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 $$ \text {Rate} =k[\text { acetylcholine receptor-toxin complex }] $$ 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 \(\mathrm{a} 0.200-\mathrm{M}\) solution of receptor-toxin complex at \(37.0^{\circ} \mathrm{C} ?\)

Short answer

The rate of the reaction for a 0.200 M solution of receptor-toxin complex at 37.0°C is approximately 2.74 × 10^-5 M/s.

Step-by-step solution

Convert the activation energy and temperature to the proper units

The activation energy (Ea) needs to be in Joules per mol (J/mol) and the temperature (T) should be in Kelvins (K). Ea = 26.2 kJ/mol × (1000 J / 1 kJ) = 26200 J/mol T = 37.0 °C + 273.15 = 310.15 K

Use the Arrhenius equation to find the rate constant (k)

The Arrhenius equation is: \(k = Ae^{\frac{-Ea}{RT}}\) where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature. Plugging in the given values: k = 0.850 s^-1 × exp(-26200 J/mol ÷ (8.314 J/mol·K × 310.15 K)) Calculate the value for k: k ≈ 1.37 × 10^-4 s^-1

Use the rate law equation to find the rate of reaction

Now that we have the rate constant (k), we can use the rate law equation to find the rate of the reaction: Rate = k[acetylcholine receptor-toxin complex] Plug in the values: Rate = (1.37 × 10^-4 s^-1) × 0.200 M Calculate the rate: Rate ≈ 2.74 × 10^-5 M/s So, the rate of the reaction is approximately 2.74 × 10^-5 M/s at 37.0°C for a 0.200 M solution of receptor-toxin complex.

Key concepts

Activation Energy

Activation energy is a crucial concept in understanding chemical reactions. It is the minimum energy required to initiate a reaction. Think of it as a barrier that reactants must overcome to transform into products. Activation energy is typically expressed in Joules per mole (J/mol) or kilojoules per mole (kJ/mol).

In a chemical reaction, the reactants need enough kinetic energy for their atoms to rearrange. Activation energy acts as this initial push. Without sufficient energy, even if reactants are present, the reaction does not proceed.

For instance, in the reaction concerning acetylcholine receptor-toxin complex, the activation energy is given as 26.2 kJ/mol, which helps determine the rate constant using another important tool: the Arrhenius equation.

Arrhenius Equation

The Arrhenius equation is a formula used to calculate the rate constant (k) of a reaction. It shows how the rate of a chemical reaction increases with temperature. This is expressed as:

\[k = A e^{-\frac{E_a}{RT}}\] where:

• \(k\) is the rate constant, which indicates the speed of a reaction.
• \(A\) is the pre-exponential factor or frequency factor, representing the likelihood of a collision between reactant molecules.
• \(E_a\) is the activation energy.
• \(R\) is the gas constant (8.314 J/mol·K).
• \(T\) is the temperature in Kelvin.

The Arrhenius equation shows that the higher the temperature, the lower the exponential function outcome, making the rate constant larger.

For our exercise, by calculating using the given activation energy, pre-exponential factor, and temperature, we find that the rate constant \(k\) is approximately \(1.37 \times 10^{-4} \text{s}^{-1}\) at \(37.0^{\circ} \text{C}\).

Acetylcholine Receptor

Acetylcholine receptors play an essential role in communication between nerve cells and muscles. They are proteins located on cell membranes, primarily on muscle tissues and nerves. When acetylcholine, a neurotransmitter, binds to these receptors, it triggers muscle contraction.

But, in the case of cobra venom, toxins mimic or interfere with acetylcholine. The venom binds to the acetylcholine receptors and blocks their normal function, causing paralysis and potentially leading to death if untreated.

Understanding this interaction helps researchers develop antidotes and treatments. By determining how toxins bind and release from these receptors, scientists can tailor antivenoms to better outcompete the toxins or reverse their effects.

In this context, knowing the rate at which the toxin releases from the receptor is key to figuring out how quickly an antidote might need to work to save someone's life.