Download 6. Macroscopic equilibrium states and state variables (Hiroshi

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6. Macroscopic equilibrium states and state variables
(Hiroshi Matsuoka)
Experiments have shown that for a macroscopic single-component system in a certain phase,
for example, in its gaseous phase, each equilibrium state of the system is uniquely specified by
the following three independent variables: the temperature T, the pressure P, and the mole
number n of the system. With a few exceptions such as work and heat, most physical quantities
in thermodynamics are functions of these variables. For example, we have found that the
volume V of the system is a function of these variables so that V = V (T,P,n ) , which we call the
equation of state of the system. In this chapter, we will carefully define equilibrium states and
introduce temperature as a quantity that classifies these states. We will also divide all the state
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variables, which are quantities uniquely specified for each equilibrium state, into two distinct
groups, extensive and intensive variables.
6.1 Equilibrium states and temperature
The operational definition of temperature
In Sec.4.1, we have adopted the following operational definition for temperature: temperature
is a macroscopic quantity that we measure using a thermometer. Below we will explain what we
mean by this definition. As the notion of temperature is derived from a notion of “equilibrium
state,” we must first carefully define the equilibrium state.
Equilibrium states of an isolated system with a constant uniform pressure
Experiments have shown that when a single-component macroscopic system is isolated for a
while so that the system cannot exchange energy with its outside (it is in fact practically
impossible to ensure this perfect blocking of energy exchange with the outside, but for now let us
assume that by some means we can reduce the amount of energy exchanged between the system
and the outside below a macroscopically detectable level), the pressure inside the system reaches
a constant value everywhere and remains at this value thereafter. This unchanging uniform state
on the macroscopic level is called “an equilibrium state.” As the pressure is uniform everywhere
inside the system, the system is in static mechanical equilibrium where there is no macroscopic
flow of substance and if we insert an imaginary piece of plane anywhere inside the system, the
forces exerted by the substance on the both sides of the plane would be balanced.
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Three independent variables, P, V, and n, to specify an equilibrium state of an isolated
single-component system
Each equilibrium state of an isolated single-component system is then specified by three
independent variables: the pressure P, the volume V, and the mole number n of the system,
(P,V,n ) .
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Thermal equilibrium between two equilibrium states
Of course, equilibrium states of an isolated system are not really interesting and in fact
useless for engineering applications where we are mostly interested in systems that exchange
energy with another. Fortunately, we can extend the notion of equilibrium state to systems that
exchange energy with others.
Suppose we have two separate isolated systems in their equilibrium states. We specify these
(
)
(
)
initial states by their pressures, volumes and mole numbers: P1( i) ,V1,n1 and P2( i ) ,V 2 ,n2 . If we
bring these systems in contact with each other while keeping their volumes and mole numbers
constant and isolating these systems as a whole from the outside, their pressures may still
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change. For example, if one of them feels hotter than the other, this will happen. We call this
type of mutual contact, where the volumes and the mole numbers of the systems in contact are
kept constant, “thermal” contact.
Experiments have shown that the pressures of these systems eventually stop changing on the
macroscopic level and settle down to constant uniform values, P1( f ) and P2( f ) . We then say that
these systems are in their equilibrium states that are in “thermal equilibrium” with each other.
Suppose we have two isolated systems in their equilibrium states. To test whether these states
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are in thermal equilibrium with each other, we only need to bring these systems into “thermal”
contact with each other and see if their pressures will change. If their pressures remain at their
initial values measured before the contact, then we know these equilibrium states are in thermal
equilibrium with each other.
Note also that after two systems reach their equilibrium states through a thermal contact, they
will remain in these equilibrium states while they maintain the contact with each other. This is a
typical situation we have in the lab, where a system of interest is in thermal equilibrium with the
air in the lab. In this text, we will be mostly focused on these equilibrium states and their
thermal properties.
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The zero-th law of thermodynamics and temperature that classifies equilibrium states
“Being in thermal equilibrium with each other” is a type of relation between two equilibrium
states of two systems. For the two equilibrium states discussed in the previous subsection,
(P ( ) ,V ,n ) and (P ( ) ,V ,n ) , we can denote this relation by
f
1
f
1
1
2
2
2
(P ( ) ,V ,n ) " (P ( ) ,V ,n ) .
f
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1
f
1
1
2
2
2
Experiments have shown that this relation of “being in thermal equilibrium with each other”
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possesses the following property: if equilibrium states of two systems A and B are in thermal
equilibrium with each other and the equilibrium state of system B is in thermal equilibrium with
an equilibrium state of a third system C, then this equilibrium state of system C is also in thermal
equilibrium with that of system A. This experimental finding is called “the zero-th law of
thermodynamics,” which we can state symbolically as
If ( PA ,VA ,n A ) " ( PB ,VB ,n B ) and ( PB ,VB ,n B ) " ( PC ,VC ,nC ) , then ( PA ,VA ,n A ) " ( PC ,VC ,nC ) .
