Question

The enzyme urease catalyzes the reaction of urea, \(\left(\mathrm{NH}_{2} \mathrm{CONH}_{2}\right)\), with water to produce carbon dioxide and ammonia. In water, without the enzyme, the reaction proceeds with a first-order rate constant of \(4.15 \times 10^{-5} \mathrm{~s}^{-1}\) at \(100^{\circ} \mathrm{C}\). In the presence of the enzyme in water, the reaction proceeds with a rate constant of \(3.4 \times 10^{4} \mathrm{~s}^{-1}\) at \(21^{\circ} \mathrm{C}\). (a) Write out the balanced equation for the reaction catalyzed by urease. (b) If the rate of the catalyzed reaction were the same at \(100^{\circ} \mathrm{C}\) as it is at \(21^{\circ} \mathrm{C}\), what would be the difference in the activation energy between the catalyzed and uncatalyzed reactions? (c) In actuality, what would you expect for the rate of the catalyzed reaction at \(100^{\circ} \mathrm{C}\) as compared to that at \(21^{\circ} \mathrm{C}\) ? (d) On the basis of parts (c) and (d), what can you conclude about the difference in activation energies for the catalyzed and uncatalyzed reactions?

Short answer

In summary, the balanced chemical equation for the urease-catalyzed reaction is \(\mathrm{NH_2CONH_2 + H_2O \rightarrow CO_2 + 2NH_3}\). Assuming the rate constant is the same at 100°C as at 21°C, the calculated difference in activation energies (\(\Delta E_\mathrm{a}\)) can be obtained using the Arrhenius equation. However, the actual rate constant for the catalyzed reaction at 100°C should be higher than that at 21°C, indicating that the enzyme urease lowers the activation energy, making the reaction proceed more quickly. Therefore, the actual difference in activation energies between the catalyzed and uncatalyzed reactions should be greater than the calculated \(\Delta E_\mathrm{a}\).

Step-by-step solution

Writing the balanced chemical equation

To write the balanced chemical equation for the reaction of urea (\(\mathrm{NH}_2\mathrm{CONH}_2\)) with water, producing carbon dioxide and ammonia, we can represent this as: \[ \mathrm{NH_2CONH_2 + H_2O \rightarrow CO_2 + 2NH_3} \] b. Calculating the difference in activation energies

Calculating activation energy difference

To calculate the difference in activation energies between the catalyzed and uncatalyzed reactions, we can use the Arrhenius equation: \[ k = A\mathrm{e}^{-E_\mathrm{a}/RT} \] We do not have enough information to find the absolute activation energies, but we can find the difference in activation energies (\(\Delta E_\mathrm{a}\)) by assuming the catalyzed reaction rate constant is the same at 100°C as it is at 21°C: \[ \Delta E_\mathrm{a} = R(T_2 - T_1) \ln{\frac{k_\mathrm{cat}(T_1)}{k_\mathrm{uncat}(T_2)}} \] Here, \(k_\mathrm{cat}(T_1) = 3.4 \times 10^4 \mathrm{~s}^{-1}\), \(k_\mathrm{uncat}(T_2) = 4.15 \times 10^{-5} \mathrm{~s}^{-1}\), \(T_1=294 \mathrm{K}\), \(T_2=373 \mathrm{K}\), and \(R = 8.314 \mathrm{J/mol \cdot K}\). By substituting the values into the equation, we can calculate the difference in activation energies. c. Rate of the catalyzed reaction at 100°C compared to 21°C

Comparing the rate constants

In general, a reaction rate constant increases with increasing temperature. Therefore, the actual rate constant for the catalyzed reaction at 100°C (\(k_\mathrm{act}(100^\circ\mathrm{C}\)) should be higher than the rate constant at 21°C (\(k_\mathrm{act}(21^\circ\mathrm{C}\)). d. Difference in activation energies for the catalyzed and uncatalyzed reactions

Concluding activation energy difference

Based on the previous discussion, we know that the actual activation energy of the catalyzed reaction at 100°C should be lower than the activation energy of the uncatalyzed reaction at 100°C. This indicates that the enzyme urease lowers the activation energy of the reaction, making it proceed more quickly. Since \(\Delta E_\mathrm{a}\) calculated in part (b) gives an approximation of how much the activation energy is lowered by the enzyme, the actual difference in activation energies between the catalyzed and uncatalyzed reactions should be greater than the calculated \(\Delta E_\mathrm{a}\).

Key concepts

Catalyzed Reaction

A catalyzed reaction is one that is accelerated by a substance known as a catalyst. In biological systems, enzymes act as natural catalysts, speeding up essential reactions that otherwise would proceed at impossibly slow rates for life processes. For example, the enzyme urease catalyzes the breakdown of urea into carbon dioxide and ammonia, a process vital to nitrogen metabolism in organisms.In the context of the exercise, urease remarkably increases the rate at which urea reacts with water. This is shown by comparing the rate constants of the reaction in the presence and absence of urease. With the enzyme, the reaction proceeds much faster, reflecting the crucial role catalysts play in increasing reaction rates without being consumed or altered permanently in the process.

Activation Energy

Activation energy, often symbolized as E_a, is the minimum amount of energy required to initiate a chemical reaction. It's like the 'hurdle' that reactants must overcome for a reaction to occur. Enzymes lower the activation energy, allowing reactions to occur more readily and at lower temperatures than would be possible without a catalyst.In the exercise, we are asked to compare the activation energy between catalyzed and uncatalyzed reactions. This comparison underlines how the enzyme, by reducing the activation energy, can greatly enhance the reaction rate. This property is harnessed in many biological processes, where reactions must be both efficient and precisely controlled.

Arrhenius Equation

The Arrhenius equation is a mathematical formula used to describe how reaction rates depend on temperature and activation energy. It is expressed as ewline \[ k = A\mathrm{e}^{-E_{\mathrm{a}}/RT} \]where k is the reaction rate constant, A is the pre-exponential factor related to the frequency of collisions and orientation of reactant molecules, E_a is the activation energy, R is the gas constant, and T is the temperature in Kelvin.By rearranging this equation, we can compare the activation energies of different reactions under various conditions, as done in the exercise. This powerful equation allows scientists to estimate reaction rates at different temperatures and understand the profound impact that catalysts, such as enzymes, have on the activation energy.

Reaction Rate Constants

Reaction rate constants, denoted by k, are numerical values that represent the speed at which a given chemical reaction proceeds. In the study of kinetics, understanding these constants is crucial for predicting the progress of reactions over time. The rate constants are influenced by factors such as temperature, pressure, and the presence of catalysts.From the exercise, we notice the dramatic difference in rate constants for the urease-catalyzed reaction compared to the uncatalyzed reaction. The enzyme significantly increases the rate constant, demonstrating how catalysts can modify the pace of reactions in biological systems. Furthermore, these constants are not static; they can change with conditions, which is why reactions will often proceed faster at higher temperatures, a trend that can be quantified using the Arrhenius equation.