Problem 63 - Chemistry: Principles and Reactions

Question

The uncoiling of deoxyribonucleic acid (DNA) is a first-order 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?

Step-by-Step Solution

Answer

Answer: The half-life of the uncoiling of DNA at 37°C is approximately 1414 minutes, and at 40°C, it is approximately 985 minutes. The rate of uncoiling increases by a factor of approximately 1.44 per °C over this temperature interval.

Step 1: Calculate the half-life at 37°C

First, we have k1 (rate constant) = 4.90 × 10^(-4) min^(-1). We can simply use the half-life equation for first-order reactions to calculate the half-life at 37°C: t_1/2 = ln(2) / k1 t_1/2 = ln(2) / (4.90 × 10^(-4) min^(-1)) ≈ 1414 min (approximately) So, the half-life of the uncoiling at 37°C is approximately 1414 minutes.

Step 2: Calculate the rate constant at 40°C

First, we need to use the Arrhenius equation. We will find a constant A and then use it to calculate k2 (rate constant) at 40°C. We are given Ea = 420 kJ/mol. k1 = Ae^(-Ea/(R × T1)) k1 = 4.90 × 10^(-4) min^(-1) T1 = 37°C + 273 = 310 K R = 8.314 J/(mol·K) Plugging these values into the equation: 4.90 × 10^(-4) = A × e^(-420000/(8.314 × 310)) Now, you can solve for A. A = 2.61 × 10^(13) min^(-1) Now, we need to calculate k2 at 40°C. T2 = 40°C + 273 = 313 K. k2 = Ae^(-Ea/(R × T2)) k2 = 2.61 × 10^(13) × e^(-420000/(8.314 × 313)) k2 ≈ 7.04 × 10^(-4) min^(-1) The rate constant at 40°C is approximately 7.04 × 10^(-4) min^(-1).

Step 3: Calculate the half-life at 40°C

Now, we can calculate the half-life at 40°C, using the rate constant k2: t_1/2 = ln(2) / k2 t_1/2 = ln(2) / (7.04 × 10^(-4) min^(-1)) ≈ 985 min (approximately) So, the half-life of the uncoiling at 40°C is approximately 985 minutes.

Step 4: Calculate the rate of uncoiling increase per °C

Next, we need to find the factor by which the rate of uncoiling increases over the temperature interval of 37°C to 40°C. We can find the rate constant ratio k2/k1: Rate increase factor = k2 / k1 = (7.04 × 10^(-4)) / (4.90 × 10^(-4)) ≈ 1.44 Therefore, the rate of uncoiling increases by a factor of approximately 1.44 per °C over this temperature interval.

Key Concepts

Activation Energy

Activation energy is a crucial concept when discussing chemical reactions, as it is the energy barrier that reactants must overcome to become products. In our scenario of the DNA uncoiling, the activation energy is given as 420 kJ/mol. This signifies the amount of energy required to initiate the uncoiling process.

Activation energy can be thought of as the 'gateway' for a reaction to occur, and it plays a pivotal role in determining how fast a reaction takes place. Higher activation energy generally means a slower reaction, as fewer molecules will have sufficient energy to overcome this barrier. Whereas, a lower activation energy allows for a quicker reaction, as more molecules can reach the necessary energy level to react. Understanding this helps us predict and control reaction rates by adjusting conditions such as temperature or adding catalysts to lower the activation energy.

Arrhenius Equation

The Arrhenius equation is a formula used to calculate the rate constant of a reaction based on temperature and activation energy. It is expressed as: \[ k = Ae^{(-Ea/(R \times T))} \] where:

• \( k \) is the rate constant,
• \( A \) is the frequency factor, representing the number of times that reactants approach the activation barrier per unit time,
• \( Ea \) is the activation energy,
• \( R \) is the universal gas constant \,\( (8.314 \text{ J/mol·K}) \),
• \( T \) is the temperature in Kelvin.

For the DNA uncoiling problem, using the Arrhenius equation is vital to calculate the change in the rate constant when the temperature changes from 37°C to 40°C.

By plugging in the known values, we determine the rate constants at these different temperatures, allowing us to understand how temperature affects the rate of the uncoiling process.

Half-Life Calculation

In first-order reactions like the uncoiling of DNA, the half-life is the time required for half of the reactant to decompose or react. The half-life \( (t_{1/2}) \) is calculated using the formula: \[ t_{1/2} = \frac{\ln(2)}{k} \] where \( k \) is the rate constant. For our particular reactions, this equation helps us compute the half-life at both given temperatures of 37°C and 40°C.

At 37°C with a rate constant of \( 4.90 \times 10^{-4} \; \text{min}^{-1}\), the half-life comes out to approximately 1414 minutes. At 40°C, with an increased rate constant of \( 7.04 \times 10^{-4} \; \text{min}^{-1}\), the half-life reduces significantly to approximately 985 minutes. This change indicates that even small increases in temperature can substantially decrease the half-life, meaning the reaction proceeds faster.

Rate Constant

The rate constant, denoted as \( k \), plays a key role in determining the speed of a chemical reaction. For a first-order reaction, the units of \( k \) are time inverse \( (\text{min}^{-1})\). In the DNA uncoiling example, \( k \) changes with temperature. At 37°C and 40°C, the constants are \( 4.90 \times 10^{-4} \text{ min}^{-1}\) and \( 7.04 \times 10^{-4} \text{ min}^{-1} \) respectively.

The variation in the rate constant with temperature is calculated using the Arrhenius equation, and it continuously increases as temperature increases. This trend is consistent with the general principle that higher temperature usually enhances reaction rates, as more molecules possess the energy needed to surpass the activation energy barrier. Understanding how \( k \) changes can help in predicting and controlling the dynamics of chemical systems by adjusting conditions such as temperature. It essentially gives chemists a tool to tweak and optimize reaction conditions for desired outcomes.