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The usual equality sign, “=,” surely satisfies this property. As another example of a relation that
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satisfies this property, we can consider a relation of “person A is at the same age as person B.”
Clearly, if John is “at the same age as” Julie and Julie is “at the same age as” Joe”, then John is
also “at the same age as” Joe.
Because of this law, we can classify all the equilibrium states of all macroscopic isolated
systems into separate groups, in each of which all member equilibrium states are in thermal
equilibrium with each other. We should then be able to label each of these groups by a number,
which we will call “temperature.” As all the equilibrium states that are in thermal equilibrium
with each other should have the same value of temperature, the relation of “being in thermal
equilibrium with each other” is equivalent to a relation of “having the same value of
temperature.” Similarly, using the relation of “being at the same age as,” we can classify people
into different age groups, in each of which all the members are at the same age. For the age
groups, the age plays the role played by temperature for equilibrium states.
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The reason why we find an equation of state for a macroscopic system
A value of temperature is then uniquely assigned for all the equilibrium states belonging to a
particular group and is therefore a function of these equilibrium states.
Moreover, for an
equilibrium state that is specified by ( P,V,n ) for a particular isolated system, the temperature
value T assigned to this equilibrium state must be a function of ( P,V,n ) :
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T = T ( P,V ,n ) ,
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which is nothing but the equation of state of this system as it can be also expressed as
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P = P (T,V,n ) . This is why we can find an equation of state for any macroscopic system.
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Thermometers and the Celsius temperature scale
A thermometer is then simply a macroscopic system that can be brought into contact with
another system whose temperature we wish to measure by establishing thermal equilibrium
between the equilibrium states of these two systems. As the temperature is a function of the
equilibrium state of the thermometer specified by its pressure, volume, and mole number or
(P,V,n ) , we can use, for example, the pressure inside the thermometer, while keeping its volume
and mole number constant, to measure the corresponding temperature.
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More specifically, we can use a pressure gauge attached to a gas in a sealed container so that
we can measure the pressure of the gas while its volume and mole number are kept constant. We
then measure the pressure of the gas when the gas is in thermal equilibrium with a piece of water
ice under the atmospheric pressure. We mark this value of pressure with a sign “0 degree
Celsius.” We can also measure the pressure of the gas when the gas is in thermal equilibrium
with boiling water under the atmospheric pressure. We mark this value of pressure with a sign
“100 degree Celsius.” For our convenience, we then divide the pressure range between these
two marks into a hundred notches, each of which corresponds to 1 degree in the Celsius
temperature scale. As mentioned in Sec.4.1, the absolute temperature scale is directly related to
the Celsius scale so that we can measure temperature in the absolute scale using this
thermometer.
We have thus introduced temperature as a quantity we measure using a
thermometer. The temperature measured by a particular thermometer is called “empirical
temperature” and each empirical temperature can be expressed in a somewhat arbitrarily chosen
scale such as the Celsius scale. We will precisely define the absolute temperature scale, the
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proper temperature scale in thermodynamics, in Ch.11, where we will also carefully define
“entropy,” a truly thermodynamic quantity.
The gas thermometer
Can we measure temperature directly in the absolute scale using a low-density gas as a
thermometer? As the ideal gas law is a good approximation for the equation of state of a lowdensity gas around room temperature, we can find its temperature in the absolute scale by
measuring the pressure P, the volume V, and the mole number n of the gas:
T=
PV
nR
Historically, the temperature scale defined by this equation was originally named “the ideal gas
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temperature scale,” which was later found to coincide with the absolute scale.
Extending the concept of equilibrium state to include mechanical equilibrium
In the above, we first defined the notion of equilibrium state for “isolated” systems, which
cannot exchange energy with another system and whose volumes and mole numbers are both
kept constant. We then extended the notion to the systems which can exchange energy with
another system through a thermal contact where the volumes and the mole numbers of the
systems in contact are still kept constant.
We now extend the concept of equilibrium state further to include a case, where a system can
not only exchange energy with another system through a mutual contact but also vary its volume
because the system is also in “mechanical” contact with the second system. When a system is in
mechanical contact with another system, these systems can exert forces on each other on the
macroscopic level because of a difference between their internal pressures and may vary their
volumes even on the macroscopic level. For example, if we leave a piece of solid in the lab, the
solid piece and the air in the lab are in mechanical contact with each other, and if we wait long
enough, they will reach mechanical or static equilibrium by adjusting their volumes so that the
pressure exerted by the solid piece on the air becomes equal to the air pressure in the lab.
In general, in order for two systems to be in mechanical equilibrium with each other, their
pressure must be equal.
