KEY POINTS FOR CHAPTER NINE
10.1 SPONTANEOUS PROCESSES
Some things happen, some things don’t: (i) Niagara Falls falls down, but not up. (ii) When left on the kitchen table, a hot cup of coffee cools to room temperature, but a glass of water at room tem- perature doesn’t heat up to 80°C. (iii) A gas under pressure expands to fill a vacuum, but a gas doesn’t spontaneously compress and move to one side of the container. (iv) Salt and water mix to form a salt solution, but a salt solution doesn’t separate into salt and pure water.
All changes can be classified as either spontaneous or non-spontaneous. A spontaneous process is one that has a natural tendency to occur; this kind of process can happen all by itself, and, in theory, is potentially capable of being harnessed to deliver work to the surroundings. A non-
spontaneous process is one that has no natural tendency to occur; this kind of process requires
some kind of work input from outside the system in order to proceed.
Some chemical reactions are spontaneous, e.g., the neutralization of an acid with a base to produce a salt plus water; others are not, e.g., H2O H2 + 1
2 O2. A reaction mixture at chemical equi- librium has no tendency to move spontaneously in either direction––no net process takes place. Note that when we say a process is a “spontaneous” process, we are referring to its tendency to occur; this doesn’t necessarily tell us that it will occur, or, if it does, at what rate it will occur. For example, H2 + 1
2O2 H2O is a spontaneous process, but if you mix some hydrogen gas together
with some oxygen gas you don’t suddenly get a tankful of water––all you get is a gaseous mixture of H2 and O2. The actual reaction rate is so slow that in practice it is zero. In order to cause this process to occur at a useful rate we need a catalyst or a source of ignition, such as a spark.
Why are some processes spontaneous and others not? Is it the tendency to move towards lower energy that decides? Consider a piece of hot copper metal and a piece of cold copper metal, each of the same mass, in contact with each other in an insulated container (Fig. 1). The hot copper spontaneously cools; but, the cold copper spontaneously heats––the cold copper seems to spontaneously move to a state of higher energy, not lower energy! Hmm.... so much for the lower energy hypothesis.
200°C 0°C hot Cu cold Cu
The tendency for a system to move towards a state of lower energy is one factor that helps to de- termine whether or not a process will occur spontaneously, but another tendency also is important:
The tendency for energy and matter to become disordered.
Ordered matter tends to become disordered matter; e.g., dissolution of a crystalline salt in water, diffusion from a region of high concentration to a region of low concentration. Concentrated energy tends to become dispersed energy; e.g., the thermal energy in a cup of hot coffee tends to dissipate to the surroundings.
Returning to the two blocks of copper: If you consider the atomic vibrations in them, it is apparent that there is a natural tendency for the concentrated vibrational kinetic energy of the hot copper at- oms to become evenly dispersed between the two blocks (Fig. 2).
Work (the movement of a force through a displacement) has direction, and therefore is an ordered type of energy flow. Heat, on the other hand, is non-directional (it disperses in all directions), and thus is a disordered type of energy flow. Because we know that ordered energy tends towards dis- ordered energy, it is not surprising that work can be converted to heat very easily, but that heat is not so readily converted to work.
10.2 ENTROPY (S)
Entropy, designated by the symbol S, is a measure of the disorder of matter and energy. But we know from experience that disorder tends to increase;1 therefore, we can generalize this observa-
tion by stating that the entropy of the universe tends to increase; this is a statement of The Second Law of Thermodynamics. A more accurate statement would be
The entropy of the universe
never decreases: Suniv 0 The Second Lawof Thermodynamics
The above statement of the Second Law is more useful for our purposes if we make use of the following mathematical formulation, which quantitatively defines entropy by its differential:
dS Q T
rev Entropy
Defined . . . [1]
where Qrev is the reversible heat for the process and T is the temperature at which the heat transfer 1 After you finish tidying up your room, does it “spontaneously” become neater or messier?
((( ))) ((( ))) ( ) ( ) concentrated weak vibrational KE vibrational KE 200°C 0°C (( )) (( )) (( )) (( )) dispersed dispersed vibrational KE vibrational KE 100°C 100°C Fig. 2
takes place.
We mentioned earlier in Chapter 6 that a reversible process is one whose direction can be reversed by an infinitesimal change in one of its thermodynamic parameters. For example, to obtain the reversible (i.e., the maximum) work from an isothermally expanding gas, we saw that the external pressure against which the gas does work continuously has to be adjusted so that it is just infini- tesimally lower than the pressure of the gas itself. If the gas pressure is 10000 bar and the external pressure is only one bar, when the pin restraining the piston is removed, will the expansion take place fast or slowly? Extremely fast, of course. The greater the P between the gas and the surroundings, the faster the piston will shoot up.
So how fast do you think the piston will rise if the difference in pressure between the gas and the external pressure is only infinitesimally small, say P = 0.0000001 bar? Obviously very, very slowly. In fact, if the pressure difference is infinitely small, as is required for a truly reversible expansion, then the piston will rise infinitely slowly.
One of the characteristics of a reversible process is that it takes place extremely slowly; the greater the rate at which a process occurs, the more irreversible the process is. How do these ideas apply to heat transfer? Well, heat transfer is driven by a difference in temperature T between the system and the surroundings. The greater the T, the faster the heat transfer. So, in order to have reversi-
ble heat transfer, it is required that the heat must flow very, very slowly, which means that T
must be small, small, small. In the limit, for truly reversible heat transfer, T 0, which means
that Tsyst Tsurr.
When heat transfers very slowly, local “hot spots” are prevented from forming; hot spots add to the irreversibility because hot spots later disperse spontaneously, adding to the entropy increase. When the heat is the reversible heat Qrev, the entropy change dS is a state function.2 It should be
noted that the definition of entropy involves Qrev/T, not just Qrev itself. It turns out that Qrev alone is not a state function.2
The total change in the entropy for the process is obtained by integrating the expression for dS:
S = Q T
rev . . . [2]
If the temperature is kept constant during the process, then the 1/T can be taken outside the integral sign to give S = Q T rev = 1 T Qrev i.e., S = Q T
rev Constant Temperature
Process . . . [3]