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 antivenins, 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 \(26.2 \mathrm{~kJ} /\) 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 for a 0.200 M solution of receptor-toxin complex at 37.0 °C is \( 3.426 \times 10^{-5} \, \mathrm{M} \cdot \mathrm{s}^{-1} \).

Step-by-step solution

Convert the temperature to Kelvin

To work with the Arrhenius equation, we need to convert our temperature from Celsius to Kelvin: \[ T = 37.0 + 273.15 = 310.15 K \]

Calculate the rate constant, k

The Arrhenius equation relates the rate constant to the activation energy and the temperature: \[ k = Ae^{\frac{-E_a}{RT}} \] where: - \( k \) is the rate constant we want to find - \( A \) is the frequency factor (0.850 s⁻¹) - \( -E_a \) is the activation energy (-26.2 kJ/mol) - \( R \) is the gas constant (8.314 J/mol·K) - \( T \) is the temperature in Kelvin (310.15 K) First, convert the activation energy from kJ/mol to J/mol: \[ E_a = 26.2 \times 1000 = 26200 J/mol \] Now, plug in the values into the Arrhenius equation: \[ k = 0.850e^{\frac{-26200}{(8.314)(310.15)}} \] Calculate k: \[ k = 1.713 \times 10^{-4} \, \mathrm{s}^{-1} \]

Calculate the rate of reaction

Using the given rate law and the calculated rate constant k, calculate the rate of the reaction with a 0.200 M solution of receptor-toxin complex: \[ \text{Rate} = k [\text{acetylcholine receptor-toxin complex}] \] Plug in the values: \[ \text{Rate} = (1.713 \times 10^{-4} \, \mathrm{s}^{-1})(0.200 \, \mathrm{M}) \] Calculate the rate of reaction: \[ \text{Rate} = 3.426 \times 10^{-5} \, \mathrm{M} \cdot \mathrm{s}^{-1} \] Therefore, the rate of reaction for a 0.200 M solution of receptor-toxin complex at 37.0 °C is \( 3.426 \times 10^{-5} \, \mathrm{M} \cdot \mathrm{s}^{-1} \).

Key concepts

Activation Energy

Activation energy is the minimum amount of energy required for a chemical reaction to occur. Think of it like a hurdle that reactants need to overcome to transform into products. For our reaction involving the acetylcholine receptor-toxin complex, the activation energy is given as 26.2 kJ/mol. This means that a certain amount of energy is needed to start the process of releasing the toxin from the receptor.
In chemical kinetics, understanding activation energy helps us realize why some reactions are slow at room temperature but can speed up when heated. Raising the temperature increases the energy of molecules, helping them overcome this energy barrier more easily. In our calculation, converting activation energy to Joules (since 1 kJ = 1000 J) was a crucial step to use it properly in the Arrhenius equation.

Arrhenius Equation

The Arrhenius Equation is a vital formula in chemistry that helps calculate the rate constant of a reaction. It shows how temperature and activation energy affect reaction rates. The equation is defined as:

• \( k = Ae^{-\frac{E_a}{RT}} \)

This formula connects the rate constant \( k \) with the activation energy \( E_a \), the frequency factor \( A \), the gas constant \( R \), and the temperature in Kelvin \( T \).
By rearranging the equation, we see how increasing temperature or decreasing activation energy can lead to a higher rate constant, meaning the reaction proceeds faster. It's like the equation is a bridge showing the direct relationship between these variables. The steps in calculating \( k = 1.713 \times 10^{-4} \text{s}^{-1} \) involve inserting the temperature, activation energy, and frequency factor values, which are essential to find how fast the receptor-toxin complex reaction occurs at 37.0 °C.

Acetylcholine Receptors

Acetylcholine receptors are proteins found in the nervous system. They play a key role in transmitting signals between nerves and muscles. When acetylcholine binds to these receptors, muscles contract. However, cobra venom binds to the same receptors, blocking them and preventing nerve signals from reaching the diaphragm muscles.
This blockage results in paralysis, which highlights why understanding the interaction between toxins and acetylcholine receptors is crucial. By studying how toxins bind and are released from these receptors, scientists can develop antivenins. This exercise involved calculating the rate at which the venom is released from the receptors, helping develop more effective treatments.

Reaction Kinetics

Reaction kinetics involves studying the speed or rate of chemical reactions. It helps us understand how different factors like concentration, temperature, and catalysts affect how quickly reactions happen. In our example, reaction kinetics was used to calculate how fast the toxin is released from the acetylcholine receptors.
Key components include:

• Rate laws, which describe the relationship between the concentration of reactants and the rate of reaction.
• The reaction rate itself, which is influenced by factors such as temperature and activation energy.

Here, applying the rate law \( \text{Rate} = k[\text{acetylcholine receptor-toxin complex}] \) allowed us to compute the rate of reaction using the calculated rate constant \( k \) and the concentration of the complex. By understanding reaction kinetics, we can predict and control reaction behaviors, which is crucial in chemistry and pharmacology.