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1 11.8 Definition of entropy and the modern statement of the second law (Hiroshi Matsuoka) In this section, we will define both entropy and the absolute temperature scale and then derive the modern statement of the second law of thermodynamics from the classical statement of the second law by Kelvin. 11.8.1 Empirical temperatures In Sec.6.1, we have defined temperature using the zero-th law of thermodynamics, but we did not precisely define the absolute temperature scale. Temperature scales such as the Celsius scale introduced in Sec.6.1 are called “empirical” temperature scales, which are not the proper temperature scale in thermodynamics. For example, thermodynamic relations such as ! Qq s = TdS are meaningful only when temperature T is measured on the absolute temperature scale. One of the main goals of this section is to define the absolute temperature scale and then qs define entropy through the relation dS = !Q T . The temperature measured by a particular thermometer is an “empirical temperature” ! and each empirical temperature can be expressed on a somewhat arbitrarily chosen scale. We will reserve the symbol T for the absolute temperature measured on the absolute temperature scale. For example, when a mercury thermometer is in thermal equilibrium with a mixture of solid ice and liquid water, we can mark the height of its mercury column to indicate “0 degree Celsius,” and when it is in thermal equilibrium with a boiling water, we can mark the height of the mercury column to indicate “100 degree Celsius.” For our convenience, we then arbitrarily divide the length of these two marks into a hundred notches, each of which corresponds to 1 degree in this Celsius temperature scale. The ideal gas thermometer and its scale We can also use n mole of low-density gas as a thermometer by keeping either its pressure P or volume V constant. The empirical temperature ! ideal measured by this “ideal gas” thermometer is defined by ! ideal " PV , nR where R is the universal gas constant, so that the ideal gas law always holds with this empirical temperature, which is measured in units of Kelvin (K) and is related to the temperature in the Celsius scale by the following equation: 2 ! ( K) = ! (°C) + 273.15 . We can show that the ideal gas temperature scale actually coincides with the absolute temperature scale (for more about this, see the appendix at the end of Sec.11.8.4). Ordering empirical temperature values Up to this point, the definition of a temperature scale is rather arbitrary, but in thermodynamics the distinction between a higher temperature and a lower temperature is important as the Clausius statement of the second law is directly focused on this distinction (i.e., “heat flows from a hot to a cold object”). In Sec.7.2, we have introduced an ordering for temperature values on an empirical temperature scale by requiring that the internal energy of any system becomes a monotonically increasing function of the empirical temperature ! if the volume and the mole number of the system are kept constant so that # !U & % ( >0. $ !" ' V ,n 11.8.2 Heat and the first law of thermodynamics Before we discuss the second law of thermodynamics, we now review how we have introduced the notion of heat and the first law of thermodynamics. Heat: energy transfer induced by a temperature difference Consider two systems whose empirical temperatures !1( i) and ! (2i) are initially different: one of the systems is hotter than the other, say !1( i) < ! 2(i) . If these systems are brought into contact with each other while their volumes and mole numbers are kept constant, they will eventually reach their equilibrium states that are in thermal equilibrium with each other so that their ( f) temperatures take the same final value ! . From experiments, we know that the hotter system cools down while the colder system warms up so that !1( i) < ! ( f ) < ! (2i ) . Since the internal energy is a monotonically increasing function of the empirical temperature, the internal energy of the initially colder system increases: ( ) U1 !1(i ),V1,n1 < U1 ! ( f ) ,V1,n1 , ( ) 3 while the internal energy of the initially hotter system decreases: ( ) U2 !2(i ),V 2 ,n2 > U2 ! ( f ),V 2 ,n2 . ( ) Applying the law of conservation of energy, we find ( ) ( U1 !1(i ),V1,n1 + U2 ! (2i) ,V2 , n2 = U1 ! ( f ) ,V1,n1 + U2 ! ( f ),V 2 ,n2 ( ) ( ) ) or ( ) ) { ( ) !U1 = U1 " ( f ) ,V1 ,n1 # U1 "1(i) ,V1,n1 = # U2 " ( f ) ,V2 ,n2 # U2 " 2(i ),V 2 ,n2 ( ( )} = #!U . 2 Therefore, the originally colder system has gained some energy (i.e., !U1 > 0 ) while the originally hotter system has lost the same amount of energy (i.e., !U2 = " !U1 < 0 ). In other words, the energy !U1 has been transferred from the hotter system to the colder system. This energy transfer is driven by the temperature difference between the two systems and is distinct from the energy transfer through work due to a volume change. To distinguish this new energy transfer process from work, we define the heat Q flowing from the originally hotter system to the originally colder system by Q ! "U1 = #"U2 . The first law of thermodynamics Now that we have defined heat as energy transfer induced by a temperature difference, we can enlarge the scope of the law of conservation of energy by including heat as another way of changing the internal energy U of a system: !U = Q + W , where !U is a change in the internal energy due to heat Q flowing into the system and work W done on the system. This is the first law of thermodynamics. 