Question

Cobra venom helps the snake secure food by binding to acetylcholine receptors on the diaphragm of a bite victim, leading to the loss of function of the diaphragm muscle tissue and eventually death. In order to develop more potent antivenoms, scientists have studied what happens to the toxin once it has bound the acetylcholine receptors. They have found that the toxin is released from the receptor in a process that can be described by the rate law Rate \(=k[\) acetylcholine receptor-toxin complex \(]\) If the activation energy of this reaction at \(37.0^{\circ} \mathrm{C}\) is \(26.2 \mathrm{~kJ} / \mathrm{mol}\) and \(A=0.850 \mathrm{~s}^{-1}\), what is the rate of reaction if you have a \(0.200 M\) solution of receptor-toxin complex at \(37.0^{\circ} \mathrm{C}\) ?

Short answer

The rate of reaction at a concentration of 0.200 M receptor-toxin complex and at 37.0\(^\circ C\) is approximately \(3.1 \times 10^{-4} M s^{-1}\).

Step-by-step solution

Convert temperature to Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \(T(K) = T(^\circ C) + 273.15\) The given temperature is: \(T = 37.0^\circ C\) So, \(T(K) = 37.0 + 273.15 = 310.15 K\)

Calculate the rate constant, k, using the Arrhenius equation

The Arrhenius equation is given as: \(k = Ae^{-\frac{Ea}{RT}}\) where k - rate constant A - pre-exponential factor Ea - activation energy (\(26.2 kJ/mol = 26200 J/mol\)) R - gas constant (\(8.314 J/mol K\)) T - temperature in Kelvin Now, plug in the values and compute k: \(k = (0.850 s^{-1})e^{-\frac{(26200 J/mol)}{(8.314 J/mol K)(310.15 K)}} \) Calculating the exponential term first: \(e^{-\frac{26200}{(8.314)(310.15)}} \approx 0.00182\) Now multiplying by A: \(k = 0.850 \times 0.00182 \approx 0.00155 s^{-1}\)

Calculate the rate of reaction at the given concentration of the receptor-toxin complex

We know that the rate law for this reaction is given as: Rate \(= k[\text{acetylcholine receptor-toxin complex}]\) Now, plug in the rate constant (k) and the given concentration (0.200 M) of the receptor-toxin complex: Rate \(= (0.00155 s^{-1})(0.200 M)\) Rate \(= 3.1 \times 10^{-4} M s^{-1}\) The rate of reaction at a concentration of 0.200 M receptor-toxin complex and at 37.0\(^\circ C\) is approximately \(3.1 \times 10^{-4} M s^{-1}\).

Key concepts

Understanding Rate Law in Reactions

In chemical kinetics, the rate law helps us understand how the concentration of substances in a reaction affects the rate of reaction.
In the exercise, the rate law is expressed as:

• Rate \(= k[\text{acetylcholine receptor-toxin complex}]\)

This indicates that the rate of the reaction is directly proportional to the concentration of the receptor-toxin complex. Here:

• \(k\) is the rate constant
• The concentration of the acetylcholine receptor-toxin complex determines how quickly the reaction proceeds

Understanding this relationship helps scientists predict how changes in concentration influence the effectiveness of antidotes.

Exploring the Arrhenius Equation

The Arrhenius equation provides insight into how temperature affects the reaction rate. It is formulated as:

• \(k = Ae^{-\frac{Ea}{RT}}\)

Here:

• \(A\) is the pre-exponential factor
• \(Ea\) is the activation energy
• \(R\) is the gas constant
• \(T\) is the temperature in Kelvin

By using this equation in the exercise, we determine the rate constant \(k\). The Arrhenius equation shows how an increase in temperature generally increases the reaction rate, revealing the complex interplay between temperature and reaction kinetics.

Importance of Activation Energy

Activation energy, \(Ea\), is a crucial concept in chemical kinetics. It refers to the minimum energy needed for a reaction to occur. In the exercise, the activation energy is given as \(26.2 \text{kJ/mol}\).

• This energy barrier determines how fast a reaction takes place
• Reactions with lower activation energies happen more quickly because less energy is required to begin the process

Understanding activation energy helps scientists design more effective antidotes by lowering this energy barrier, making reactions proceed more swiftly.

Role of Concentration

Concentration plays a vital role in determining the speed of chemical reactions. In the context of this exercise:

• The concentration of the receptor-toxin complex is \(0.200 M\)

This concentration directly impacts the reaction rate as expressed by the rate law. Increased concentration usually leads to a faster reaction because:

• More molecules are available to react
• This results in a higher probability of effective collisions

By managing concentrations, scientists can control how quickly reactions occur and improve the design of antidotes.

Understanding the Rate of Reaction

The rate of reaction is an essential aspect of understanding how quickly a reaction proceeds. In this exercise, it is calculated as \( 3.1 \times 10^{-4} \text{M s}^{-1} \). This calculation reflects:

• The change in concentration of reactants per unit time
• How efficiently the receptor-toxin complex forms and decomposes

By knowing the rate of reaction, scientists can evaluate the effectiveness of antivenoms, allowing them to make adjustments for better results.