ENTROPY AND THE SECOND LAW
5.1 ENTROPY
5
HOT COLD WORK
FIGURE 5.1 An Engine.
thermodynamics, which, like the first law, can be stated in many ways. It follows that S=b
a dS=b
a dq/T is the entropy change of a system carried reversibly over an arbitrary path from a to b and is independent of the path. This powerful definition constitutes the second of the two great pillars of thermodynamics. If we can devise a way of calculatingS for a reversible chemical reaction, we shall know it for all chemical reactions having the same initial and final states (reactants and products) because of path independence.
Clausius expanded upon the concept of entropy by writing the complete statement as
dS≥ dq T
which takes both reversible and irreversible changes into account. The irreversible change dSirr>dqirr/T is the real case, a change that takes place in finite time.
If we attempt to take an engine around an irreversible cycle to reproduce its initial state, we shall fall short. We have received a certain amount of work from the engine, but when it comes to the payback (in heat) we see the following with regard to the second law:
dS≥ dq
T implies that dqirr<TdSirr
The system will not be returned to its original state, violating the principal stipulation that the system operate around a cyclic path. We shall have to take some heat from the hot reservoir in Fig. 5.1 to complete the cycle and bring the entropy back to its initial value. Where does the extra heat over and above the reversible heat eventually end up? It can go only one place. Since it hasn’t done any work, it must have passed through the engine and gone directly to the low-temperature reservoir. The efficiency, work out relative to heat in, of a real engine operating irreversibly is less than 1.0 because some heat is doing work and some is not. The important concept is that, of the heat taken from the hot reservoir, not all of it can do work. Some heat must pass through the engine from the hot reservoir directly to the cold reservoir doing nothing
ENTROPY 73 but restoring the system entropy to its original state. Because of this necessity, some of the heat drawn from the hot reservoir in an irreversible cycle is unavailable to do work.
5.1.1 Heat Death and Time’s Arrow
Because there is always heat transferred to the surroundings in an irreversible cycle, the entropy of the system plus the surroundings always increases. If we take the universe as the surroundings, then since all real processes are irreversible we have a consequence of the second law: The entropy of the universe tends to a maximum.
When the universe has reached its maximum entropy, no more irreversible change will be possible. The driving force of change will be gone. This is called the heat death of the universe. In case you are worried, it is calculated to be in the far distant future.
The entropy of the universe must be greater after an irreversible change has occurred than it was before the change, so we have a thermodynamic definition of the direction of time (which the first law doesn’t give). Time must go from before the change to after the change; it cannot go in the reverse direction.1 Entropy is sometimes called “time’s arrow.”
5.1.2 The Reaction Coordinate
Prigogine has defined a reaction coordinateξwhich progresses as a chemical reaction takes place. Starting with pure reactant A, the reaction coordinate increases as product B is produced:
A→B
At some point, the time derivative ofξbecomes zero and the reaction stops insofar as macroscopic concentration measurements are concerned.2When the time derivative of the reaction coordinate is zero, the system consisting of nA+nBis at equilibrium.
Because there are no macroscopic concentration changes, the ratio of the mole num- bers of reactant and product nB/nAis constant. It is called the equilibrium constant Keq:
∂ξ
∂t
T,p
=0 Keq=nB/nA
It would be appealing to think that the reaction coordinate is determined solely by the energy U or enthalpy H of the system flowing from a high level to a low level,
1Classical thermodynamics does not include QED.
2There are microscopic exchanges between species A and B, but they are equal and opposite on average so they do not bring about measurable concentration changes.
but this is not the case. There is more to think about in a chemical or physical change than just minimization of U or H. There is the question of order and disorder of the reactant state and the product state.
5.1.3 Disorder
When we look at a chemical reaction
A→B or an analogous physical change
A(l) → A(g)
we must look at the driving force that produces the change. Part of that force comes from the tendency to seek a minimum (water flows downhill), but part of it comes from the universal tendency of thermodynamic systems to seek maximum disorder.
A familiar example is vaporization of a liquid such as water.
The liquid state, though not perfectly ordered, is held together by strong inter- molecular forces. These are the very forces that we say are nonexistent or negligible in the vapor state. The entropy change for many liquids is about 88 J K−1mol−1, a rule known as Trouton’s rule, that has been verified many times over for liquids as diverse as liquid Cl2, HCl, chloroform, and the n-alkanes. Liquids that deviate from this rule do so, not because of any failure of the entropy concept for vaporization, but because of abnormal forces in the liquid state. An example is water, which deviates a little due to hydrogen bonding, and hydrogen fluoride HF, which deviates a lot.