Spontaneous Change and Equilibrium

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Free Energy.

Diamond turns spontaneously into graphite; yet we may have to wait longer than the lifetime of the universe to see such a change.  Hydrocarbons are spontaneously oxidized into carbon dioxide and water; yet gasoline requires a source of heat before it burns.

Spontaneity

In thermodynamics, the term spontaneity has a very specific meaning. Dictionary definitions include the following.

From Dictionary.Com::

1. coming or resulting from a natural impulse or tendency; without effort or premeditation; natural and unconstrained; unplanned: a spontaneous burst of applause.
2. (of a person) given to acting upon sudden impulses.
3. (of natural phenomena) arising from internal forces or causes; independent of external agencies; self-acting.

From Merriam Webster:

1.  proceeding from natural feeling or native tendency without external constraint
2. arising from a momentary impulse
3. controlled and directed internally : self-acting
4. produced without being planted or without human labor : indigenous
5. developing or occurring without apparent external influence, force, cause, or treatment
6. not apparently contrived or manipulated : natural

Notice that even the dictionary definition does not necessarily imply a time-frame, i.e., it does not have to happen right away to be spontaneous, even though in common parlance the word is often used in contexts that imply that an event occurred immediately.

The  thermodynamic definition is perhaps closest to the definition that states "arising from internal forces or causes."

A caveat is necessary, however.  Some changes that are spontaneous, in a thermodynamics sense, can require an external impetus to initiate them.  The change arises from forces inherent in the system, but an external nudge may be necessary to overcome an activation barrier.

Consider a glass sitting on a table.  If it falls to the floor and shatters, the occurrence arose from the force of gravity that was always present in the system.  Of course, it may require a clumsy person, or a curious cat to provide a nudge to get over an activation barrier, but after that initial nudge, the energy was already stored in the system.

In thermodynamics, a spontaneous change is just such a change, in which the driver for change is internal to the system.  We do not care how fast the change happens, or whether something external might help it get going.

A ball spontaneously rolls down a hill; it does not spontaneously roll up a hill.  The fact that it may need a nudge to get started is actually a feature of kinetics.  


Kinetics focuses on how fast a change will occur, thermodynamics is only concerned with the direction in which the net change occurs.

A change that is not spontaneous can still happen.  It is possible to roll a ball up a hill.  It is possible to pick a glass up off the floor and put it on a table.  It is possible to transfer heat from a cold reservoir to a hot reservoir.   All these non-spontaneous changes, require something to drive them.

The driver for a non-spontaneous change, in fact, must be a change that is spontaneous. Spontaneous changes  provide the energy that is used to drive a non-spontaneous change.

Of course, we must be careful, when we talk about providing energy.  Energy is conserved. Energy cannot strictly be used up.

It is the distribution of energy that changes during a spontaneous change.  The useful energy comes from changing the way that energy is distributed in the system and the surroundings.

At constant pressure, a change is spontaneous if the change in Gibbs Free Energy is negative.

It is possible to drive a change, in which the free energy is positive by a bigger change in which the free energy is negative.


Equilibrium

If you were truly at equilibrium, you'd be dead.  We live in a world that is constantly in flux.  Perhaps, at the heat death of a closed universe, equilibrium will be reached.  Equilibrium is a useful concept, but we must understand equilibrium to be both local and temporary.

Consider a glass of ice water.  In this glass, I have placed enough water and enough ice that they can come to local equilibrium.  That equilibrium is reached at the freezing point of water (32 °F, 0 °C, 273.15 K).

Of course, if we wait long enough the glass will warm to the temperature of the room and all the ice will melt.  Let's assume that the glass is well insulated from the room so that heat from the room cannot be transferred to the glass and vice versa.  At least for the short term, it is possible to obtain equilibrium in the glass.

At equilibrium, ice is melting to become liquid water and absorbing heat, but liquid water is freezing on the surface of the ice cubes and giving off heat.

These processes are in balance, water becomes ice, and ice becomes water.  If the glass is truly at equilibrium, the ratio of ice to water remains the same as these changes occur.

