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7.2 The first law of thermodynamics: an internal energy change due to heat and work
(Hiroshi Matsuoka)
Two types of energy transfer: heat and work
In thermodynamics, we are mainly interested in two types of energy transfer between a
macroscopic system and another: heat and work. Using the law of conservation of energy, we
will define heat as energy transfer due to a temperature difference between two systems. Before
we define heat, we must define an ordering of temperature values so that we can tell which of
two systems have a higher temperature? Historically, the ordering of temperature values was
originally chosen to coincide with our sensation of hot and cold. However, human perception is
not always reliable and we should have a more precise way of determining the ordering of
temperature values. For this goal, we first need to introduce notions of adiabatic wall and
adiabatic work with no net volume change.
Insulating materials and adiabatic walls
Consider two systems whose temperatures are initially different. If these systems are brought
into contact with each other, they will eventually reach their equilibrium states that are in thermal
equilibrium with each other so that their temperatures will take the same value. However,
experiments have also shown that if we initially separate these systems by a wall made of an
“insulating” material, then the original equilibrium states of these systems hardly change even if
they are only separated by the wall. Such insulating walls are also called “adiabatic walls.”
Adiabatic work with no net volume change
Suppose that we cover every surface of a system with an adiabatic wall so that we can isolate
the system from other systems nearby. We can still come up with various ways to do some
mechanical work on the system.
For a system in its solid phase, we can slowly compress or expand the system with an
external pressure slightly different from the pressure inside the system to do some mechanical
work on the system. If the system is a liquid, we can place the liquid in an insulated box with a
paddle inside. By turning the paddle, we can do some mechanical work on the liquid. If it is a
gas, we can put it inside an insulated cylinder fitted with an insulated piston. By pushing or
pulling the piston, we can do some mechanical work on the gas.
Work W done on a system that is covered by adiabatic walls is called “adiabatic work,” and
according to the law of conservation of energy, such work changes the internal energy U of the
system by its amount: !U = W .
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Furthermore, we can have a positive amount of adiabatic work done on a system whose
volume takes the same value before and after the work is done. For example, we can slowly
compress a system covered by adiabatic walls to a smaller volume and then let it expand in a
vacuum back to its original volume. Turning a paddle in a liquid in an insulated box is another
example of adiabatic work with no net change of volume.
Ordering temperature values
In Sec.6.1, we have defined an empirical temperature as a quantity we measure by using a
thermometer. For example, we can use a low-density gas in a sealed container with a fixed
volume as a thermometer so that we can measure the temperature of a system that is in thermal
equilibrium with the gas by measuring the pressure of the gas using a pressure gauge attached to
it. As the volume and the mole number of the gas are both kept constant, the pressure of the gas
is in a one-to-one correspondence with its temperature. However, in Sec.6.1, we did not discuss
how we decide on an ordering of temperature values, or more specifically, which of two
equilibrium states of a thermometer should have a higher temperature.
Suppose that an isolated system initially in an equilibrium state, P ( i) ,V,n , ends up in
(
(f)
(
)
)
another equilibrium state, P ,V,n , after a positive amount (i.e., W adiabatic > 0 ) of adiabatic
work with no net change of volume is done on the system. As mentioned above, applying the
law of conservation of energy, this adiabatic work increases the internal energy U of the system
by !U = W adiabatic > 0 . For any given pair of equilibrium states that share the same values of
volume and mole number, we can then have a positive amount of adiabatic work done on the
system so that the system starts in one of the states with a lower value of the internal energy and
ends up in the other state with a higher value of the internal energy.
If the gas thermometer initially in an equilibrium state, P ( i) ,V,n , ends up in another
(
(
)
(f)
)
equilibrium state, P ,V,n , after a positive amount of adiabatic work (i.e., W adiabatic > 0 ) with
no net change of volume is done on the thermometer, then we assign a higher value of
temperature to the final state:
(
) (
)
T ( i) ! T P ( i) ,V,n < T P ( f ) ,V,n ! T ( f ) .
According to this way of ordering temperature values, a positive amount ( W adiabatic > 0 ) of
adiabatic work with no net change of volume always raises both the internal energy and the
temperature of any system: for T ( i) < T ( f ) ,
(
)
(
)
(
)
U T ( i) ,V,n < U T ( f ) ,V,n = U T ( i) ,V,n + W adiabatic .
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The internal energy of any system is therefore a monotonically increasing function of its
temperature T while its volume and the mole number are kept constant. Therefore, for any
system, we always find
" !U %
$ ' >0
# !T &V ,n
and
" !u %
$ ' > 0,
# !T & v
where U = nu(T,v ) . In fact, we are ordering temperature values so that the internal energy of
any system becomes a monotonically increasing function of its temperature.
Ordering temperature values and the law of conservation of energy
You may wonder if the above way of ordering temperature values works for every system. If
it did not work for some systems, we would end up with a contradiction with the conservation of
energy as follows.