When two systems 1 and 2 are in both thermal and mechanical
equilibrium with each other, then the temperatures and pressures of these systems must be equal:
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T1 = T2
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and
P1 = P2 .
(the conditions for thermal and mechanical equilibrium)
When a system is in thermal and mechanical equilibrium with the outside, we simply state that
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the system is in its equilibrium state.
How do we know if a system is in its equilibrium state?
To see if a system is in its equilibrium state, we check the following two conditions on its
temperature T, pressure P, and mole number n on the macroscopic level:
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T, P, and n are constant in time.
•
T and P are spatially uniform.
(when a system is subject to an external (e.g., electric, magnetic, or gravitational) field, T
and P may not be spatially uniform. In this text we will focus on systems that are not
subject to an external field or for which the effects of an external field can be neglected.
For example, we can neglect the effect of the earth’s gravitational field on the pressure of
a gas in a container whose height h is less than 10 m since the pressure difference
between the top and the bottom parts of the gas is at most
"P = #gh ~ (1 kg m3 )(10 m s2 )(10 m) = 100 Pa << 10 5 Pa ,
where " is the density of the gas.)
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The first condition ensures that there is no net macroscopic exchange of energy (through heat or
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work) between the system and its outside, while the second condition guarantees that when we
divide the system into a set of subsystems that are each macroscopic in size, these subsystems
are all in thermal and mechanical equilibrium with each other.
Non-equilibrium states
Systems found in nature or used in engineering applications are mostly not in their
equilibrium states. They are typically in their non-equilibrium states, where the temperature T,
the pressure P, and the mole number n of each of these systems are not constant in time and T
and P are not spatially uniform (i.e., T (r,t ) , P (r,t ) , n ( t ) ). A prime example of a system always
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in its non-equilibrium state is a real heat engine. Of course, in the lab, we can come up with a
way to maintain some systems approximately in their equilibrium states during an observation
time. As non-equilibrium states are very difficult to study and still not completely understood,
we will confine ourselves to equilibrium states in this text.
6.2 State variables
Physical quantities that are well defined for each equilibrium state of a macroscopic system
are called “state variables.” State variables that we have discussed so far include temperature T,
pressure P, volume V, mole number n, molar volume v, compressibility factor Z, coefficient of
thermal expansion " , and isothermal compressibility " T . State variables that we will encounter
later in this text include heat capacity at constant pressure CP , heat capacity at constant volume
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Helmholtz free energy F, Gibbs free energy G, and
CV , internal
! energy U, entropy S, enthalpy H,
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their corresponding molar variables (e.g., molar heat capacity at constant pressure c P , molar
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internal energy u, etc.) including chemical potential µ , which coincides with molar Gibbs free
energy for a single-component system.
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Quantities that are not state variables: heat and work
There are two important quantities in thermal physics that are not state variables: heat Q
flowing into a system and work W done on a system. Both the heat and the work depend on a
process that the system undergoes: they are process-dependent quantities. Even when the system
starts with the same initial equilibrium state and ends with the same final equilibrium state, if
processes connecting these two states are different, then the amount of heat and work involved in
each process would be different. Both heat and work are therefore functions of a process so that
their values are assigned to a series of macroscopic states sampled along the process rather than
specified for a single equilibrium state.
Extensive and intensive state variables
There are two types of state variables: extensive and intensive variables. Extensive variables
are those that scale with the size of a system while the temperature and pressure of the system
remain unchanged. For example, the volume and the mole number of a system are extensive.
On the other hand, variables such as temperature and pressure are not extensive. Instead, they
are “intensive” and do not scale with the size of the system.
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Extensive variables
What do we exactly mean by “scale with the system size?” Imagine that we have two
identical systems with the same temperature and pressure. If we put together these two systems,
then both the total volume and the total mole number would be doubled while the temperature
and pressure inside the combined system remain unchanged.
The key condition on this
combining procedure is that we combine these two systems in order that their original
equilibrium states are preserved. More generally, the volume as a function of T, P, and n must
satisfy
V (T,P, "n ) = "V (T,P,n ) ,
where " is a positive real number. Setting " to be 2, we get the above example of combining
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two identical systems. In general, if we multiply all the extensive variables among the
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independent variables by a factor of " , any other extensive variable would be multiplied by the
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same factor " .
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Intensive variables
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What about intensive variables? Intensive variables are invariant or unchanged with respect
to a change in system size. For example, even after we put together two identical systems with
the same temperature and pressure, the temperature and the pressure of the combined system
would be the same as the temperature and the pressure of the original two systems. More
generally, the pressure as a function of T, V, and n must satisfy
P (T, "V, "n ) = P (T,V,n ) .
In general, if we multiply all the extensive variables by a factor of " , then any intensive variable
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would remain unchanged.
