The Concept of Entropy

It is a unique property of heat that it always flows in one direction, viz, from a body at higher temperature to one at lower temperature when they are in contact with each other either directly or through some intermediate conductor. To show this, let us heat a piece of metal, say iron, of mass mi and specific heat ci to a temperature Ti and place it in water of mass m2, specific heat c2 and temperature T2, with T1 > T2. In this case, heat will flow from the metal to the water and soon an equilibrium temperature T will be reached, which is given by the relation v = (E/p)1/2 = (p/p)1/2

In the above process, an amount of heat Q was drawn from the metal at higher temperature T1 and delivered to the water at lower temperature T2 till they were both at the same equilibrium temperature T.

Max Planck divided thermo-dynamical processes, in general, into three distinct categories: natural, unnatural and reversible. The above-mentioned example of heat transfer from a warmer to a cooler body is a natural process. An unnatural process would be one in which heat would flow in the opposite direction, i.e., from a cooler to a warmer body. Experience tells us that the latter never happens. In most naturally-occurring processes, heat flows one-way only, i.e., down the temperature gradient till an equilibrium is reached. Unnatural processes move away with equilibrium and never occur. Sometimes, however, a process can be reversible and equilibrium can be achieved by only a slight change in external conditions. Consider, for example, evaporation from a liquid in contact with its vapour under an external pressure P. Let p be the saturation vapour pressure of the liquid. If P < p, some additional liquid will evaporate as a natural process, while if P > ps evaporation will be an unnatural process. In fact, under latter condition, some vapour will condense. At P = p, the process can go either way. A slight decrease (increase) in P will cause evaporation (condensation). Such a process is clearly reversible.

To provide a quantitative basis to such naturally-occurring processes as mentioned above, classical thermodynamics first introduced a function called entropy which may be defined as

where S is entropy and / §Q is a quantity of heat that is added to, or subtracted from, a working substance at temperature T.

Since the boundary value of S is not known and specified, S is usually expressed as a differential between two thermodynamic states. Its relationship with the other thermodynamic variables, derived from the First law of thermodynamics (3.2.1) and the equation of state (2.4.12), explains why dS should be an exact differential and not a total quantity like §Q to which it is related and which denotes only a change in heat content.

Since the expressions on the right-hand side of (3.5.3) are all exact differentials, dS must be an exact differential. Thus, (1/T) in (3.5.2) is simply an integrating factor and 5Q = T dS.

It follows from (3.5.3) that the change of entropy of a system may be expressed as functions of either T and V or T and p, as stated below:

8Q = TdS = cp dT - V dp = cp dT - T (dV/dT)p dp (3.5.5)

If the constant in (3.5.2) is taken as S0, where S0 represents the zero energy state of the system at temperature T0 of the thermodynamic scale, then S may be regarded as the absolute energy state at temperature T. However, since, in most cases, we are only concerned with the energy input or output between two thermodynamic states, we can get a measure of this energy simply by multiplying dS by T in a cyclic process, as in a T-S diagram shown in Fig. 3.1.

In Fig. 3.1, the entropy (S) is shown along the ordinate and the temperature T along the abscissa. If we visualize that the working substance is taken through a reversible cycle represented by the curve ABCD, then the energy drawn by it from a heat source in moving from P to P is TdS (represented by the hatched area) and in going through the whole cycle is /T dS. The total amount of heat is treated as positive during the stage when entropy increases, i.e., in moving from A to B along the curve and negative when the entropy decreases, i.e., in moving from B to A, if B is taken at the top of the curve and A at the bottom T-S diagrams are routinely used in many meteorological services for estimation of available heat energy in the atmosphere (For further information on these diagrams, see Appendix-3).

It can be shown that in all naturally-occurring thermodynamical processes, entropy increases. Take, for example, the case of the heat transfer from the warmer to the cooler body, discussed earlier (3.5.1). Here, the warmer body loses an entropy, -Q/T1, while the cooler body gains an entropy, +Q/T2, so the net entropy of the system is Q {(1/T2) - (1/T1)}, which is clearly positive. Here, the transfer of heat takes place fast till a steady state is reached. This is an irreversible process. A reversible process is also a naturally-occurring process which passes through a continuous sequence of equilibrium states and takes place slowly so that the entropy remains constant. Thus, in all naturally-occurring processes, dS > 0 (3.5.6)

Fig. 3.1 Representation of energy in a temperature (T) -entropy(S) diagram

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