The definition of Spontaneous in thermodynamics?

Question

The definition of spontaneous is often briefly glossed over in most of the thermodynamics texts that I own. Peter Atkins in Physical chemistry defines spontaneous as follows

Some things happen naturally, some things don’t. Some aspect of the world determines the spontaneous direction of change, the direction of change that does not require work to bring it about.

Later in the text and in many others you then find that

The entropy of an isolated system increases in the course of a spontaneous change: $\Delta S_{tot} > 0$

These two definitions of spontaneity lead me to two conclusions:

1: That net work must be done on a system in order to bring about a non-spontaneous process. Alternatively, systems may only do net work on their surroundings through spontaneous processes.

2: That there is no such thing as a non-spontaneous process in this universe.

My question pertains to my second conclusion. Is this conclusion valid? The thought process that leads me to this conclusion is as follows: The universe is an isolated system. Any spontaneous change in an isolated system increases the entropy of that system. The second law dictates that the only processes possible are those that increase the entropy of the universe. Thus, only spontaneous processes are possible. As an example, a fridge which transfers energy from a cold sink to a heat sink must be spontaneous process. This is because in order for this to occur, we must perform work on the fridge and that work is coming from a heat engine(i.e a power plant) which is spontaneous. The ultimate result is that the refrigerator increases the entropy of the universe and thus is spontaneous. Is this correct?

I guess my question could be more succinctly stated as follows: Is the set of possible spontaneous processes equal to the set of all thermodynamically allowable processes? That is, if a process is non spontaneous, is it forbidden to occur by the laws of thermodynamics?

Any help on this issue would be greatly appreciated!

Answer 1

You can see it like this: if you consider the universe as a whole, "of course" each process is "spontaneous" in the sense that the total entropy of the universe entropy always increases. After all, if something happens somewhere in your universe, then it must be allowed to happen: otherwise, it just would not happen!

However, what you ususally care about is that your fridge stays cold, and that is why thermodynamics focuses on sub-systems: it is not enough to say "somewhere in the universe there is the possibility of a fridge". You want to know about your fridge (and your electrical bill at end of the month and the natural resources of your planet!).

A question about a spontaneous process would be: if I buy a fridge, would it get cold? And the answer is: no (unless you plug it to an external power source)! If you turn the question to "do cold fridges exist in the universe without any external assumption except the existence of the universe" then the answer is, of course, yes.

So depending on the scale you look at things (fridge < electrical bill < planet < universe) things can be considered spontaneous or not. You need the current for your fridge, you need a power plant for the current, you need the sun for the power plant, you need the big bang for the sun, etc.). So you need to choose: at what scale do I look at the system: would a fridge work without current? Would current work without the sun? Would the sun work without the univse?

Can you say this in more scientific words?

The universe as a whole only has processes which increase the total entropy. Now, let us focus on the entropy of a sub-system! Here is the tricky point: a sub-system of the universe, if you consider it as an individual object, could locally decrease its entropy if you supply work to it, i.e. it could to something you would NOT expect to happen unless you considered it as part of a bigger system.

So, let's say I have a subsystem, I see its evolution. Then the question is:

do I have to resort to the presence of an outside energy source in my system to explain its behavior?

If yes, then the process is not spontaneous. If not, then the process is.

A spontaneous process inside a system the you are observing is one that you can explain without resorting to an outside energy source.

Extended discussion with a simple example

Imagine you have a system which is not in an equilibrium state. How did it get there is irrelevant.

The simplest example is two identical boxes with a perfect gas inside, one at temperature $T_1$, one at temperature $T_2$ with $T_1>T_2$. If they are put somehow in contact, then, without any additional external work, heat will flow from the hot to the cold bath ($T_1\to T_2$) until the two temperatures are the same at the equilibrium temperature $T_e=(T_1+T_2)/2$.

You did not do anything: just put the two gases in contact and they will spontaneously change their temperature to the equilibrium state.

Now, imagine you want the cold gas to stay cold (a fridge) despite its hot sourrounding: to counter the spontaneous process you need to remove heat from the cold gas to keep it cold. To do that, you need some work (the energy given by the current to the fridge).

In this latter case, the entropy of the universe will increase, but the entropy of the sub-system "fridge" will not, because you are artificially keeping it constant. You are basically removing entropy from the fridge at the expense of the universe. But you need to assume that there is an outisde universe that is providing the extra energy required!

