Question

Given that \(\Delta G_{\mathrm{sys}}=-T \Delta S_{\text {univ }},\) explain how the sign of \(\Delta G_{\mathrm{sys}}\) correlates with reaction spontaneity.

Short answer

A negative \( \Delta G_{\text{sys}} \) indicates a spontaneous reaction.

Step-by-step solution

Understand the relationship

The equation \(\Delta G_{\text{sys}} = -T \Delta S_{\text{univ}}\)relates the Gibbs free energy change of the system (\( \Delta G_{\text{sys}} \)) with the universal entropy change (\( \Delta S_{\text{univ}} \)). Here, \(T\) is the absolute temperature.

Analyze the equation components

Identify what each term represents:- \( \Delta G_{\text{sys}} \): Change in Gibbs free energy of the system.- \( T \): Absolute temperature (always positive).- \( \Delta S_{\text{univ}} \): Change in universal entropy.

Consider the sign of \( \Delta S_{\text{univ}} \)

For a process to be spontaneous, the universal entropy change (\( \Delta S_{\text{univ}} \)) must be positive. This means the system plus surroundings are increasing in entropy.

Apply the sign of \( \Delta S_{\text{univ}} \) to the equation

Given that \( \Delta S_{\text{univ}} \) is positive for spontaneous processes, plugging this into the equation \(\Delta G_{\text{sys}} = -T \Delta S_{\text{univ}}\)because \( T \) is always positive, results in a negative \( \Delta G_{\text{sys}} \).

Conclude the correlation

If \( \Delta S_{\text{univ}} \) is positive (spontaneous process), the sign of \( \Delta G_{\text{sys}} \) will be negative. Hence, a negative \( \Delta G_{\text{sys}} \) indicates that a reaction is spontaneous.

Key concepts

Gibbs free energy

Gibbs free energy is a thermodynamic potential that measures the maximum reversible work a system can do at constant temperature and pressure. It is symbolized as \(\triangle G_{\text{sys}}\). The equation is derived as: \[ \triangle G_{\text{sys}} = \triangle H_{\text{sys}} - T \triangle S_{\text{sys}}, \] where \(\triangle H_{\text{sys}}\) is the change in enthalpy, \(\triangle S_{\text{sys}}\) is the change in entropy, and \(\triangle G_{\text{sys}}\) is the change in Gibbs free energy.
This quantity helps determine if a process or reaction will proceed spontaneously. A negative \(\triangle G_{\text{sys}}\) indicates that the process can occur without needing additional energy supply, emphasizing its importance in understanding chemical reactions.

Universal entropy

Universal entropy change \( \triangle S_{\text{univ}} \) is the sum of the entropy changes of both the system and its surroundings. It's represented as: \[ \triangle S_{\text{univ}} = \triangle S_{\text{sys}} + \triangle S_{\text{surr}}. \] In thermodynamics, the second law states that for any spontaneous process, the universal entropy tends to increase. This means \(\triangle S_{\text{univ}}\) must be positive.
Spontaneity ties closely with entropy as processes that increase universal entropy are naturally favorable. This relationship is crucial because it connects macroscopic thermodynamic quantities to probabilistic behaviors at the microscopic level.

Reaction spontaneity

Reaction spontaneity refers to whether a chemical process occurs under a set of given conditions without external intervention. According to the equation \[ \triangle G_{\text{sys}} = -T \triangle S_{\text{univ}}, \] if \(\triangle S_{\text{univ}}\) is positive, then \(\triangle G_{\text{sys}}\) becomes negative, signifying a spontaneous process. Conversely, a positive \(\triangle G_{\text{sys}}\) indicates a non-spontaneous reaction.
Furthermore, the absolute temperature \( T \) always remains positive, ensuring that the product of \(-T\) amplifies the direction indicated by \(\triangle S_{\text{univ}}\). Therefore, the negativity of Gibbs free energy directly implies that the reaction is favorable and will proceed spontaneously under the defined conditions.