Download 1 11.8 Definition of entropy and the modern statement of the second

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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:
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! ( 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 ,
(
)
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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 .
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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
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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.
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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
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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.”
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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
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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.