Problem 92 - Chemistry: An Atoms-Focused Approach

Question

Melting DNA strands. a. What is the sign of \(\Delta S\) for the separation process? b. The DNA double helix re-forms as the system cools. What is the sign of \(\Delta S\) for the process by which two single strands re-form the double helix? c. The melting point of DNA is defined as the temperature at which \(\Delta G=0 .\) At that temperature, the melting reaction produces two single strands as fast as two single strands recombine to form the double helix. Write an equation that defines the melting temperature (T) of DNA in terms of \(\Delta H\) and \(\Delta S\).

Step-by-Step Solution

Answer

Answer: The change in entropy for the separation of DNA strands is positive (\(\Delta S > 0\)) because the system becomes more disordered. On the other hand, the change in entropy for the reformation of the double helix is negative (\(\Delta S < 0\)) because the system becomes more ordered. The melting temperature of DNA is given by the equation \(T = \frac{\Delta H}{\Delta S}\), where T is the temperature at which the Gibbs free energy (\(\Delta G\)) is equal to zero.

Step 1: Understand the separation process

When DNA strands separate, they become two single strands, increasing the total number of possible configurations they can have. Entropy is a measure of disorder; therefore, when the system becomes more disordered, it has a positive change in entropy.

Step 2: Determine the sign of \(\Delta S\)

Since separating DNA strands increases the disorder of the system, the change in entropy is positive: \(\Delta S > 0\). #b. Sign of \(\Delta S\) for the process of re-forming the double helix#

Step 3: Understand the re-formation process

When two single strands re-form the double helix, the system becomes more ordered, decreasing the total number of configurations they can have.

Step 4: Determine the sign of \(\Delta S\)

Since re-forming the double helix decreases the disorder of the system, the change in entropy is negative: \(\Delta S < 0\). #c. The equation that defines the melting temperature of DNA#

Step 5: Review the Gibbs free energy equation

Gibbs free energy (\(\Delta G\)) is related to the enthalpy (\(\Delta H\)) and entropy (\(\Delta S\)) of a system by the equation \(\Delta G = \Delta H - T\Delta S\).

Step 6: Find the melting temperature equation

The melting temperature (T) of DNA is defined as the temperature at which \(\Delta G = 0\). In this case, we can set the Gibbs free energy equation to zero and solve for T: 0 = \(\Delta H - T\Delta S \Rightarrow T = \frac{\Delta H}{\Delta S}\) So, the melting temperature of DNA is given by the equation \(T = \frac{\Delta H}{\Delta S}\).

Key Concepts

Entropy Change

Entropy is a measure of disorder or randomness within a system. In thermodynamics, it indicates how energy is dispersed or spread out at a particular temperature. Imagine DNA strands: when they separate, they turn into two single strands, which increases the number of possible configurations or arrangements. This increase leads to more disorder in the system. Therefore, the separation process of DNA strands results in a positive change in entropy, denoted as \( \Delta S > 0 \).

Conversely, when two single DNA strands recombine to form a double helix, the system becomes more ordered. The possibility for different configurations decreases, resulting in decreased randomness. Hence, during this re-formation process, the change in entropy is negative, indicated as \( \Delta S < 0 \). Understanding the direction of entropy change helps us comprehend the fundamental nature of processes, whether they lead to more disorder (positive entropy change) or order (negative entropy change).

In summary, entropy changes reflect the movement between order and disorder within a system. Recognizing these changes can help in predicting the likelihood and direction of a chemical or physical process.

Gibbs Free Energy

Gibbs free energy is a central concept in thermodynamics that helps predict whether a process is spontaneous. It combines the system's entropy, enthalpy, and the temperature to give an overarching measure of a system's energy availability to do work.

The equation for Gibbs free energy is given by:
\[ \Delta G = \Delta H - T\Delta S \]
where:

• \( \Delta G \) is the change in Gibbs free energy
• \( \Delta H \) is the change in enthalpy
• \( \Delta S \) is the change in entropy
• \( T \) is the temperature in Kelvin

Understanding this equation helps determine whether a reaction will proceed without any external intervention. If \( \Delta G \) is negative, a process or reaction is spontaneous under constant pressure and temperature.

Specifically, in the context of DNA melting, the equation highlights that the melting point is reached when \( \Delta G = 0 \). This means energy contributions from enthalpy and entropy balance out perfectly at a certain temperature, known as the melting temperature \( T \). At this melting point, the formation and separation rates of the DNA double helix are equal.

Enthalpy

Enthalpy is a measure of the total energy of a thermodynamic system, often associated with heat content. It embodies the total internal energy plus the energy required to make room for the substance by displacing its environment. In simpler terms, it indicates the amount of heat absorbed or released during a reaction under constant pressure.

In the case of DNA, during processes such as melting or reformation of the helix, the enthalpy change \( \Delta H \) reflects how much heat energy is taken in or released. When DNA strands separate, they typically absorb energy, resulting in a positive \( \Delta H \). This energy intake is necessary to break the hydrogen bonds holding the strands together.

When discussing the melting point of DNA, \( \Delta H \) plays a crucial role in determining the required heat input at which the melting process is balanced with reformation. The balance of enthalpy, entropy, and temperature at melting point allows for an equal rate of separating strands and reformation into the double helix. Being familiar with enthalpy's role in these biological processes aids in understanding how heat energy influences molecular interactions.