# Truly Understanding the Second Law of Thermodynamics and Entropy

There are some seemingly contradictory things I’d like cleared up:

Second Law states: $\mathrm dS_\text{univ} = \mathrm dS_\text{sys} + \mathrm dS_\text{surr}$, and $\mathrm dS_\text{univ}$ is always positive in a spontaneous process.

First off, what does “spontaneous” truly mean? “It means the reaction occurs without any external energy input” – I’ve heard this one before, but it doesn’t make sense. In an equilibrium there is a spontaneous direction (the side that is favored) but the reverse reaction also occurs, though at a slower rate, than the spontaneous reaction. What is the most fundamental definition, intuitive and universally applicable definition of spontaneous?

Moving on, according to ChemWiki the Second Law states that the entropy change of the universe can never be negative, even though my textbook says that this is only true for spontaneous processes. Who is right and why (explain in quantum terms, since I understand that in the long run of course the entropy of the universe increases by the laws of probability)?

An example to work through:

In a living cell, large molecules are assembled from simple ones. Is this process consistent with the second law of thermodynamics?

Book’s solution: To reconcile the operation of an order-producing cell with the second law of thermodynamics, we must remember that $\mathrm dS_\text{univ}$, not $\mathrm dS_\text{sys}$, must be positive for a process to be spontaneous. A process for which $\mathrm dS_\text{sys}$ is negative can be spontaneous if the associated $\mathrm dS_\text{surr}$ is both larger and positive. The operation of a cell is such a process.

First off, are we to consider the operation of a cell in this problem spontaneous? As I recall from biology, life processes require ATP, the carrier of energy, so these would technically “require external energy input,” would they not? However, the book treats it as a spontaneous process. Again, this confuses me because the definition of spontaneity is inadequate as I now semi-understand it.

Sorry for such a long post, but these are very important and fundamental ideas in spontaneity, entropy, and free energy, and I don’t want to read on until these ideas are well ingrained, and correctly so.

• +1 for the research. Spontaneity means a reaction will happen without any, say, intervention. Also, please take a look at this question which explains the [seemingly deviating] contradiction between life and the second law. Apr 12, 2015 at 19:17
• Gibbs enthalpy is for spontaneity - it seems your textbook is wrong Apr 12, 2015 at 19:39
• Oh, I think I get it, "It means the reaction occurs without any external energy input" - it's closed system in definition and that's all. Apr 12, 2015 at 20:52

Be careful in citing equilibrium as an example, as it fully integrates entropy in its definition. See this article for the details, but in a spontaneous reaction the entropy is increasing, and part of reaching equilibrium is that the energy gain and loss among products and reactants is balanced. Also, most experimental reactions are (nearly) closed systems for short periods of time, and if no energy is gained or lost (again, for short periods), you can have an equilibrium where there is no change in entropy. If you let your experiment sit and the heat of the system changes (for example), the equilibrium will change as well. And eventually you end your experiment, clean out your flask, and thus increase the entropy of the universe again!

@MARamezani's comment pointing to the article on Biology looks like a good answer to your biology questions. Earth is most definitely not a close system with that giant energy source 93 million miles away.

First off, what does “spontaneous” truly mean? “It means the reaction occurs without any external energy input” – I’ve heard this one before, but it doesn’t make sense. In an equilibrium there is a spontaneous direction (the side that is favored) but the reverse reaction also occurs, though at a slower rate, than the spontaneous reaction. What is the most fundamental definition, intuitive and universally applicable definition of spontaneous?

"Spontaneous" is being replaced in some syllabi by "thermodynamically favoured", which is a mouthful, but maybe more accurate! It means that $$\Delta G < 0$$ for the forward reaction, in other words, that we move to a lower $$G$$ if the forward reaction runs and, overall, converts some reactants to products.

