Suppose that, in the absence of a catalyst, a certain biochemical reaction occurs x times per second at normal body temperature 37 °C. In order to be physiologically useful, the reaction needs to occur 5000 times faster than when it is uncatalyzed. By how many kJ/mol must an enzyme lower the activation energy of the reaction to make it useful?

Verified step by step guidance

1. Step 1: Understand the Arrhenius equation, which relates the rate constant of a reaction to the activation energy and temperature: k = A * e^(-Ea/(RT)), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin.
2. Step 2: Convert the given temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature: T(K) = 37 + 273.15.
3. Step 3: Recognize that the reaction rate needs to be 5000 times faster with the enzyme, which implies that the catalyzed rate constant (k_cat) is 5000 times the uncatalyzed rate constant (k_uncat). Therefore, k_cat = 5000 * k_uncat.
4. Step 4: Use the Arrhenius equation for both the uncatalyzed and catalyzed reactions: k_uncat = A * e^(-Ea_uncat/(RT)) and k_cat = A * e^(-Ea_cat/(RT)). Set up the equation k_cat/k_uncat = 5000 = e^((Ea_uncat - Ea_cat)/(RT)).
5. Step 5: Solve for the difference in activation energy (Ea_uncat - Ea_cat) by taking the natural logarithm of both sides: ln(5000) = (Ea_uncat - Ea_cat)/(RT). Rearrange to find Ea_uncat - Ea_cat = R * T * ln(5000).

Key Concepts

Activation Energy

Activation energy is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to transform into products. Lowering the activation energy increases the rate of the reaction, making it more likely to occur at a given temperature. In biochemical reactions, enzymes often lower this energy barrier, facilitating faster reactions.

Enzyme Catalysis

Enzymes are biological catalysts that speed up chemical reactions by lowering the activation energy. They achieve this by providing an alternative reaction pathway and stabilizing the transition state. The efficiency of an enzyme is often quantified by its turnover number, which indicates how many substrate molecules one enzyme can convert to product per second. Understanding enzyme kinetics is crucial for determining how much the activation energy must be reduced.

Arrhenius Equation

The Arrhenius equation describes the temperature dependence of reaction rates and relates the rate constant of a reaction to the activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation helps in calculating how changes in activation energy affect reaction rates, which is essential for determining the required reduction in activation energy for the reaction to occur at the desired rate.