11.8.3 The classical statements of the second law of thermodynamics The classical statements of the second law and irreversible processes Each of the classical statements of the second law such as the ones by Clausius and Kelvin selects a particular adiabatic process and declares it to be irreversible. The main message of the 4 second law is therefore that in nature as well as in the lab, there exist adiabatic irreversible processes. Be sure to keep in mind that the second law is concerned with a particular set of processes, adiabatic processes. This restriction is not so limiting as it seems because we can make almost any process adiabatic by placing all the systems involved in the process inside a “box” with adiabatic walls so that there is no heat transfer between these systems and the outside. Notable exceptions are large-scale geological or astronomical systems such as the earth’s environments, the solar system, and any part of the universe, for which electromagnetic radiations always allow heat transfer between theses systems and the “outside” of these systems. The point is that when we claim a certain process to be an adiabatic irreversible process, we should be able to clearly define a region in space that contains the systems involved in this process and make sure that there is no heat transfer between the inside of this region and its outside. Reversibility versus retraceability What are adiabatic irreversible processes? To answer this question, it is best to define adiabatic reversible processes because adiabatic irreversible processes are adiabatic processes that are not reversible. An adiabatic reversible process in a macroscopic system is an adiabatic process for which we can construct another adiabatic process that restores the initial equilibrium states for the system and all other systems involved in the original process. In this definition of reversible processes, the adiabatic process that is to restore the initial state of the system does not have to be a process that retraces the states which the system has undergone in the original process. For example, consider a low-density gas that undergoes a quasi-static adiabatic expansion inside an insulated cylinder with an insulated piston. We let the volume of the gas increase quasi-statically by slowly pulling the piston. We can, of course, restore the initial equilibrium state through the reverse process of this quasi-static adiabatic expansion. But we can also restore the initial state by a different quasi-static process that does not retrace the original expansion as follows. First, we insert a partition inside the cylinder to divide it into two chambers. This partition is fastened to a fixed position while it allows for heat transfer between the chambers so that the gases in the two chambers will be in thermal equilibrium with each other and therefore will have the same temperature. By slowly pushing the piston to compress the gas in one of the chambers, we can then adjust the total volume of the two chambers to match the initial volume of the entire gas. We then slowly move the partition to bring the pressures of the gases in the chambers to a common value so that the gases in the two chambers are at the same temperature and at the same pressure. Finally, we remove the partition, which does not change the equilibrium states of the gases and restores the initial state of the entire gas. 5 Reversibility therefore appears to be a weaker condition than retraceability so that reversible processes do not have to be retraceable whereas retraceable processes are always reversible as with quasi-static processes that are retraceable because we can always construct their reverse processes. However, all adiabatic reversible processes are in fact indistinguishable from quasistatic adiabatic processes as far as thermodynamics is concerned, because if we have an adiabatic reversible process between two equilibrium states of a macroscopic system, according to the modern statement of the second law that we will show at the end of this chapter, there exists a quasi-static adiabatic process that connects these equilibrium states. Being adiabatic (i.e., with no heat transfer between the system and the outside), in both the original adiabatic reversible process and the corresponding quasi-static adiabatic process, the same amount of work must be done on the system since according to the first law, we have !U = W and the internal energy difference !U between the two states depends only on these states and not on how these processes proceed. Thermodynamics is only concerned with how much heat and work is accepted by a system during a process, and therefore the original adiabatic reversible process and the corresponding quasi-static adiabatic process are “equivalent” within thermodynamics. Thus, we use the terms, “adiabatic reversible process” and “quasi-static adiabatic process,” interchangeably. The irreversible free expansion of a low-density gas As an example of an adiabatic irreversible process, let us consider a free expansion of a lowdensity gas. In this adiabatic process where the gas initially confined in a smaller insulated chamber expands into a larger insulated chamber. After the gas fills the larger chamber and is settled down to an equilibrium state, we find the temperature of the system is the same as its initial temperature (see Sce.11.4 for more detail). We can easily restore the initial equilibrium state for the gas through a quasi-static isothermal compression of the gas, in which the gas gives some heat Q to another system A and receives some work W from a third system B. For a quasistatic isothermal process in a low-density gas, we find Q = W. To keep this entire process to be adiabatic, we need to place the gas and the auxiliary systems, A and B, inside an insulated box with adiabatic walls. We must also make sure that both the auxiliary systems, A and B, return to their initial equilibrium states, which would be possible if we could draw heat Q out of the system A and fully convert it into work W done on the system A. As we will discuss below, according to Kelvin’s statement of the second law, such a full conversion of heat into work is not possible. Alternatively, we can also attempt to restore the initial state for the gas by a quasi-static adiabatic compression of the gas with an insulated piston so that there is no heat transfer during the compression. According to the first law, the internal energy of the gas will then increase because of the work done by the piston so that the temperature of the gas becomes higher than its 6 initial value. To restore the initial state of the gas without causing any net change in the equilibrium states of the systems involved in this compression, we must make sure: (i) to decrease the temperature of the gas back to its original value; (ii) to move the piston back to its