Entropy

46 The basic criterion for irreversibility is given by the second principle of thermodynamics through the statement on the limitation of entropy production. This law postulates the existence of two distinct state functions: 0 the absolute temperature and S the entropy with the following properties:

1. 0 is a positive quantity.

2. Entropy is an extensive property, i.e. the total entropy is in a system is the sum of the entropies of its parts.

where ds(e) is the increase due to interaction with the exterior, and ds(i) is the internal increase, and ds(e) > 0 irreversible process (6.50-a)

48 Entropy expresses a variation of energy associated with a variation in the temperature.

6.5.1.1 Statistical Mechanics

49 In statistical mechanics, entropy is related to the probability of the occurrence of that state among all the possible states that could occur. It is found that changes of states are more likely to occur in the direction of greater disorder when a system is left to itself. Thus increased entropy means increased disorder.

50 Hence Boltzman's principle postulates that entropy of a state is proportional to the logarithm of its probability, and for a gas this would give

where S is the total entropy, V is volume, 0 is absolute temperature, k is Boltzman's constant, and C is a constant and N is the number of molecules.

6.5.1.2 Classical Thermodynamics

51 In a reversible process (more about that later), the change in specific entropy s is given by ds =( 0) (6.52)

52 If we consider an ideal gas governed by pv = R0 (6.53)

where R is the gas constant, and assuming that the specific energy u is only a function of temperature 0, then the first principle takes the form du = dq — pdv (6.54)

and for constant volume this gives du = dq = cv d0 (6.55)

wher cv is the specific heat at constant volume. The assumption that u = u(0) implies that cv is a function of 0 only and that du = cv (0)d0 (6.56)

53 Hence we rewrite the first principle as dv dq = cv (0)d0 + R0— (6.57)

v or division by 0 yields rP,v rd

which gives the change in entropy for any reversible process in an ideal gas. In this case, entropy is a state function which returns to its initial value whenever the temperature returns to its initial value that is p and v return to their initial values.

rP,v rd

6.5.2 Clausius-Duhem Inequality

54 We restate the definition of entropy as heat divided by temperature, and write the second principle dtl VpS

y Internal production

Rate of Entropy I

r = 0 for reversible processes, and r > 0 in irreversible ones. The dimension of S = psdV is one

J v of energy divided by temperature or L2MT-20-1, and the SI unit for entropy is Joule/Kelvin.

55 The second principle postulates that the time rate of change of total entropy S in a continuum occupying a volume V is always greater or equal than the sum of the entropy influx through the continuum surface plus the entropy produced internally by body sources.

56 The previous inequality holds for any arbitrary volume, thus after transformation of the surface integral into a volume integral, we obtain the following local version of the Clausius-Duhem inequality which must holds at every point

57 We next seek to express the Clausius-Duhem inequality in terms of the stress tensor,

thus ds 1 „ 1 _ „ pr pdS Vq+qV0 + pr but since 0 is always positive, where — V-q + pr is the heat input into V and appeared in the first principle Eq. 6.47

0 0

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