Experiments have shown that after a hotter system and a colder system are brought into
contact with each other, the temperature of the hotter system always decreases while the
temperature of the colder system always increases. Suppose that for one of the systems, say the
hotter one, the internal energy is not a monotonically increasing function of the temperature.
Then after the contact between the systems, the internal energy of the hotter system does not
decrease as its temperature decreases while the internal energy of the colder one increases as its
temperature increases. The total internal energy of the two systems then increases after the
contact, which violates the law of conservation of energy. We would also end up with violating
the law of conservation of energy, if we assume that the internal energy of the colder system is
not a monotonically increasing function of the temperature.
Heat: macroscopic energy transfer induced by a temperature difference
Consider two systems whose temperatures T1( i) and T2( i) are initially different from each other:
one of the systems is hotter than the other, say T1( i) > T2( 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 temperatures take the same final value: T1( f ) = T2( f ) = T ( f ) .
Experiments have shown that the hotter system cools down while the colder system warms
up: T1( i) > T ( f ) > T2( i) . Since the internal energy of any system is a monotonically increasing
function of its temperature, the internal energy of the initially colder system 2 increases after the
systems reach their final equilibrium states:
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(
)
(
)
U 2 T ( f ) ,V2 ,n 2 > U 2 T2( i) ,V2 ,n 2 .
Similarly, the internal energy of the initially hotter system 1 decreases:
(
)
(
)
U1 T ( f ) ,V1,n1 < U1 T1( i) ,V1,n1 .
Applying the law of conservation of energy, we also find
(
)
(
)
(
(
)
(
) { (
)
(
U1 T1( i) ,V1,n1 + U 2 T2( i) ,V2 ,n 2 = U1 T ( f ) ,V1,n1 + U 2 T ( f ) ,V2 ,n 2
)
so that
)
(
)}
!U 2 = U 2 T ( f ) ,V2 ,n 2 " U 2 T2( i) ,V2 ,n 2 = " U1 T ( f ) ,V1,n1 " U1 T1( i) ,V1,n1 = "!U1 .
Therefore, the colder system gains some energy (i.e., !U 2 > 0 ) while the hotter system loses the
same amount of energy (i.e., !U1 = "!U 2 < 0 ). In other words, the amount of energy Q ! "U 2
has been transferred from the hotter system to the colder system. We call this transferred energy
Q the “heat” flowing into the originally colder system or the “heat” flowing out of the originally
hotter system. The energy transfer through heat is thus driven by the temperature difference
between the two systems.
Heat on the microscopic level: energy exchange between atoms
Atoms on the surface of the hotter system are moving faster with more kinetic energy than
the atoms on the opposing surface of the colder system. As the atoms on the surface of the hotter
system collide with the atoms on the surface of the colder system, they, on average, give energy
to the atoms on the surface of the colder system. The atoms on the surface of the colder system,
in turn, give energy to nearby atoms further inside the colder system and thus trigger a process of
heat conduction inside the colder system. Similarly, the atoms on the surface of the hotter
system receive energy from nearby atoms further inside the hotter system and thus trigger a
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process of heat conduction inside the hotter system. Heat conduction is a process where a
system undergoes non-equilibrium states and is at present not completely understood on the
microscopic level.
The first law of thermodynamics: the law of conservation of energy
When some heat Q flows into a system and some work W is done on the system, the internal
energy U of the system changes by
!U = Q + W .
This restatement of the law of conservation of energy involving heat and work is called the first
of law of thermodynamics and becomes a very powerful tool when we analyze thermal processes
such as those in a heat engine.
Note that with our definition for heat and work, when some energy is transferred into a
system through heat or work, the heat or work takes a positive value so that if Q > 0 and/or
W > 0 , then the internal energy of the system increases or !U > 0 .
Quasi-static processes
In thermodynamics, we are interested not only in equilibrium states but also in processes
along each of which a system is always in its equilibrium state. Real processes we observe in
nature and in the lab are not such processes because they involve non-equilibrium states.
However, these idealized processes called “quasi-static processes” are still very useful because
they serve as a kind of limit for real processes.
A quasi-static process is basically a slow succession of equilibrium states: a system is
changing its macroscopic state so slowly that it stays in its equilibrium states throughout the
process.
A nice feature of these processes is that all the state variables, being uniquely
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determined for equilibrium states, have well-defined values throughout these processes so that
we can calculate the amount of energy exchanged between the system and the outside through
heat and work during each quasi-static process.