original position. Suppose that we could come up with an adiabatic process to achieve these. We can then let the gas expand its volume back to that of the larger chamber through a quasistatic isothermal expansion, where the gas does some work on the outside while it absorbs some heat, which implies that we have managed to fully convert heat into work while the gas returns to the equilibrium state in the larger chamber. As we will discuss below, according to Kelvin’s statement of the second law, such a full conversion of heat into work is not possible, which implies, in turn, that the free expansion of the low-density gas is irreversible. Clausius’ statement of the second law of thermodynamics Clausius’ statement of the second law summarizes what we have found about heat transfer: “heat flows spontaneously only from a hot to a cold system if these systems are insulated from their environments.” In other words, heat conduction between two systems initially at different temperatures is an irreversible process. More precisely, Clausius’ statement asserts: “it is impossible to extract some energy from a heat reservoir by allowing only heat transfer between the reservoir and its outside and to deposit the same amount of energy to a hotter heat reservoir by allowing only heat transfer between the reservoir and its outside with no net change of state in all the systems (except for the reservoirs) involved in this energy transfer process.” A full conversion of work into heat and Kelvin’s statement of the second law Although both heat and work are energy transfer processes, they are different in an essential way. For example, we can fully convert work into heat flowing into a heat reservoir by having a system do some mechanical work to run an electrical generator (e.g., your hand turning a coil inside the generator) to have an electrical current flow in a wire attached to the heat reservoir so that some heat will flow from the wire to the system. One important point is that after this process both the generator and the wire will return to their original states while the system from which the mechanical work is extracted and the heat reservoir that receives heat from the wire will change their states. In contrast to the conversion of work into heat, we cannot fully convert heat into work so that a part of the original heat must turn into heat flowing into another heat reservoir. In other words, a full conversion of work into heat is an irreversible process. Kelvin’s statement of the second law claims exactly that: “it is impossible to extract some energy from a heat reservoir by allowing only heat transfer between the reservoir and its outside and to convert that energy completely into work done on some system with no net change of state in all the systems (except for the reservoir and the system receiving the work) involved in this energy transfer process.” 7 Clausius’ statement and Kelvin’s statement are equivalent to each other We can show that Clausius’ statement is equivalent to Kelvin’s statement as follows: 1. From Clausius’ statement to Kelvin’s statement: assume that Kelvin’s statement is false so that we can fully convert heat from a heat reservoir into work, which we then fully convert into heat into another heat reservoir whose temperature is higher than the first one. We have thus shown the falsity of Clausius’ statement. See the figure below on the left. 2. From Kelvin’s statement to Clausius’ statement: assume that Clausius’ statement is false so that we can extract some energy as heat from a heat reservoir and deposit it as heat to another heat reservoir whether the first reservoir is hotter than the second one or not. Suppose that we extract heat Q from a heat reservoir and convert it into work W done on some system plus heat Q! into another heat reservoir so that Q = W + Q! . As we assume that Clausius’ statement is false, we can then extract heat Q! from the second reservoir and deposit it to the first reservoir. The net effect is that heat Q ! Q" from the first reservoir is fully converted into the work W, which implies the falsity of Kelvin’s statement. See the figure below on the right. Thot Thot Q Q Q! W W Q! Q Tcold Tcold 8 SUMMARY FOR SEC.11.8.1 THROUGH SEC.11.8.3 1. We define an empirical temperature scale ! so that the internal energy of any system becomes a monotonically increasing function of the empirical temperature if the volume and the mole number of the system are kept constant: # !U & % ( >0. $ !" ' V ,n 2. Each of the classical statements of the second law such as the ones by Clausius and Kelvin selects a particular adiabatic process and declares it to be irreversible. The main message of the second law is therefore that in nature as well as in the lab, there exist adiabatic irreversible processes. 3. An adiabatic reversible process in a macroscopic system is an adiabatic process for which we can construct another adiabatic process that restores the initial equilibrium states for the system and all other systems involved in the original process. 4. An adiabatic reversible process between two equilibrium states and a quasi-static adiabatic process between the same states are “equivalent” within thermodynamics. We thus use the terms, “adiabatic reversible process” and “quasi-static adiabatic process,” interchangeably. 5. Clausius’ statement asserts: “it is impossible to extract some energy from a heat reservoir by allowing only heat transfer between the reservoir and its outside and to deposit the same amount of energy to a hotter heat reservoir by allowing only heat transfer between the reservoir and its outside with no net change of state in all the systems (except for the reservoirs) involved in this energy transfer process.” 6. Kelvin’s statement claims: “it is impossible to extract some energy from a heat reservoir by allowing only heat transfer between the reservoir and its outside and to convert that energy completely into work done on some system with no net change of state in all the systems (except for the reservoir and the system receiving the work) involved in this energy transfer process.” 7. Clausius’ statement of the second law and Kelvin’s statement of the second law are equivalent to each other.