We often consider this under standard conditions, where all concentrations are $$1M$$ so that concentration effects on the favourability of a reaction are ignored and the intrinsic stabilities of the reactants and products can be seen. This also gives us an idea of where an equilibrium will lie, to the right (mostly products) gives a large $$K$$ or to the left (mostly reactants) gives a small $$K$$. Overall the relationship is given by $$\Delta G^o = -RTlnK$$ In other words. A process will be thermodynamically favoured ($$\Delta G^o < 0$$) under standard conditions ($$Q = 1$$) if the equilibrium position is to the right of standard conditions (more products than reactants, $$K > 1$$).

To me, this is the real definition of "spontaneous". If a system is to the left of the equilibrium position (more reactants compared to products than we see at equilibrium) then it will naturally happen that the system will shift towards equilibrium as the forward reaction runs. This involves a decrease in $$G$$ as you move towards the more stable, equilibrium composition. http://people.cst.cmich.edu/teckl1mm/PChemI/Chm351Ch9aF01.htm

Interestingly, this phenomenon can also be seen in terms of $$S$$ rather than $$G$$. The difference is that an equilibrium position involves the highest $$S_{TOT}$$ as this actually corresponds to the lowest $$G$$. Another way of thinking of $$G$$ is the opposite of the entropy of the universe. Processes tend to involve $$G$$ decreasing and $$S_{TOT}$$ increasing. Either one of these can be thought of as a statement of the Second Law.

Your analysis of the kinetics is correct - if a system is shifting to the right to come to equilibrium, then $$forward~rate > reverse~rate$$ this is by definition and it explains, from a kinetics point of view, what is happening. However the overall thermodynamics depend only on the state before and after, not the path taken in terms of rates.

Moving on, according to ChemWiki the Second Law states that the entropy change of the universe can never be negative, even though my textbook says that this is only true for spontaneous processes. Who is right and why (explain in quantum terms, since I understand that in the long run of course the entropy of the universe increases by the laws of probability)?

$$\Delta S_{TOT}$$ is always positive for a spontaneous change in isolation. However, by coupling a change that has a negative $$\Delta S_{TOT}$$ (non-spontaneous) to another process that has a more positive $$\Delta S_{TOT}$$ (spontaneous) we can get a situation where a "non-spontaneous" change is carried out but the overall process, including both changes, is spontaneous. You can argue the same point from the point of view of $$\Delta G$$ except that the signs are reversed.

See if the example that you gave to work through makes any more sense now. You have all the pieces needed in order to understand it.

Some concepts in thermodynamics are unclear due to historic issues.

At first: "In an equilibrium there is a spontaneous direction (the side that is favored) but the reverse reaction also occurs". Thermodynamics arose as a macroscopic subject, in the XIX century two elemental reactions in opposite directions were not obvious. (Also, analyze your statement cautiously).

You should understand "spontaneous" in a macroscopic sense. Of course, you may find contradictions in the literature (because of the mixing of the macroscopic and microscopic worlds because of pedagogical reasons?)

that the entropy change of the universe can never be negative, even though my textbook says that this is only true for spontaneous processes

The first one is referring entropy changes associated to processes that are real. I mean, non considering results of calculations for arbitrary processes (which are very unlikely to have happened).

About the example. As most theories, thermodynamics has applicability limits. Specially the restricted version treated in most text books. The example is invalid because of it is completely incorrect to apply the thermodynamic of you book to a non-homogeneous system (or a system composed by homogeneous parts). There is a strange tendency to apply thermodynamic to anything. (I do not know exactly what is your book but I can be sure of the kind of treatment because of your cites).

In the other side, there the author is trying to relate entropy with order/disorder. This is only valid for total energy constant processes (I'm referring to the tendence to disorder). My advice is: be patient till you take a rigorous course (statistical physics or statistical thermodynamics or thermodynamics from another perspective), learn to repeat the examples for your test and forget it all as fast as you can. Or remember this while keeping in mind that maybe it is not exactly this way.

If you want to start with a relatively simple but more serious thermodynamics book, you can check out this:

Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.