Infinitesimal quasi-static heat and work
To calculate the heat Qqs flowing into the system and the work W qs done on the system
during a quasi-static process, we divide the process into a set of infinitesimal quasi-static
processes. During each infinitesimal quasi-static process, an infinitesimal amount of heat !Qqs
flows into a system and an infinitesimal amount of mechanical work !W qs is done on the system
so that we can sum up or integrate !Qqs and !W qs to evaluate Qqs and W qs :
Qqs =
# !Q
qs
and
W qs =
i" f
# !W
qs
.
i" f
Below, we will show that !W qs is given by
!W qs = "PdV ,
where P is the pressure inside the system while dV is an infinitesimal change in the volume of
the system. This expression for !W qs is convenient if we monitor the temperature T, the volume
V, and the mole number n of the system along the quasi-static process and measure the pressure
P as a function of T, V, and n so that P = P (T.V .n ) . If we decide instead to monitor the
temperature T, the pressure P, and the mole number n of the system along the quasi-static
process, then we use dV = V!dT " V# T dP in !W qs = "PdV so that
!W qs = "PV#dT + PV$ T dP .
In Ch.11, we will show that !Qqs is given by
!Qqs = TdS ,
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where T is the temperature of the system while dS is an infinitesimal change in the entropy of the
system. Note that both !W qs and !Qqs are expressed as a product of an intensive variable, ! P
or T, and an infinitesimal change in an extensive variable, V or S. In Ch.8, we will also show
that
dS =
CV
!
dT +
dV
T
"T
and
dS =
CP
dT ! V"dP ,
T
where CV and CP are state variables called heat capacity at constant volume and heat capacity at
constant pressure and we will define them below. !Qqs is then given by
!Qqs = CV dT + T
"
dV
#T
and
!Qqs = CP dT " TV#dP .
The first expression for !Qqs is convenient if we monitor the temperature T, the volume V, and
the mole number n of the system along the quasi-static process whereas the second expression is
convenient if we decide instead to monitor the temperature T, the pressure P, and the mole
number n of the system along the quasi-static process.
Infinitesimal quasi-static work
During an infinitesimal quasi-static process, we may slowly perform a small amount of
mechanical work !W qs on a system by squeezing the system with an external pressure that is
slightly higher than the pressure P inside the system. The quasi-static work !W qs is then given
by
!W qs = "PdV ,
where dV is the infinitesimal volume change of the system and is negative because the system is
compressed.
Later when you try to remember this equation, you may not be quite sure what sign, positive
or negative, appears in front of PdV . In that case, you can remind yourself that when the system
is compressed (i.e., dV < 0 ), a positive amount of work must be done on the system (i.e.,
!W qs > 0 ) so that an extra amount of energy is transferred into the system and thereby the
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internal energy of the system increases. Since the pressure P is always positive, you can then
conclude the sign in front of it must be negative.
To derive the above equation, we consider a system, say a gas, inside a cylinder with a piston
as shown below. The small amount of work done on the gas by the external force !F is
!W qs = F dL = ( PA) dL = P ( A dL ) = "PdV ,
where A is the cross-sectional area of the piston and we have used A dL = !dV > 0 (note again
that dV < 0 because we are compressing the gas).
Using this expression for !W qs or
!W qs = "PdV , we can evaluate the work done on a system during a quasi-static process if we
know the equation of state P = P (T,V,n ) of the system as shown in the example below.
Example: Work done on a low-density gas during an isothermal expansion
We can calculate the work done on n mole of low-density gas during an isothermal
expansion at temperature T by the following integral:
Vf
W
qs
T ,n
=
" !W
Vi
qs
=
Vf
Vf
Vi
Vi
" #P (T,V,n)dV = # "
$V '
nRT
V
dV = #nRT [lnV ]V f = #nRT ln& f ) ,
i
V
% Vi (
where Vi and V f are initial and final volumes of the gas and we have used the ideal gas law
P = nRT V as the equation of state for the gas.
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SUMMARY FOR SEC.7.2
1. Work W done on a system that is covered by insulating or adiabatic walls is called “adiabatic
work,” and according to the law of conservation of energy, such work changes the internal
energy U of the system by its amount: !U = W .
2. We define a temperature scale so that after a positive amount of adiabatic work with no net
change of volume is done on a system, its final temperature always has a higher value than
the initial temperature.
3. We define an ordering of temperature values so that the internal energy U of a system is
always a monotonically increasing function of its temperature T while its volume and mole
number are kept constant, which implies
" !U %
$ ' > 0.
# !T &V ,n
4. Heat is an amount of energy transferred between two systems in a thermal contact because of
a temperature difference between the two: it flows from a hotter to a colder system.
5. The first law of thermodynamics is the law of conservation of energy, according to which
when some heat Q flows into a system and some work W is done on the system, the internal
energy U of the system changes by
!U = Q + W .
When some energy is transferred into a system through heat or work, the heat or work takes a
positive value so that if Q > 0 and/or W > 0 , then the internal energy of the system increases or
!U > 0 .
6. A quasi-static process is a slow succession of equilibrium states and serves as an idealized
limit for a real process. Along a quasi-static process, the system is always in an equilibrium
state so that the state variables of the system are well defined throughout the process.
7. An infinitesimal amount of work !W qs done on a system during an infinitesimal quasi-static
process is proportional to the volume change dV of the system by
!W qs = "PdV ,
where dV is negative when the system is compressed.