Let's reverse the reasoning. You see two gases together, at $T_1$ and $T_2$. You observe them and see that their temperature does not change. Then it must mean that there must be somewhere some work given to the system to maintain them as they are. On the other hand, if you work at the two gases and see that they equilibrate, then you can assume the system is behaving as it should be if there is no work done on it. That a process is spontaneous means you can describe it without resorting to an external energy source: that is how two isolated gas should behave.

If now you include the whole universe then of course you can explain why the fridge stays cold: the universe already contains the energy source, so there is no inconsistency. And because the universe is isolated, you don't see any process inside the universe that requires an energy source outside your system: it's already somewhere in your system.

Summing up, (not-)"spontaneous" actually can only be applied to sub-systems. If you consider only a subsystem, which can exchange heat with the rest of the universe (so not an isolated system) you can make it evolve towards states which the same subsystem, if isolated, would never evolve to. We call spontaneous processes those that would happen also if the system was completely isolated.

In order to have a non - spontaneous process, you need some sort of energy flux (work, heat..) from the outside. In the big picture of the universe, on the other hand, everything is spontaneous, in a way.

It really depends on your point of view when describing the system.

Answer 2

The two definitions you cite do not mean that non-spontaneous processes decrease entropy, if that is what you are implying. In other words, the fact that all spontaneous (natural) processes increase entropy does not mean all non-spontaneous processes should decrease entropy.

The refrigeration/heat pump cycle is a non-spontaneous transfer of heat from cold to hot, but as you know a net work input is required to do so and that results in an increase in entropy for a real cycle (zero change for an ideal reversible cycle), not a decrease. If no net work input were required, there would be a decrease in entropy and a violation of the Clausius's Statement of the Second Law:

No refrigeration or heat pump cycle can operate without a net work input.

Hope this helps.

Answer 3

Two types of processes:

• There are thermodynamics processes that change the state of some variables like temperature and entropy.

• In statistical mechanics there are microscopic processes which move particles between microstates, this change may move the system closer to the equilibrium macrostate or not.

There is no such thing as a spontaneous thermodynamic process, in the sense of a system spontaneously doubling its volume.

There are spontaneous and stimulated (non-spontaneous) microscopic processes.

Spontaneous processes tend to move the system to the equilibrium macrostate, because they allow energy to be distributed ergodically across all degrees of freedom of the particles involved.

For example, a laser operates by the competition of spontaneous emission and simulated (non-spontaneous) emission processes.

In a laser, energy flows into a system, initially, this causes the spontaneous processes to dominate, as optical modes are populated more or less equally. After a certain time stimulated processes will dominate: photons begin to populate a single optical mode. This is a highly ordered macrostate and is only maintained by energy flowing into the system.

I’ve tried to add how I would approach thinking about this, it was too long for a comment.

Answer 4

These are interesting questions.

1: That net work must be done on a system in order to bring about a non-spontaneous process. Alternatively, systems may only do net work on their surroundings through spontaneous processes.

The first statement might be considered true* but it does not rank as a definition; the second is wrong.

(* Note that if a process "has been made" spontaneous, it cannot strictly be considered as the same process anymore. So this rather common way of stating that "a non-spontaneous process becomes spontaneous", even though frequently encountered, is actually improper.)

Think about a system at a lower pressure than the exterior. It will spontaneously contract, and thereby receive work: $dW = -P_{\rm{ext}}dV > 0$. Inversely, the non-spontaneous expansion would correspond to the system delivering work.

And more generally, since one would get the reverse results by following a similar reasoning with a system at a higher pressure than the exterior, it follows that whether work is received or delivered is not a criterion of spontaneity.

2: That there is no such thing as a non-spontaneous process in this universe.

This is actually the logical negative ($\neg$) equivalent form of the definition of spontaneous, it is excessive to call it a "conclusion", rather a reformulation of the definition of the term.

Is the set of possible spontaneous processes equal to the set of all thermodynamically allowable processes? That is, if a process is non spontaneous, is it forbidden to occur by the laws of thermodynamics?

Yes, by definition. With the caveat regarding whether reversible processes qualify as "spontaneous" and "allowable" since they are limiting cases...

(A side note: "possible spontaneous" can be seen as a pleonasm here depending on the meaning of the term "possible" at this place in the sentence...)

And yet, all this still doesn't give an operational definition (i.e. a criterion) that would allow one to determine whether a given process should be qualified as spontaneous or non-spontaneous...