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Entropy - an extensive parameter. In Study Unit 2, we discussed how the state of a system can be represented as points on both the thermal and the mechanical planes of representation. Both of these planes have an intensive and an extensive axis. For example, the mechanical plane of representation for a gas has axes indicating its pressure (the intensive parameter) and its volume (the extensive parameter). In this Study Unit we are concentrating on the thermal plane of representation. If we identify the thermal intensive parameter with temperature, we are left with the problem of what is the corresponding extensive parameter. It is given the name entropy; and the entropy of a system, being an extensive parameter, is the sum of the entropies of its parts.

For any transfer of energy to or from a system in conditions near equilibrium, we can use the following expressions:

dZ or LZ is the amount of energy transferred. Y is the intensive parameter and X is the corresponding extensive parameter. For example, in magnetizing a paramagnetic specimen the work done is:

H is the field intensity and I the magnetization. I is the initial and I the final magnetization. During the process, H changes in magnitude.

In the case of a transfer of energy by heat to a system, we would expect a similar expression to those describing the transfer of energy by work. In fact we write:

T is the absolute temperature and S the entropy where Q is the quantity of heat involved. When T remains constant during the transfer of energy we have the special case:

Q = TS

Transfers of energy involving changes in entropy are transfers of thermal energy.

The significance of thermal equilibrium. Equations [3.1] and [3.2] are valid only for conditions of equilibrium. This is where we have a difference between ordinary mechanics and thermodynamics. The requirement of equilibrium can be qualitatively explained by an illustrative case. If a gas is being compressed "violently" then there will be localized variations in pressure, especially near the moving surface, so that the actual work done cannot be expressed in the form W = -PdV, where the P is supposed to be the pressure of the system during compression. In fact, W > -PdV. On the other hand, if we have a sudden transfer of thermal energy to a system then Q < TdS. The reason for this will emerge later, but there we can say that any deviation from equilibrium conditions means additional production of entropy in the system. So we should write equation [3.2] in a more general way:

Q TdS [3.3]

We drop the sign in front of Q because Q is obviously a finite quantity related to the change in energy of the system. When we use the form [3.2] we are assuming a quasi-static process (a process at all times "near equilibrium", a "slow" process, etc.). After this brief formal exercise, we now look at some situations where entropy comes in as a useful concept.

The separation of different adiabatics.
An adiabatic can be defined as "a continuum of states accessible to one another by work alone". In terms of fig. 3.1.- which represents three adiabatics for a gas-any point on A can be reached from any other point on A by doing work on, or taking work out of, the system; but no point on B or C can be reached from any point on A by changes of energy due to work

on or by the system. What is it that separates A, B and C from each other? It is that each represents a different magnitude of the entropy of the system. In other words, each adiabatic has a constant entropy- work transfer does not change the entropy of a system.

On the thermal plane, the adiabatics appear as straight lines corresponding to different entropies (see fig. 3.2).

Quantity of heat converted into work in a reversible cycle.
If a system goes through a cycle, such as that shown in fig. 3.3, returning periodically to its original state and in such a way that the whole process could be exactly reversed (the importance of this proviso will be discussed later), then the effective result is that a certain amount of energy taken in as heat is given out as useful work. The corresponding representation on the thermal plane is shown in fig. 3.4.

Now the amount of work produced is represented by the shaded area on the mechanical plane:

W = PdV

where means integration over the cycle. The amount of thermal energy which has entered by heat and been used up in the cycle is represented by the shaded area on the thermal plane:

Q = TdS

By the First Law: Q = U + W
U is the change in the internal energy of the system over a whole cycle and since we are saying that it returns in a cycle to its original state:

U = 0 and then Q = W
T d S = P d V




What happens to the energy used up in a change of phase?
When a liquid evaporates, there is no change in its temperature; yet energy is needed to bring about the transition of the mass of material involved from the liquid to the vapour phase (latent heat of vaporization). The heat transfer is not increasing the temperature of the system, but it is increasing its entropy. We write:

Q = T S or S = Q / T

where S is the finite increase in entropy at the constant temperature T. Entropy is measured in units of energy degrees , such as calories per degree centigrade. It is significant that a change of phase, such as from liquid to vapour, involves a breakdown in the internal cohesion of the system. The solid phase has its own rigid structure which gives way to the fluid properties of the liquid phase, and this in its turn gives way to the "randomness" of the vapour phase, in which the system can maintain no definite volume by itself. The transitions from solid to liquid, and from liquid to vapour, are marked by increases in entropy. So we can associate increase of entropy with disordering.

The change involved in heat transfer.
It is an irreducible feature of experience that, whenever no work is involved, heat always flows from the hotter to the cooler and never the reverse.

