Question

When heated, the DNA double helix separates into two random coil single strands. When cooled, the random coils re-form the double helix: double helix \(\Longrightarrow 2\) random coils. (a) What is the sign of \(\Delta S\) for the forward process? Why? (b) Energy must be added to break \(\mathrm{H}\) bonds and overcome dispersion forces between the strands. What is the sign of \(\Delta G\) for the forward process when \(T \Delta S\) is smaller than \(\Delta H ?\) (c) Write an expression for \(T\) in terms of \(\Delta H\) and \(\Delta S\) when the reaction is at equilibrium. (This temperature is called the melting temperature of the nucleic acid.)

Short answer

a) \Delta S > 0. b) \Delta G > 0. c) T = \frac{\Delta H}{\Delta S}

Step-by-step solution

Identify the Process

The forward process involves the DNA double helix separating into two random coil single strands. This means the orderly structure (double helix) becomes more disordered (random coils).

Sign of \(\Delta S\) for the Forward Process

\(\text{S}\) represents entropy. Since the ordered double helix transitions to two random coils, disorder increases. Hence, \( \Delta S \ > 0 \).

Sign of \(\Delta G\)

For the forward process, \( \Delta G = \Delta H - T \Delta S \). If \( T \Delta S \) is smaller than \( \Delta H \, \) \( \Delta G \) remains positive (\[ \Delta G > 0 \]).

Expression for \( T \) at Equilibrium

At equilibrium, \( \Delta G = 0 \. \) Therefore, \( \Delta H = T \Delta S \. \) Solving for \( T \), we get \( T = \frac{\Delta H}{\Delta S} \. \)

Key concepts

DNA melting

When we heat DNA, the double helix separates into two single strands. This process is called DNA melting or denaturation. It involves breaking the hydrogen bonds and van der Waals forces that hold the two strands together. When the double helix melts, the highly ordered structure transforms into a more disordered state. This is like opening a coiled spring into a loose, flexible coil.

The reverse process occurs when these single strands cool down and reanneal to form the double helix again, recovering the ordered structure. Both melting and reannealing are crucial in biological processes such as replication and transcription.

Entropy Change

Entropy (\text{S}\text{S}) measures the disorder or randomness in a system. For the DNA melting process, the disorder increases as the ordered double helix becomes two separate, random coils. Hence, the entropy change (\text{ΔS}\text{ΔS}) for this process is positive.

The formula for entropy change in this context is: \( \text{ΔS > 0} \). This positive value indicates that the system has become more disordered. Understanding entropy helps explain why certain processes occur spontaneously. Systems naturally evolve toward higher entropy.

Gibbs Free Energy

Gibbs Free Energy (\text{ΔG}\text{ΔG}) combines both enthalpy (\text{ΔH}\text{ΔH}) and entropy (\text{ΔS}\text{ΔS}) to determine if a process is spontaneous. The relation is given by: \[ \text{ΔG} = \text{ΔH} - T\text{ΔS} \], where \[ T \] is the temperature in Kelvin.

For the DNA melting process, if \[ T\text{ΔS} \] is smaller than \[ \text{ΔH} \], then \[ \text{ΔG} \] is positive, which implies the process is not spontaneous without an external energy input. ΔG indicates how energy should be managed in biological systems.

Equilibrium Temperature

The equilibrium temperature, also known as the melting temperature (T_m\text{T_m}), is when forward and reverse processes (melting and reannealing of DNA) occur at the same rate. At this point, the system is at equilibrium, and \[ \text{ΔG} = 0 \].

Using the equilibrium condition, we can derive the expression for temperature: \[ \text{ΔH} = T\text{ΔS} \]. Solving for T, we get \[ T = \text{ΔH}/\text{ΔS} \]. This temperature depends on the specific entropy change and enthalpy change of the DNA involved, defining the stability of the double helix structure.