhenius Equation

The Arrhenius equation relates the temperature, the activation energy, and the reaction rate constant (k). The equation is: k = A * e^{(-E_a / RT)} where: - k is the reaction-rate constant - A is the frequency factor (pre-exponential factor) - E_a is the activation energy - R is the universal gas constant (8.314 J/(mol * K)) - T is the temperature in Kelvin

Convert the temperature to Kelvin

The temperature given is \(25^{\circ}C\). To convert this to Kelvin, we add 273.15: T = 25 + 273.15 = 298.15 K

Calculate the reaction-rate constants for the uncatalyzed and catalyzed reactions

Since the frequency factors (A) are the same for both reactions, we can use the ratios of the reaction-rate constants to find the factor by which the reaction-rate constant increases. Let's denote k1 and k2 as the reaction-rate constants for the uncatalyzed and catalyzed reactions, respectively. k1 = A * e^{(-215000 / (8.314 * 298.15))} k2 = A * e^{(-206000 / (8.314 * 298.15))}

Calculate the factor by which the reaction-rate constant increases

To find the factor by which the reaction-rate constant increases, we need to divide k2 by k1: Factor = k2 / k1 Factor = (A * e^{(-206000 / (8.314 * 298.15))}) / (A * e^{(-215000 / (8.314 * 298.15))}) Since A is the same for both reactions, we can simplify this expression: Factor = e^{(215000 - 206000) / (8.314 * 298.15)} Factor = e^{9000 / (8.314 * 298.15)} Factor ≈ e^{0.0305} ≈ 1.031 So, the reaction-rate constant increases by a factor of approximately 1.031 when the catalyst is used.

Key concepts

Activation Energy

Understanding the concept of activation energy is crucial when studying chemical reactions. Activation energy, often denoted as \(E_a\), is the minimum amount of energy required for reactants to transform into products during a chemical reaction. Think of it as the 'energy barrier' that reactants must overcome for a reaction to occur.

This barrier is crucial because it determines the speed at which a reaction will proceed. A higher activation energy means that fewer molecules at any given time have enough energy to react, making the reaction slower. Conversely, a lower activation energy means more molecules can overcome the barrier, leading to a faster reaction.

In the exercise provided, we observe how a catalyst affects this barrier by reducing the activation energy from \(215 \text{ kJ/mol}\) to \(206 \text{ kJ/mol}\), which leads to an increase in the reaction rate, as further explained by the Arrhenius equation.

Reaction Rate Constant

The reaction rate constant, represented by \(k\), is a proportionality factor that links the reactant concentration to the rate of reaction in the rate law equation. It's one of the central parameters in the Arrhenius equation, which mathematically expresses the relationship between the reaction rate constant and several factors including temperature \(T\) and activation energy \(E_a\).

The Arrhenius equation is given as: \[ k = A \times e^{(-E_a / RT)} \] where \(A\) is the frequency factor, indicating the number of times reactants approach each other per unit time, and \(R\) is the universal gas constant. When activation energy decreases, as seen in the presence of a catalyst, the exponent term becomes less negative, leading to an increased value of \(k\).

In the given exercise, by knowing both the original and the catalyzed activation energies and keeping \(A\) the same, we calculated the extent to which the catalyst increased the reaction-rate constant at a constant temperature.

Catalysis in Chemistry

Catalysis is a process by which the rate of a chemical reaction is increased by a substance called a catalyst, which is not consumed by the reaction itself. Catalysts function by lowering the activation energy, facilitating a faster reaction without being permanently altered. In many cases, reactions that would otherwise be slow or may not even proceed in practical timeframes can be accelerated by catalysis.

The importance of catalysts lies in their ability to increase reaction rates while being used in small amounts and eventually being recovered. Catalysis is vital in both industrial processes and biological systems. For example, enzymes are natural catalysts that drive biological reactions in our bodies at room temperature and ambient pressure conditions, which would otherwise require extreme conditions.

Referring back to our exercise, the catalyst reduced the activation energy by \(9 \text{ kJ/mol}\), which, though seemingly minimal, resulted in a noticeable increase in the reaction-rate constant, illustrating the profound effect that even a small decrease in activation energy has on reaction rates due to catalysis.