Question

In the limit where the diffusion coefficients and radii of two reactants are equivalent, demonstrate that the rate constant for a diffusion controlled reaction can be written as $$k_{d}=\frac{8 R T}{3 \eta}$$

Short answer

In the limit where the diffusion coefficients and radii of two reactants are equivalent, we can use the Smoluchowski equation, \(k = 16 \pi Dr\), and the Stokes-Einstein equation, \(D = \frac{k_BT}{6\pi\eta r}\). By substituting the expression for D and simplifying the equation, we can derive the rate constant for a diffusion-controlled reaction as \(k_d = \frac{8RT}{3\eta}\), where R is the gas constant, T is the temperature, and η is the viscosity of the medium.

Step-by-step solution

Understand the background concepts

Diffusion-controlled reactions are chemical processes that depend on the motion and collisions of reactants. In such reactions, the rate of reaction is limited by the rate at which reactants find each other through diffusion. The rate constant is a proportionality factor that directly relates the reaction rate to the concentration of reactants. In a bimolecular reaction, it can be expressed in terms of diffusion coefficients and radii of reactants. Given the limit in which the diffusion coefficients and radii are equivalent, we need to derive an expression for the rate constant.

Use the Smoluchowski equation

The Smoluchowski equation is an expression that relates the rate constant of a bimolecular reaction to the diffusion coefficients and radii of the reactants. It is given by: \[k = 4 \pi (D_1 + D_2)(r_1 + r_2)\] Here, k is the rate constant, \(D_1\) and \(D_2\) are the diffusion coefficients, and \(r_1\) and \(r_2\) are the radii of the reactants. Since we are considering the limit where \(D_1 = D_2\) and \(r_1 = r_2\), let's call them D and r, respectively. The Smoluchowski equation then becomes: \[k = 4 \pi (2D)(2r)\] Simplifying this expression gives: \[k = 16 \pi Dr\]

Relate the diffusion coefficient to temperature and viscosity

To derive the required expression, we need to relate the diffusion coefficient to temperature and viscosity. The Stokes-Einstein equation allows us to do this. It states that: \[D = \frac{k_BT}{6 \pi \eta r}\] Here, \(k_B\) is the Boltzmann constant, T is the temperature, and η is the viscosity of the medium. Now, we will substitute this expression of D into the equation for k: \[k = 16 \pi \left(\frac{k_BT}{6 \pi\eta r}\right)r\]

Simplify the expression for the rate constant

By simplifying the expression and canceling out some terms, we obtain: \[k = \frac{16 \pi k_BTr}{6 \pi\eta}\] Notice that the gas constant (R) is related to the Boltzmann constant by R = \(N_Ak_B\), where \(N_A\) is Avogadro's number. We will replace \(k_B\) with \(\frac{R}{N_A}\) and simplify: \[k = \frac{16 \pi \left(\frac{R}{N_A}T\right)r}{6 \pi\eta}\] Next, we cancel out the \(\pi\) terms: \[k = \frac{16 \frac{RT}{N_A}r}{6\eta}\] Finally, we simplify the expression and introduce the notation of \(k_d\) for the rate constant: \[k_d = \frac{8RT}{3\eta}\] This is the desired expression for the rate constant for a diffusion-controlled reaction under the specified conditions.

Key concepts

Understanding Rate Constant in Diffusion-Controlled Reactions

In chemistry, the rate constant is a fundamental concept, especially when discussing reaction kinetics. It serves as a proportionality factor that helps quantify how fast a reaction occurs by relating the reaction rate to the concentrations of reactants. In diffusion-controlled reactions, the rate at which reactants come together is influenced by their movement through a medium, making the reaction rate dependent on diffusion.

These reactions happen so fast that the only thing limiting their speed is how quickly molecules can diffuse through the solvent to meet one another. The rate constant (designated by \(k_d\)) for these reactions signifies how effectively diffusion is occurring under specified conditions. The unit of \(k_d\) is typically \([\text{m}^{3}/\text{mol} \, \text{s}]\), reflecting the bimolecular nature of these reactions. Understanding this constant is crucial because it helps predict how efficiently reactants find each other and how reaction conditions, such as temperature and medium viscosity, affect this process.

The Role of the Smoluchowski Equation

The Smoluchowski equation plays a vital role in determining the rate constant for diffusion-controlled reactions. This equation links the rate constant \(k\) to the sum of diffusion coefficients and the sum of the radii of the reactants involved.
It is presented as:
\[k = 4 \pi (D_1 + D_2)(r_1 + r_2)\]
where:

• \(D_1\) and \(D_2\) are the diffusion coefficients of the reactants
• \(r_1\) and \(r_2\) are the radii of the reactants

In scenarios where the reactants have identical diffusion coefficients and radii, the expression can be simplified, resulting in a modified form. This simplification helps derive more particular expressions for \(k_d\) based on specific conditions.
Through the exercise, by replacing and simplifying terms, students can witness how theoretical models accurately describe actual chemical processes.

The Significance of the Stokes-Einstein Equation

The Stokes-Einstein equation provides a pivotal link between the diffusion coefficient \(D\), temperature \(T\), and viscosity \(\eta\) of the medium. It is represented by the formula:
\[D = \frac{k_BT}{6 \pi \eta r}\]
Here:

• \(k_B\) stands for the Boltzmann constant
• \(T\) is the absolute temperature
• \(\eta\) denotes the medium's viscosity
• \(r\) is the radius of the particle

Incorporating this equation into the context of diffusion-controlled reactions allows the transformation of the Smoluchowski equation into a form that depends directly on temperature and viscosity. This underscores how environmental factors influence reaction rates.

With the help of the Stokes-Einstein equation, students can observe the direct impact of physical conditions on molecular behavior, facilitating a deeper comprehension of how intrinsic properties like diffusion coefficients underpin chemical kinetics.