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The molar variable of an extensive variable is intensive
A ratio between an extensive variable A and the mole number n of a system is intensive
because if we multiply both A and n by a factor of " , then we find
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"A A
= .
"n n
We call this ratio A n the “molar” variable a corresponding to A:
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a"
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A
,
n
from which we also find
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A = na.
For example, the molar volume v " V n is intensive and V = nv .
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Classifying state variables
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We can now classify all the state variables mentioned above into two categories:
Extensive variables: V, n, CP , CV , U, S, H, F, G
Intensive variables:
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T, P, Z, " , " T , v, c P " CP n, c v " CV n , u " U n s " S n ,
h " H n, f " F n , µ " G n
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An intensive variable is a function of two intensive variables
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The condition that an intensive variable must remain unchanged when all the extensive
variables are multiplied by a factor of " puts a restriction on the functional form of the intensive
variable. For example, by setting " = 1 n in P (T, "V, "n ) = P (T,V,n ) , we get
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P (T,v = V /n,1) = P (T,V,n ) ,
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which implies that pressure must be a function of two intensive variables, the temperature T and
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the molar volume v:
P = P (T,v ) ,
as we mentioned before. The ideal gas law does satisfy this equation:
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"n%
RT
P = $ 'RT =
.
#V &
v
Generally, an intensive variable must be a function of intensive variables only. In a single!
component system, every intensive variable is therefore a function of two intensive variables.
An extensive variable expressed as a product of the mole number and its molar variable
The extensive variable A can be now expressed as a product of the mole number n and the
molar variable a, which must be a function of two intensive variables:
A = na(2 intensive variables) ,
which also implies that if we find a state variable to be proportional to the mole number of the
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system, then this state variable must be an extensive variable. For example, the volume can be
expressed in terms of the molar volume v as
V = nv (T,P ) .
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SUMMARY FOR CH.6
1. A macroscopic isolated system that does not exchange energy with its outside settles down to
an unchanging state called an equilibrium state after being left alone for some time. Each
equilibrium state an isolated system is uniquely specified by its pressure P, volume V, and
mole number n.
2. When two systems are in mutual contact with each other while their volumes and mole
numbers are kept constant, they will reach unchanging states or equilibrium states that are in
thermal equilibrium with each other. The mutual contact between the two systems whose
volumes and mole numbers are kept constant is called “thermal contact.”
3. The relation of “being in thermal equilibrium with” satisfies the zero-th law of
thermodynamics: if an equilibrium state ( PA ,VA ,n A ) of system A “is in thermal equilibrium
with” an equilibrium state ( PB ,VB ,n B ) of system B while the equilibrium state ( PB ,VB ,n B ) of
system B “is in thermal equilibrium with” an equilibrium state ( PC ,VC ,nC ) of system C, then
the equilibrium state ( PA ,VA ,n
!A ) of system A “is in thermal equilibrium with” the
equilibrium state!( PC ,VC ,nC ) of system C.
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If ( PA ,VA ,n A ) " ( PB ,VB ,n B ) and ( PB ,VB ,n B ) " ( PC ,VC ,nC ) , then ( PA ,VA ,n A ) " ( PC ,VC ,nC ) .
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4. The zero-th law implies that we can classify all the equilibrium states of all isolated systems
into distinct groups, in each of which the equilibrium states in the group are all in thermal
equilibrium with each other. We can then label each group by a number called
“temperature.” As the temperature is uniquely assigned for all the equilibrium states in
group, for a particular system, the temperature becomes a function of its equilibrium state so
that T = T ( P,V ,n ) , which is nothing but the equation of state for the system.
5. A thermometer is a system that can be brought into thermal contact with another system
whose temperature we wish to measure by establishing thermal equilibrium between the
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equilibrium states of these two systems.
6. We can further extend the notion of equilibrium state by allowing two systems to be in both
thermal and “mechanical” contact with each other so that the systems can exert forces on
each other and vary their volumes. When two systems are in both thermal and mechanical
equilibrium, the temperatures and pressures of these systems must become equal.
7. Physical quantities that are defined for each equilibrium state of a system is called “state
variables.” An amount of heat flowing into a system and an amount of work done on a
system are both process-dependent quantities and are therefore not state variables.
8. There are two types of state variables: extensive and intensive variables.
9.
When we combine two identical systems at the same temperature and pressure, an extensive
variable gets doubled while an intensive variable remains unchanged. The volume and the
mole number of a system are extensive, whereas its temperature and pressure are intensive.
10. The molar variable of an extensive variable is intensive. For example, the molar volume
v " V n is intensive.
11. An intensive variable is a function of two intensive variables. For example, P = P (T,v ) .
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12. An extensive variable can be expressed as a product of mole number and its corresponding
molar variable that is a function of two intensive variables: A = na(2 intensive variables) .
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For example, V = nv (T,P ) .
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