The same occurs in a chemical reaction.  Consider a simple reaction, in which two reactants, A and B react to form two products C and D.

     A + B  →  C + D

Suppose that this reaction is spontaneous, i.e., ΔG < 0 for the reaction.  Still, it is necessary to consider the reverse reaction, which after all can be driven by energy from somewhere in the system.

     C + D  →  A + B

At equilibrium, a solution of the reactants and products will have all four species.  The amount of the constituents is determined by an equilibrium constant, K.

     K = [C]*[D]/[A]*[B]

Here the square brackets indicate the concentration.  It is simplest to take the concentrations as molar concentrations.

The equilibrium constant K is related to the standard change in Gibbs energy for the reaction.

     K = exp (-ΔGrxn/RT )

Exp is the exponential function. ΔG rxn is the standard molar change in Gibbs energy  for the reaction.  R is the universal gas constant, and T is the thermodynamic temperature.

At equilibrium the forward change is balanced by the backward change.  It does not mean there is no change; it means that the change balances.

Energy Flow

The same is true of energy flow from one body to another body.  Two bodies in equilibrium with each other are at the same temperature (the so-called zeroth law of thermodynamics).  If two bodies are at the same temperature, it is not the case that no energy is flowing between the bodies.

Rather the energy flowing between the two bodies exactly balances.  There is no net flow of energy because the flow of energy cancels.

Now consider a hot body and a cooler body.  It is important to realize that energy flows in both directions.  It flows from the hot body to the cool body and from the cool body to the hot body.  It is not a violation of the second law of thermodynamics.

More energy flows from the hot body to the cool body, and the net flow of energy is from the hot body to the cool body.

Consider a black body at 100 K.  It radiates according to its characteristic radiation curve.

Figure source

That body at 100K does not know whether it is radiating toward empty space or whether there may be another body there that could be cooler or hotter.

Now imagine that there is another hotter black body at 300 K.  It is a black body; so it will absorb any photons that come its way.  It does not know the source of the photons is cooler than it is.  There is no magic here; the bodies do not somehow know to stop radiating to a hotter body.

However, the body at 300K radiates more than the body at 100 K.  More photons, and thus more energy flow from the hot body to the cool body than vice versa.  The net flow of energy is from hot to cold, but energy flow both ways.

If allowed to come to equilibrium (and ignoring the influence of other bodies), the hot body will cool as it radiates,  the cool body is absorbing radiation from the hot body and it will warm.  As the hot body cools, it radiates less.  As the cool body warms it radiates more.  When they reach the same temperature, the flows of energy will be equal, resulting in no net flow of energy.

If it did not work this way, there would be no use putting blankets over you at night.  It is an exercise for the reader to consider why a blanket that is colder than one's body can nevertheless keep one warm.

The next post is entitled The Second Law, Radiative Transfer and Global Warming.

Sources

• Dictionary.Com: Spontaneous
• Merriam Webster: Spontaneous
• Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
• McQuarrie, Donal d A., Statistical Thermodynamics,  University Science Books, Mill Valley, CA, 1973 
• Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
• Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
• http://commons.wikimedia.org/wiki/File:BlackbodySpectrum_loglog_150dpi_en.png

Contents

1. Introduction
2. What the Second Law Does Not Say
3. What the Second Law Does Say
4. Entropy is Not a Measure of Disorder
5. Reversible Processes
6. The Carnot Cycle
7. The Definition of Entropy
8. Perpetual Motion
9. The Hydrogen Economy
10. Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
11. The Second Law and Swamp Coolers
12. Entropy and Statistical Thermodynamics
13. Fluctuations
14. Partition Functions
15. Entropy and Information Theory
16. The Second Law and Creationism
17. Entropy as Religious, Spiritual, or Self-Help Metaphor
18. Free Energy
19. Spontaneous Change and Equilibrium
20. The Second Law, Radiative Transfer, and Global Warming
21. The Second Law, Microscopic Reversibility, and Small Systems
22. The Arrow of Time
23. The Heat Death of the Universe
24. Gravity and Entropy
25. The Second Law and Nietzsche's Eternal Recurrence
26. Conclusion

1. Conclusion