In the special case of a substance isolated from external heat supply, dH=o, the change of entropy is zero in a reversible process, but must be positive if the process is not reversible.
Writing formulae (3r) and (33) for the energy and entropy with indeterminate constants A and B, instead of taking them between limits, we obtain the following expressions for the thermodynamic functions in the case of the vapour: " =Solog e 0 - R log e p - ncp/D+A".
If we write h=sot+dh, where so is a selected constant value of the specific heat of the liquid, and dh represents the difference of the actual value of h at t from the ideal value sot, and if we similarly write q5 = sologe(6/90)+dcp for the entropy of the liquid at t, where do represents the corresponding difference in the entropy (which is easily calculated from a table of values of h), it is shown by Callendar (Proc. R.S.
The adiabatics are lines of constant entropy, and are also called Isentropics.
In the case of an ideal gas, dp/d9 at constant volume =R/v, and dvld6 at constant pressure =R/p; thus we obtain the expressions for the change of entropy 0-4)0 from the state poeovo to the state pev, log e e/eo+R logev/vo =S log e 9/00-R (32) In the case of an imperfect gas or vapour, the above expressions are frequently employed, but a more accurate result may be obtained by employing equation (17) with the value of the specific heat, S, from (29), which gives the expression 4-¢o = Sologe0/00 - R logep/po-n(cp/B-copo/Bo)
(33) The state of a substance may be defined by means of the temperature and entropy as co-ordinates, instead of employing the pressure and volume as in the indicator diagram.
Changes of entropy must be calculated in terms of quantities of heat, and must be interpreted in a similar manner.
In practice, however, there is always some frictional dissipation, accompanied by an increase of entropy and by a fall of pressure.
The entropy tends to a maximum, and the state is one of stable equilibrium when the value of the entropy is the maximum value consistent with the conditions of the problem.
Since the condition of heat-isolation is impracticable, the condition of maximum entropy cannot, as a rule, be directly applied, and it is necessary to find a more convenient method of expression.
If 0', E', v'; and 4)", E", v", refer to unit mass of the substance in the first and second states respectively in equilibrium at a temperature 0 and pressure p, the heat absorbed, L, per unit mass in a change from the first to the second state is, by definition of the entropy, equal to 0(4)"-4)'), and this by the first law is equal to the change of intrinsic energy, E" - E', plus the external work done, p(v" - v'), i.e.
0, Entropy of vapour or liquid.
In passing along an adiabatic there is no change of entropy, since no heat is absorbed.
(29) (30) The expression for the change of entropy between any two states is found by dividing either of the expressions for dH in (8) by 0 and integrating between the given limits, since dH/B is a perfect differential.
The 0, 4) diagram is useful in the study of heat waste and condensation, but from other points of view the utility of the conception of entropy as a " factor of heat " is limited by the fact that it does not correspond to any directly measurable physical property, but is merely a mathematical function arising from the form of the definition of absolute temperature.
In all such cases there is necessarily, by Carnot's principle, a loss of efficiency or available energy, accompanied by an increase of entropy, which serves as a convenient measure or criterion of the loss.
If the change is not reversible, but the final state is the same, the change of entropy, do, is the same, but it is no longer equal to dII/B.
The entropy cannot diminish.
Any change involving decrease of entropy is impossible.
The total entropy of the system is found by multiplying the entropy per unit mass of the substance in each state by the mass existing in that state, and adding the products so obtained.
This quotient is called the change of entropy, and may be denoted by (4,"-0').
In virtue of relations (2), the change of entropy of a substance between any two states depends only on the initial and final states, and may be reckoned along any reversible path, not necessarily isothermal, by dividing each small increment of heat, dH, by the temperature, 0, at which it is acquired, and taking the sum or integral of the quotients, dH/o, so obtained.
In any small reversible change in which the substance absorbs heat, dH, from external sources, the increase of entropy, d0, must be equal to dH/9.