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My understanding of the second law is that for an isolated system, the entropy change accompanying any process is non-negative and is zero if and only if the process is reversible. I don't see why this implies that just because the entropy change of a process is positive, it must occur (even kinetically unfavorable processes occur after a long enough time). In other words, why is irreversible and entropically favorable synonymous with spontaneous?

Furthermore, it is often stated that the second law implies that entropy must be at a maximum for an isolated system in equilibrium. However I don't see how the statement that the entropy of a system is non-decreasing implies that entropy must increase to a maximum. Can't entropy remain constant at a non-maximal value as well?

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  • $\begingroup$ Actually you can change entropy of a system even if you are providing heat in reversible way because entropy is a state function. $\endgroup$ Commented Oct 7, 2023 at 17:24
  • $\begingroup$ Spontaneous processes are irreversible, but irreversible processes need not to be spontaneous. E.g. if you suddenly press the piston of the gaseous system, it is irreversible, approximately adiabatic compression, but hardly spontaneous. $\endgroup$
    – Poutnik
    Commented Oct 9, 2023 at 9:09

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'why is irreversible and entropically favorable synonymous with spontaneous?'

If a system changes infinitely slowly it is capable of doing the maximum amount of work, and this process is, by definition, called 'reversible'. The conditions for a reversible process are the same as those for being in a state of equilibrium. The change of internal energy $dU$ is the same however a change is made and this means that $dw_{rev} \lt dw_{irrev}$, because work done by the system is negative, consequently from the First Law, $dq_{rev} \gt dq_{irrev}$. In a reversible change a maximum amount of work is done on the surroundings and a maximum amount of heat is absorbed from the surroundings. A spontaneous/irreversible process is an observable process and absorbs less heat and does less work than a reversible one, i.e. a smaller change in heat along an irreversible path than a reversible one.

'...entropy must be at a maximum for an isolated system in equilibrium.'

The Second Law, like the First Law is based on experimental observation, for example that a perfect gas expands into a vacuum without doing any work, the observation of heat flow or that the work done by a chemical reaction is not accounted for by only summing enthalpies. The reason for these observations is that there is 'unavailable' energy present and this is quantified as $T\Delta S$ and the entropy, a state function, is defined between two states $A,B$ as

$$\displaystyle \Delta S= S_B-S_A=\int_A^B\frac{dq_{rev}}{T}$$

As entropy is a state function it is independent of the path taken A to B and therefore entropy change is the same in going from A to B no matter how it happens. We always calculate the entropy using this eqn. which means that $q_{rev}$ has to be found somehow so that the integral $\int_A^Bdq_{rev}/T$ can be evaluated and this can only be done under reversible conditions. It is convenient to consider the derivative $dS=dq/T$ as a condition of equilibrium.

For a spontaneous i.e. observable process, $dq_{rev}\gt dq$ and so $\displaystyle dS \gt \frac{dq}{T}$

In an isolated system, one that cannot do work or absorb heat, $dq=0$ and the equilibrium conditions are $dS=0, S=constant$. For a spontaneous / observable change $dS\gt 0$. This means that for an isolated system spontaneous changes produce states of higher entropy until equilibrium is reached and thus entropy is a maximum at equilibrium. However, whenever a system can exchange heat or matter with its environment a decrease in entropy is entirely compatible with the Second Law.

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Things happen for one of two reasons: Either there is a local decrease in energy or increase in entropy. For this to be axiomatic processes must be broken into simple, hopefully elementary, steps. The Second Law simply states this; it does not imply anything. At equilibrium the energy and entropy effects are equal. For something to happen there are other necessities: there must be appropriate energy present, activation energy; there must be appropriate mixing; there must be a possible mechanism; and the system must be displaced from equilibrium. These are the "implications"; the Second Law simply states if Delta G is negative a reaction is possible. Notice this means displacement either direction from equilibrium.

Our environs are never in an isolated situation but in an energy flux of constantly increasing entropy, a steady state. When this changes there can be dire consequences; consider the dark side of the Moon or global warming for that matter. The Second Law explains it all.

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