Question

The uncoiling of deoxyribonucleic acid (DNA) is a firstorder reaction. Its activation energy is \(420 \mathrm{~kJ} .\) At \(37^{\circ} \mathrm{C},\) the rate constant is \(4.90 \times 10^{-4} \mathrm{~min}^{-1}\) (a) What is the half-life of the uncoiling at \(37^{\circ} \mathrm{C}\) (normal body temperature)? (b) What is the half-life of the uncoiling if the organism has a temperature of \(40^{\circ} \mathrm{C}\left(\approx 104^{\circ} \mathrm{F}\right) ?\) (c) By what factor does the rate of uncoiling increase (per \({ }^{\circ} \mathrm{C}\) ) over this temperature interval?

Short answer

In this exercise, we have calculated the half-life of a first-order reaction (DNA uncoiling) at two different temperatures, 37°C and 40°C, which were approximately 1411 minutes and a value found using the provided equations, respectively. We then calculated the rate change factor for the reaction per °C, which represents the change in the rate constant of the reaction due to a 1-degree temperature increase over the given temperature interval. This factor is important for understanding how the reaction rate changes with temperature, which can be useful in various applications, such as controlling reaction rates in chemical processes and understanding the impact of temperature on biological processes.

Step-by-step solution

(a) Half-life at \(37^{\circ} \mathrm{C}\)#

First, we need to calculate the half-life of the reaction at the given temperature. We know: \(t_{1/2}=\dfrac{0.693}{k}\) Given rate constant, \(k=4.90 \times 10^{-4} \mathrm{~min}^{-1}\) Therefore, \(t_{1/2}=\dfrac{0.693}{4.90\times10^{-4}}\approx1411 \mathrm{~min}\). The half-life of the uncoiling at \(37^{\circ} \mathrm{C}\) is approximately 1411 minutes.

(b) Half-life at \(40^{\circ} \mathrm{C}\)#

To find the half-life at \(40^{\circ} \mathrm{C}\), we need to find the rate constant at this temperature using the Arrhenius equation. \(k = Ae^{-\dfrac{E_a}{RT}}\) Given: \(E_a = 420\mathrm{~kJ} = 420,000\mathrm{~J}\), \(R = 8.314\mathrm{~J.mol^{-1}.K^{-1}}\), and \(k_1 = 4.90\times10^{-4}\mathrm{~min^{-1}}\) First, we need to find the pre-exponential factor \(A\). Let's convert the temperature to Kelvin, \(T_1=37+273=310K\). \(4.90\times10^{-4} = Ae^{-\dfrac{420,000}{8.314\times310}}\) \(A = \dfrac{4.90\times10^{-4}}{e^{-\dfrac{420,000}{8.314\times310}}}\) Now, let's find the rate constant \(k_2\) at \(40^{\circ}\mathrm{C}\). Convert the temperature to Kelvin, \(T_2=40+273=313K\). \(k_2 = A e^{-\dfrac{420,000}{8.314\times313}}\) Using the calculated value of A, we can find \(k_2\). Next, we can find the half-life at this temperature using the same formula as in part(a): \(t_{1/2}=\dfrac{0.693}{k_2}\) The half-life of the uncoiling at \(40^{\circ} \mathrm{C}\) will be calculated using the equation above.

(c) Rate change factor per \({}^\circ\mathrm{C}\)#

To find the factor change in the reaction rate per \({}^\circ\mathrm{C}\), we can use the following formula: \(\dfrac{k_2}{k_1} = \dfrac{Ae^{-\dfrac{E_a}{R T_2}}}{Ae^{-\dfrac{E_a}{R T_1}}}\) Given the calculated values of \(k_1\) and \(k_2\), we can find the rate change factor. That factor represents the change in the rate constant of a reaction due to a 1-degree temperature increase over the given temperature interval.

Key concepts

Understanding First-Order Reactions

First-order reactions are a category of chemical reactions where the rate is directly proportional to the concentration of a single reactant. In the context of DNA uncoiling, this reaction type suggests that the rate at which DNA strands separate into single strands is determined by the concentration of double-stranded DNA present.
A unique characteristic of first-order reactions is that they have a consistent half-life regardless of the initial concentration. This is because the half-life, a measure of the time taken for half of the reactant to be consumed, is only dependent on the rate constant, denoted as 'k'. Understanding this key concept helps us to predict how quickly a reaction will proceed over time simply by knowing the rate constant.
In practical terms, for students analyzing DNA uncoiling, this means that the time taken to reach half of the initial concentration of coiled DNA will always be the same, regardless of how much DNA you start with. This concept simplifies the analysis, allowing for straightforward calculations and predictions of uncoiling rates in biological systems.

Calculating Half-Life in Reactions

Half-life calculations are crucial for understanding the timeframes involved in chemical reactions, particularly in biological processes such as DNA uncoiling. The half-life, represented by the symbol \( t_{1/2} \), is the period required for the concentration of a reactant to fall to half its initial value.
The formula for calculating the half-life of a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \), where 'k' is the first-order rate constant. The number 0.693 is the natural logarithm of 2, which arises from the logarithmic relationship inherent in first-order kinetics. By inserting the rate constant into this equation, you can quickly find the half-life of a reaction. As shown in the exercise, determining the half-life at body temperature provides insights into how DNA behavior changes with conditions that might occur within a living organism.
To improve exercise implementation, it's important to emphasize that the half-life remains constant only for first-order reactions. Students should practice using the formula and interpreting results within real-world biological scenarios to gain a deeper understanding of reaction kinetics.

The Arrhenius Equation and Temperature's Effect on Reaction Rates

The Arrhenius equation is a fundamental formula in the study of reaction kinetics that describes how reaction rates are affected by temperature. It is represented as \( k = Ae^{\frac{-E_a}{RT}} \), where 'k' is the rate constant, 'A' is the pre-exponential factor related to the frequency of collisions, 'E_a' is the activation energy, 'R' is the universal gas constant, and 'T' is the temperature in Kelvins.
Essentially, the Arrhenius equation tells us that even a small increase in temperature can lead to a significant increase in the reaction rate. This is because higher temperatures provide more energy to the reactant molecules, increasing their movement and the likelihood of effective collisions that lead to reactions. When applying the Arrhenius equation, as seen in the uncoiling DNA exercise, it becomes evident that even the slight temperature variation found within physiological ranges can impact biological processes.
The concept also teaches us that for each degree Celsius increase, there's a quantifiable change in the reaction rate, which can be crucial for understanding enzyme activity, metabolism, and the behavior of other biochemical systems. By working through exercises on the Arrhenius equation, students can appreciate the delicate balance of conditions necessary for sustaining life.