Total entropy change determining spontaneity

[Angela 1K]

I was working on homework problem 9.73, and it asks how _TOT could help assess spontaneity, and according to the solutions manual, the total change in entropy has to be positive.

Can someone explain why, both mathematically and logically, a positive change in entropy signifies spontaneity?

Thanks.

[GabrielGarciaDiscussion1i]

I can help LOGICALLY (and sort of mathematically), think about a firecracker. The deltaS for a firecracker exploding is extremely positive when ignited because it leads to a tremendous amount more of disorder, where S represents entropy (disorder). However, think about the reverse process, an exploded firecracker will NEVER turn back into a new firecracker because the deltaS for the reverse rxn is EXTREMELY negative and logically, that just wouldn't make sense. We have never seen that, nor will we, in real life.

[Jonathan Tangonan 1E]

If you look at the equation delta G= deltaH-TdeltaS, we can determine why we want entropy to be positive for a spontaneous reaction. For a reaction to be spontaneous we want delta G to be a negative value because a negative value indicates spontaneity. There's a table in the textbook Table 9.5 on page 349 which shows the factors that favor spontaneity when looking at this equation.

[melissa carey 1f]

Since total entropy of an isolated system must always increase (2nd law) - then ∆S_tot must always be positive. Order cannot come from disorder without an input on energy. For example, think about freezing ice. Liquid > solid is a decrease of energy, but you must put in MORE energy (electrical from the fridge) to cause this process to occur, creating a larger increase in S than the decrease that is happening to the water. ∆S is positive.