Question

The enzyme carbonic anhydrase catalyzes the reaction \(\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}^{+}(a q) .\) In water, without the enzyme, the reaction proceeds with a rate constant of 0.039 \(\mathrm{s}^{-1}\) at \(25^{\circ} \mathrm{C}\) . In the presence of the enzyme in water, the reaction proceeds with a rate constant of \(1.0 \times 10^{6} \mathrm{s}^{-1}\) at \(25^{\circ} \mathrm{C}\) . Assuming the collision factor is the same for both situations, calculate the difference in activation energies for the uncatalyzed versus enzyme-catalyzed reaction.

Short answer

The difference in activation energies for the uncatalyzed versus enzyme-catalyzed reaction is approximately -43.8 kJ/mol. This result indicates that the enzyme lowers the activation energy required for the reaction to proceed, which is consistent with the role of enzymes as catalysts.

Step-by-step solution

Write down the Arrhenius equation

The Arrhenius equation relates the rate constant (k) of a reaction to its activation energy (Ea) and temperature (T), as follows: \[ k = Ae^{-\frac{Ea}{RT}} \] where A is the pre-exponential factor, R is the universal gas constant (8.314 J/mol K), and T is the temperature in Kelvin.

Convert the given temperature to Kelvin

We are given the temperature in Celsius (25°C). To use the Arrhenius equation, we need the temperature in Kelvin. To convert Celsius to Kelvin, add 273.15: \[ T = 25 + 273.15 = 298.15 \ \mathrm{K} \]

Set up two separate Arrhenius equations for uncatalyzed and enzyme-catalyzed reactions

We have the rate constants (k) for both the uncatalyzed and enzyme-catalyzed reactions. We will set up two separate Arrhenius equations, one for each reaction: \[ k_{1} = Ae^{-\frac{Ea_{1}}{RT}} \text{ (for the uncatalyzed reaction)} \] \[ k_{2} = Ae^{-\frac{Ea_{2}}{RT}} \text{ (for the enzyme-catalyzed reaction)} \] where \(k_{1}\) and \(k_{2}\) are the rate constants for the uncatalyzed and enzyme-catalyzed reactions, respectively, while \(Ea_{1}\) and \(Ea_{2}\) are their activation energies.

Divide one equation by the other and solve the resulting equation for the difference in activation energies

We want to find the difference in activation energies (\(Ea_{2} - Ea_{1}\)). To do this, divide the equation for the enzyme-catalyzed reaction (k2) by the equation for the uncatalyzed reaction (k1): \[\frac{k_{2}}{k_{1}} = \frac{Ae^{-\frac{Ea_{2}}{RT}}}{Ae^{-\frac{Ea_{1}}{RT}}} \] The pre-exponential factors (A) cancel out: \[\frac{k_{2}}{k_{1}} = e^{\frac{Ea_{1}-Ea_{2}}{RT}} \] We will now solve this equation for \(Ea_{2} - Ea_{1}\). First, take the natural logarithm of both sides: \[\ln{\left(\frac{k_{2}}{k_{1}}\right)} = \frac{Ea_{1} - Ea_{2}}{RT} \] Now, solve for \(Ea_{2} - Ea_{1}\): \[ Ea_{2} - Ea_{1} = -RT\ln{\left(\frac{k_{2}}{k_{1}}\right)} \]

Substitute the given values and calculate the difference in activation energies

Using the given rate constants (\(k_{1} = 0.039 \ \mathrm{s}^{-1}\) and \(k_{2} = 1.0 \times 10^6 \ \mathrm{s}^{-1}\)), the temperature in Kelvin (298.15 K), and the gas constant R (8.314 J/mol K): \[ Ea_{2} - Ea_{1} = -8.314 \ \mathrm{J/mol \ K} \times 298.15 \ \mathrm{K} \times \ln{\left(\frac{1.0 \times 10^6 \ \mathrm{s}^{-1}}{0.039 \ \mathrm{s}^{-1}}\right)} \] \[ Ea_{2} - Ea_{1} \approx -43.8 \ \mathrm{kJ/mol} \] The difference in activation energies for the uncatalyzed versus enzyme-catalyzed reaction is approximately -43.8 kJ/mol. This result indicates that the enzyme lowers the activation energy required for the reaction to proceed, which is consistent with the role of enzymes as catalysts.

Key concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental principle that helps understand how temperature affects reaction rates. It's given by the formula: \[ k = Ae^{-\frac{Ea}{RT}} \] where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( Ea \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. By examining this formula, you can see how the rate constant \( k \) increases when the activation energy \( Ea \) decreases or the temperature \( T \) increases.
In the context of chemical reactions, this equation proves crucial for predicting how quickly a reaction will proceed under different conditions. Lower activation energies or higher temperatures lead to faster reactions. Understanding the Arrhenius Equation is key to manipulating and controlling chemical processes, especially when designing experiments or industrial processes.

Enzyme Catalysis

Enzymes are remarkable proteins that speed up chemical reactions in biological systems. This process is known as enzyme catalysis. Enzymes work by providing an alternative reaction pathway with a lower activation energy than the uncatalyzed reaction.

• Enzymes bind to the reactants, known as substrates, to form a temporary complex.
• This binding helps position the reactants for better interaction, often altering their structure temporarily.
• As a result, it requires less energy to initiate the reaction, thereby increasing the rate at which products are formed.

The reduction in activation energy is crucial as it allows reactions that are necessary for life to occur at a significant rate. Without enzymes, most biochemical reactions would be too slow to sustain life processes. In the given problem, carbonic anhydrase is the enzyme that dramatically increases the reaction rate by lowering the activation energy significantly.

Rate Constant

The rate constant, symbolized as \( k \), is a parameter that quantifies the speed of a chemical reaction. Within the framework of the Arrhenius Equation, it directly links to the activation energy and temperature. A higher rate constant means a faster reaction and vice versa.
In the example problem, two rate constants are provided: one for an uncatalyzed reaction \( k_1 = 0.039 \ \text{s}^{-1} \) and another for the enzyme-catalyzed reaction \( k_2 = 1.0 \times 10^6 \ \text{s}^{-1} \). The dramatic increase in the rate constant when the enzyme is used showcases the powerful effect of the enzyme in speeding up the reaction.
Consider that even tiny changes in activation energy can cause large variations in rate constants. Therefore, accurately determining the rate constant is vital for predicting how quickly a specific reaction will proceed, making it an essential tool for chemists and biochemists alike. Understanding this concept is foundational for both laboratory research and industrial applications.