Question

An experiment is performed in which the rate constant for electron transfer is measured as a function of dis- tance by attaching an electron donor and acceptor to pieces of DNA of varying length. The measured rate constant for electron transfer as a function of separation distance is as follows: $$\begin{array}{lcccc} \boldsymbol{k}_{e x p}\left(\mathbf{s}^{-1}\right) & 2.10 \times 10^{8} & 2.01 \times 10^{7} & 2.07 \times 10^{5} & 204 \\ \text { Distance (Á) } & 14 & 17 & 23 & 32 \end{array}$$ a. Determine the value for \(\beta\) that defines the dependence of the electron transfer rate constant on separation distance. b. It has been proposed that DNA can serve as an electron \(" \pi\) -way" facilitating electron transfer over long distances. Using the rate constant at 17 Á presented in the table, what value of \(\beta\) would result in the rate of electron transfer decreasing by only a factor of 10 at a separation distance of \(23 \AA ?\)

Short answer

Also, find the value of β that would result in the rate of electron transfer decreasing by only a factor of 10 at a separation distance of 23 Å. Answer: The value of β that defines the dependence of the electron transfer rate constant on separation distance is approximately 1.32 Å⁻¹. The value of β that would result in the rate of electron transfer decreasing by only a factor of 10 at a separation distance of 23 Å is approximately 0.439 Å⁻¹.

Step-by-step solution

Choose two data points

Choose any two data points from the given data, for example, the first and the second data points: $$k_1 = 2.10 \times 10^8 \,\mathrm{s}^{-1} \;\; \text{ and } \;\; r_1 = 14 \AA$$ $$k_2 = 2.01 \times 10^7 \,\mathrm{s}^{-1} \;\; \text{ and } \;\; r_2 = 17 \AA$$

Set up the exponential equation

Using the equation \(k = k_0 e^{-\beta r}\) for the two data points, we obtain: $$ k_1 = k_0 e^{-\beta r_1} \\ k_2 = k_0 e^{-\beta r_2} $$

Solve for \(\beta\)

By dividing the first equation by the second one, we obtain: $$\frac{k_1}{k_2} = e^{-\beta (r_1-r_2)}$$ Now, we can solve for β: $$\beta = \frac{\ln{\frac{k_1}{k_2}}}{r_1 - r_2}$$

Calculate the value of \(\beta\)

Plug in the values of \(k_1, k_2, r_1, \text{and} \: r_2\) and solve for \(\beta\): $$\beta = \frac{\ln{\frac{2.10 \times 10^8}{2.01 \times 10^7}}}{14-17}$$ $$\beta \approx 1.32 \, \AA^{-1}$$ So, the value of β that defines the dependence of the electron transfer rate constant on separation distance is approximately 1.32 Å⁻¹. #b. Finding the value of \(\beta\) that results in a rate decrease by a factor of 10 at 23 Å#

Set up the equation

We know that the rate constant at 17 Å is \(k_3 = 2.01\times10^7 s^{-1}\). Using the same equation, \(k = k_0 e^{-\beta r}\), we need to find the value of \(\beta\) that would result in a rate constant \(k_4 = k_3/10\) at 23 Å: $$k_4 = k_0 e^{-\beta r_4}$$

Use the rate constant at 17 Å

First, let's set up the equation for 17 Å to find the value of \(k_0\): $$k_3 = k_0 e^{-\beta r_3}$$

Solve for \(k_0\)

We are using the value of \(\beta\) found in part a to find the value of \(k_0\). Thus: $$k_0 = k_3 e^{\beta r_3} = (2.01 \times 10^7) e^{(1.32)(17)}$$

Solve for new \(\beta\)

Now we can find the value of \(\beta\) that would result in a rate constant \(k_4 = k_3 / 10\) at 23 Å: $$k_4 = k_0 e^{-\beta r_4}$$ $$\beta = \frac{-\ln{\frac{k_4}{k_0}}}{r_4}$$ Plug in the values of \(k_0, k_4 \text{ and } r_4\): $$\beta = \frac{-\ln{\frac{2.01 \times 10^6}{(2.01 \times 10^7) e^{(1.32)(17)}}}}{23}$$ $$\beta \approx 0.439 \, \AA^{-1}$$ The value of β that would result in the rate of electron transfer decreasing by only a factor of 10 at a separation distance of 23 Å is approximately 0.439 Å⁻¹.

Key concepts

Rate Constant

The rate constant, denoted as \( k \), is a crucial parameter in chemical kinetics that quantifies the speed at which a reaction occurs. In the context of electron transfer in DNA, the rate constant tells us how quickly electrons move between a donor and an acceptor molecule as they are separated by the DNA structure. This value is typically measured in \( ext{s}^{-1} \), indicating the number of electron transfer events per second. It provides insight into the efficiency of electron transfer through comparing different sets of conditions, such as varying separation distances.
To find the rate constant in practical situations, scientists use experimental data to set up mathematical models, often involving exponential equations. In our case, understanding how the rate constant changes with separation distance allows researchers to infer properties of the DNA's ability to facilitate electron transfer.

Separation Distance

Separation distance refers to the space between two points, specifically between an electron donor and acceptor in the DNA chain, often measured in Ångströms (Å). It plays a vital role in determining how efficiently electrons can transfer across molecular systems.
The experiment mentioned measures the electron transfer rate at different separation distances to understand the efficiency of DNA in conducting electrons. Generally, as the separation distance increases, the rate constant decreases, illustrating that electrons have more difficulty traveling longer distances. This relationship emphasizes the importance of separation distance as a control variable in analyzing electron transfer phenomena.
The observed trend in electron transfer efficiency helps scientists develop models to predict electron behavior in similar biological contexts, enhancing our understanding of molecular electronic processes.

Mathematical Modelling

Mathematical modeling is the process of using mathematical equations and expressions to represent real-world phenomena. In our context, it involves the use of exponential functions to describe the relationship between the rate constant \( k \) and the separation distance \( r \) in electron transfer through DNA.
These models often employ the equation \( k = k_0 e^{-\beta r} \), where \( k_0 \) is a pre-exponential factor, \( \beta \) is the decay constant, and \( r \) is the separation distance. Plugging in experimentally obtained data, scientists can solve for parameters like \( \beta \), which indicates how quickly the rate of electron transfer decreases with distance.

• Choosing appropriate data points is essential to derive meaningful equations.
• Solving for \( \beta \) involves algebraic manipulations, using known values to find unknown factors.

The accuracy of such models helps researchers predict and manipulate electron transfer behaviors in various biological and synthetic systems.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In electron transfer in DNA, it characterizes how the transfer rate constant diminishes as the separation distance between the donor and acceptor increases.
This concept is mathematically expressed using exponential functions, often in the form \( k = k_0 e^{-\beta r} \). In this expression, \( \beta \) is the exponent that dictates the rate of decay. A larger \( \beta \) would mean a faster decline in the transfer rate with distance.
This decay behavior is significant for understanding how well electrons can "jump" along the DNA. Intuitively, it reflects the diminishing likelihood of successful electron transfer events across growing distances, making it a key consideration in biological and physical-based electron transfer studies.