Question

The enthalpy change in the denaturation of a certain protein is \(125 \mathrm{~kJ} / \mathrm{mol}\). If the entropy change is \(397 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol},\) calculate the minimum temperature at which the protein would denature spontaneously.

Short answer

The minimum temperature is approximately 314.62 K.

Step-by-step solution

Identify the Formula

To find the minimum temperature at which the reaction is spontaneous, we use the Gibbs free energy change formula: \[ \Delta G = \Delta H - T \Delta S \] For spontaneity, \( \Delta G \leq 0 \). Therefore, set \( \Delta G = 0 \) for the minimum temperature.

Rearrange and Substitute

Rearrange the equation to find the temperature:\[ T = \frac{\Delta H}{\Delta S} \]Substitute the values: - \( \Delta H = 125,000 \) J/mol (converted from kJ to J)- \( \Delta S = 397 \) J/K·mol

Calculate the Minimum Temperature

Plug in the values to find the temperature:\[ T = \frac{125,000}{397} \approx 314.62 \text{ K} \]This is the minimum temperature for the protein to denature spontaneously.

Key concepts

Enthalpy Change

In chemistry, enthalpy change (\( \Delta H \)) refers to the heat absorbed or released during a reaction at constant pressure. It's an essential concept for understanding protein denaturation because it shows how much energy is required to break the bonds that maintain a protein's structure.
- In the context of the exercise, the enthalpy change is given as 125 kJ/mol. This is the amount of energy needed to denature the protein, causing it to lose its structure.
- We express enthalpy changes in kilojoules per mole, which makes it easier to relate the energy changes to the amount of substance in the reaction.
Understanding enthalpy change helps us know whether a process releases or absorbs heat, providing insight into the energy dynamics of molecular transformations, such as protein unfolding.

Entropy Change

Entropy change (\( \Delta S \)) is another crucial concept. It's the measure of disorder or randomness in a system. When a protein denatures, its highly ordered structure unfolds, leading to increased disorder, hence higher entropy.
- In the example given, the entropy change during denaturation is 397 J/K·mol. This aligns with the intuitive concept that when a protein's structure is disrupted, the disorder is increased significantly.
- Entropy is measured in joules per kelvin per mole, reflecting how much disorder energy is distributed over the system at a specific temperature.
By considering entropy change, we understand how the energy distribution within a system changes, influencing whether a reaction occurs spontaneously.

Gibbs Free Energy

Gibbs free energy (\( \Delta G \)) is a vital thermodynamic quantity that predicts whether a chemical reaction will proceed spontaneously. It combines both enthalpy and entropy changes in its calculation.
- The equation \( \Delta G = \Delta H - T\Delta S \) is central to thermodynamics, and it lets us assess the trade-offs between energy absorption and increase in disorder.
- A negative \( \Delta G \) indicates a spontaneous process, meaning the reaction can proceed without any added energy.
In our context, setting \( \Delta G = 0 \) helps determine the threshold condition where denaturation just starts to become spontaneous, helping us find the minimum temperature.

Minimum Temperature for Spontaneity

To determine the minimum temperature for the spontaneity of protein denaturation, we rely on the rearranged Gibbs free energy equation:\( T = \frac{\Delta H}{\Delta S} \). This calculation provides the lowest temperature at which the enthalpy and entropy conditions allow the reaction to proceed without external energy input.
- In the original exercise, substituting \( \Delta H = 125,000 \) J/mol and \( \Delta S = 397 \) J/K·mol gives a temperature of approximately 314.62 K.
- This temperature marks the point where the energy and disorder required for the protein to denature align in a way that the process happens spontaneously.
Understanding this concept is crucial in biochemistry, as it helps predict protein stability under varying temperatures, which is essential for enzyme activity, food processing, and medical applications.