The second law of thermodynamics can be divided into three parts. The first part gives universal measures of entropy and temperature in terms of quasistatic heat. The second part associates the one-way adiabatic accessibility relation with the proper direction of the entropy scale, and the "hotterthan" relation with the proper direction of the temperature scale. The third part is the principle of increase of entropy for spontaneous transitions in unbalanced coupled systems; this is deduced from the first two parts.
The First Part of the Second Law consists of the following two postulates.
(1) On a quasistatic path in an isentropic set Q = 0.
(2) Let two different environments be given. Choose any thermodynamic system and any pair of isentropic sets in the system. Then the quasistatic heats Q1 and Q2 on isothermal quasistatic paths in thermal equilibrium with the two environments from the first isentropic set to the second are always in the same positive ratio Q1/Q2.
Since Q1/Q2 depends only on the environments, and not on the thermodynamic system chosen, nor on the isentropic sets chosen, we define absolute temperatures T1 and T2 for the two environments by means of the equation
T1/T2 = Q1/Q2,
with a standard value T = 273.16 kelvins at the triple point of water. The kelvin defined in this way is the universal thermal unit of thermodynamics.
Since the standard value of T at the triple point of water is positive, it follows from postulate (2) that: All absolute temperatures are positive.
We can now define an entropy potential S for each isentropic set such that for the two isentropic sets chosen
S2 - S1 = Q1/T1 = Q2/T2.
The first part of the second law and the definitions of absolute temperature and entropy imply that:
Along a quasistatic path we have dQ = TdS.
An outline of the proof is as follows: Consider two points close together on the path. Suppose (without loss of generality) that T is non-decreasing and S is monotonic along the given path from point 1 to point 2. Choose a three-step path from point 1 to point 2 as follows:
(a) An isentropic step with entropy S1 from temperature T1 to temperature T such that T1 < T < T2,
(b) an isothermal step at temperature T from entropy S1 to entropy S2,
(c) an isentropic step with entropy S2 from temperature T to temperature T2.
The temperature T is chosen so that the quasistatic work along the three-step path is the same as the quasistatic work W along the given path. By the first law of thermodynamics, we have along the given path U2 - U1 = W + Q, where Q is the quasistatic heat. Therefore, since U2 - U1 is the same for any path, the quasistatic heat along the three-step path is also Q. By postulate (1) in the first part of the second law, the quasistatic heat is zero in steps (a) and (c). By the definition of entropy, the quasistatic heat is T(S2 - S1) in step (b). Therefore Q = T(S2 - S1). By making point 1 and point 2 closer and closer together we obtain in the limit
dQ = TdS.
Note that in a transition from a given initial state to a given final state the entropy difference S2 - S1 between the states is the same for all possible processes. In particular, along any quasistatic path from the initial state to the final state
S2 - S1 = Integral: dQ/T.
The Second Part of the Second Law consists of the following two postulates.
(3) A state with entropy S2 is one-way adiabatically accessible from a state with entropy S1 if and only if S1 < S2.
(4) A state with absolute temperature T2 is hotter than a state with absolute temperature T1 if and only if T1 < T2.
A corollary of postulate (3) is that: On a quasistatic path along which dQ>0 the initial state is adiabatically inaccessible from the final state.
To prove this, note that along any quasistatic path from the initial state to the final state S2 - S1 = Integral: dQ/T. Since T is always positive, dQ>0 implies that S1 < S2. Therefore, by postulate (3) above, the final state is one-way adiabatically accessible from the initial state. In other words, the initial state is adiabatically inaccessible from the final state.
A corollary of postulate (4) is that: A system with a fixed volume at a higher absolute temperature than its environment loses heat to the environment when placed in thermal contact with it.
To prove this, note that, by postulate (4), the system is initially hotter than the environment. Therefore, by the definition of "hotter than", the initial state is one-way adiabatically accessible from a final state in thermal equilibrium with the environment. It follows from the corollary of postulate (3) just proved (with the initial and final states interchanged) that the system loses heat when placed in thermal contact with the environment.
Imagine two closed thermodynamic systems surrounded by adiabatic walls and coupled by a movable adiabatic wall so that macroscopic changes in volume DV of the two systems satisfy the equation DV1 + DV2 = 0. Each system may be regarded as the environment of the other system. Suppose that initially the pressures are unequal with P1 < P2, and after the movable wall is released it comes to rest with the final pressures equal. It follows from the laws of mechanics that DV1 < 0, and DV2 > 0.
We can imagine quasistatic paths from the initial states to the final states. In these quasistatic paths the total quasistatic work W1 + W2 is negative because P1 < P2, DV1 < 0, and DV2 > 0. But the total macroscopic change in internal energy DU1 + DU2 is zero. Therefore the total quasistatic heat is positive. Since the walls are adiabatic, it follows from the equation dQ = TdS that the sum of the entropies of the two systems increases.
In this example the quasistatic heat is produced by friction, not by heat transfer, because both systems are surrounded by adiabatic walls.
Imagine two systems with fixed volumes in thermal contact with each other through a diathermal wall, but thermally isolated from the general environment by adiabatic walls. Each system may be regarded as the local environment of the other system. Suppose that initially the temperatures are unequal with T1 < T2. It follows from the second part of the second law that system 1 gains heat and system 2 loses heat until the two systems reach thermal equilibrium at the same temperature. Therefore, by the first part of the second law, DS1 > 0, and DS2 < 0.
We can imagine quasistatic paths from the initial states to the final states. On these quasistatic paths the quasistatic work is zero, because the volumes are fixed, and the total change in internal energy DU1 + DU2 is zero; therefore the quasistatic heat gained by system 1 equals the quasistatic heat lost by system 2. But along the path between the initial and final states we have T1 < T2; therefore, by the equation dQ = TdS, we have |DS1| > |DS2|. Since DS1 > 0 and DS2 < 0, it follows that the sum of the entropies of the two systems increases.
The same result can be proved for two systems thermally isolated from the environment and coupled by a movable diathermal wall.
These "thought experiments" prove the following statement, which we call the Third Part of the Second Law:
When two coupled systems separated from the environment by adiabatic walls pass spontaneously from an unbalanced condition to mutual equilibrium, the sum of the entropies of the two systems increases.
The general Principle of Increase of Entropy is based on this law. Spontaneous natural processes involve changes towards mutual equilibrium in unbalanced coupled systems. By the third part of the second law these changes cause the total entropy of the combined systems to increase.
By R. H. B. Exell, 2001. King Mongkut's University of Technology Thonburi.
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