We imagine a system 1 transferring energy by heat to system 2, as shown in fig. 3.5. Taken as a whole, what is changing? The sum of the energies of the two systems remains the same, and we can suppose the changes in temperature resulting from the heat transfer to be negligible. But when we calculate the total change in entropy S we find that it has increased.

S = - ( Q / T1 ) + ( Q / T 2 )
but ( Q / T1 ) < ( Q / T2 )
S > 0

For all "natural processes" S > 0, and for perfectly reversible cycles S = 0.

There is a general principle here, which can be expressed as follows: in all processes not involving work the total entropy of the systems involved must always increase. This is one expression of the Second Law of Thermodynamics, which we now go on to consider in more detail.

Heat engines and heat pumps.
The basic operations involved in a heat engine are three: (i) taking in energy by heat from a source (Q), (ii) losing a lesser amount of energy by heat to a sink (Q), (iii) performing mechanical work (W). The cycle shown in Jigs. 3.3 and 3.4 can be treated as a representation of the operations of a heat engine. This is a complementary type of apparatus to the heat engine: this is the heat pump, which is a heat engine operating in reverse. The two kinds of apparatus are shown in figs. 3.6 and 3.7.

The heat engine gets work out of heat; the heat pump works to get heat out of a cooler region into a hotter region. The point of introducing these apparatuses is this: we cannot know from heat transfer alone what could happen if work were also involved in the process; in particular, we do not know whether the entropy might decrease.

By an indirect argument, we can in fact establish that the entropy can never decrease, and also that S = 0 expresses the ideal limit of performance for a heat engine in terms of the quantity of work that can be obtained from a given input of heat working between two absolute temperatures. The trick is to consider a heat engine and a heat pump coupled together so that the work produced by the engine drives the pump. In that way, the work is zero for the total system of engine + pump and all we look at are heat transfers. We take it that the W in fig. 3.6 and fig. 3.7 is the same in both cases.

We have Q - Q'leaving the source at temperature T, and Q - Q'entering the sink at temperature T. According to what we know of heat transfer, Q - Q' > 0 and Q - Q' > 0: that is, thermal energy flows from the hotter to the cooler. Then by the conservation of energy we also have Q - Q' = Q - Q'. What can we say of the entropy changes?

Change of entropy of source = - ( (Q1 - Q'1) / T 1)

Change of entropy of sink = + ( ( Q2 - Q'2 ) / T2 )
= ( ( Q1 - Q'1 ) / T2 )

The total change of entropy is
S = (Q1 - Q'1) . ( (1/T2) - (1/T1) )

Since T > T then S 0; for if the cycles involved are perfectly reversible, Q = Q' and S = 0. S can only he negative if we have, effectively, heat flowing from a cooler to a hotter body.

More detailed argument than is appropriate here goes on to prove rigorously that, if either the engine or the pump were working so that for one of them S > 0, then the net result would be that heat would flow from a cooler to a hotter body without any work being involved. So we form the general law (another expression of the Second Law of Thermodynamics): S 0
For all "natural processes" S > 0 and for perfectly reversible cycles S = 0.

The heat engine produces work by the "transport" of entropy.


If a charged particle falls through a potential drop it acquires kinetic energy. Mechanical energy is in general acquired by the transport of something, such as a charge or a mass, across a difference of potential. Fig. 3.8 shows this in terms of intensive (related to potential) and extensive (such as charge and mass) quantities:

mechanical energy acquired = YX = (Y1 - Y2) X
In the case of the ideal heat engine (perfectly reversible) the work is produced by the transport of entropy across the thermal gap between source and sink:

W = T.S = (T1 - T2 ) S

We can derive this relation from one which we have already worked out for a perfectly reversible cycle:

W = TdS

There are only two changes in the energy which we have to consider-that at the source and that at the sink. The change in entropy in both cases is the same and S is the extensive quantity transported across the temperature gap.

The fundamental equation of thermodynamics. The First Law of Thermodynamics can now be rewritten to produce an expression known as the fundamental equation of thermodynamics:

dQ = dU + dW (First Law)
becomes TdS = dU + PdV

This fundamental equation involves the Second Law of Thermodynamics. It has one very important qualification. The equation is only valid for processes which are reversible. In ordinary situations, we call these quasi-static processes because they approximate at all stages to conditions of thermal equilibrium. One of the most important reasons for this qualification can be stated very simply: if entropy is a parameter of state of a system, then any changes in it must still correspond to conditions of equilibrium. The two conditions of reversibility and equilibrium coincide, though in practice any actual change or cycle can only approximate at all stages to equilibrium conditions.

Now since we have the general law:
S 0

it follows that the entropy of a system will increase until it reaches a maximum value (when dS = 0). There must be a change of entropy if the system is not in equilibrium, and this can only be positive. Thus conditions of equilibrium for a system correspond to a maximum entropy for that system.