Not all processes in which the system increases in entropy are spontaneous. How can this observation be consistent with the second law? Provide an example and explain your answer in complete sentences.

Entropy is a fundamental concept in the realm of thermodynamics, often symbolized by the letter 'S'. It is a measure of the disorder or randomness within a system, and the second law of thermodynamics states that the entropy of an isolated system tends to increase over time. This can sometimes be misunderstood to mean that systems always move towards disorder, but in reality, entropy is one factor in a larger thermodynamic puzzle.

For example, when ice melts to form water, the entropy of the system increases; the organized structure of ice breaks down into the more disorganized form of liquid water. However, it's essential to emphasize that while the second law notes that entropy tends to increase, it doesn't directly determine whether a process will occur spontaneously—that's where Gibbs free energy comes into play.

Gibbs Free Energy, denoted as 'G', combines the concepts of energy and entropy to provide a comprehensive understanding of a system's thermodynamics. The Gibbs free energy equation, \(\Delta G = \Delta H - T\Delta S\), relates the change in free energy (\(\Delta G\)) to the enthalpy change (\(\Delta H\)), the temperature (T), and the entropy change (\(\Delta S\)) of a system.

A negative value for \(\Delta G\) indicates a spontaneous process. This doesn't necessarily require an increase in entropy; a decrease in system enthalpy can also drive spontaneity. For instance, the freezing of water into ice is spontaneous under the right conditions, even as the system entropy decreases, because the release of heat (enthalpy change) is sufficient to make \(\Delta G\) negative.

Spontaneous processes are natural occurrences that do not require external energy to proceed. They can be misleadingly thought of as processes that happen 'by themselves,' but this description lacks the precision of the thermodynamic definition. In thermodynamics, spontaneity is all about Gibbs free energy.

If the change in Gibbs free energy (\(\Delta G\)) for a process is negative, the process is considered spontaneous. This takes into account both entropy and enthalpy changes, as well as the temperature at which the process occurs. The negativeness of \(\Delta G\) captures the essence of spontaneity, but it's crucial to recognize that even spontaneous processes may not happen instantly—they may require a catalyst or may proceed at a slow rate.

Thermodynamic equilibrium is the state at which a system experiences no net change over time. This occurs when the system has reached a balance in energy and particle distribution, and as such, no further spontaneous processes can occur. This balance also implies that the system's entropy has reached a maximum relative to the constraints of the system, and the Gibbs free energy is at its minimum.

At equilibrium, a system's forward and reverse processes occur at the same rate, leading to no observable changes in the system's properties. For example, in a room-temperature sealed container with water and its vapor, an equilibrium is achieved when the rate of water evaporating equals the rate of vapor condensing back into water. Understanding the fine balance of thermodynamic equilibrium is essential in fields like chemistry, physics, and engineering.