CHAPTER 4

GENERAL RELATIONSHIPS BETWEEN STATE VARIABLES

OF HOMGENEOUS SUBSTANCES

INTRODUCTION

The first law of thermodynamics relates the differential change in internal energy of a system.Ito the differential amount of energy crossing the system boundary in the form of heat and work according to the

equation

.I œ U� [$ $ (4.1)

The internal energy is a variable (for an ideal gas it is related to the temperature of the system),I state only

whereas and depend upon the specific . We have discovered that for processes, the$ $U [ process reversible

process-dependent heat and work can both be written in terms of . During quasi-staticchanges in state variables

(reversible) processes, we can write

$ $U œ X .W [ œ T .Z</@ </@ and (4.2)

This allows us to write the first law solely in terms of state variablesfor reversible processes

.I œ X .W � T .Z (4.3)

The last equation is usually refered to as equation, and serves as the startingthe combined first and second law

point for discussing any problem in thermodynamics.

The ability to write the first law in terms only of state variables allows us to calculate many useful

relationships between the thermodynamic properties of a system. This is because the first law, written in terms of

quasi-static processes, is actually applicable , since each of the variables in this equation are in general state

variables. This means that . Thisrelationships which we can derive from this equation hold true for processany

is because the system will be in some unique state some arbitrary process, and at the end of that process, itbefore

will be in some unique state. The actual process the system undergoes may not be quasi-static, but the again end

points of the process are on the surface of the -T@X =C=>/7 and that's all that matters!

To illustrate this last point, consider the combined first and second law, where we consider and asZ Xindependent variables. The changes in internal energy, then, can be expressed in terms of and by theZ Xequation

$

$

U œ .I � T .Z œ .Z � .X � T .Z`I `I

`Z `X

U œ � T .Z � .X`I `I

`Z `X

(4.4)Œ � Œ �” • Œ �Œ �

X Z

X Z

Now, if the volume were to remain constant during some process, the first term on the right-hand side is zero, and

we are left with

$U œ .X`I

`XZ Z

ZŒ � (4.5)

Now the heat capacity at constant volume is to bedefined

G ´ œU `I

.X `XZ

Z ZŒ � Œ �$

(4.6)

This relationship is valid, for constant volume processes, ; i.e., we can always replace thenot only but in general

quantity / with no matter what kind of process is being considered. This is because the internal� �`I `X GZ Z

energy is a function of the of the system, not the I state process.

Chapter 4: General Relationships between State Variables 2

In the special case where we are dealing with an ideal gas the internal energy of this gas is solely aßfunction of the temperature. This last equation, then, becomes

G œ 8 - œ.I

.XZ Z (4.7)

which means that we can calculate the change in the internal energy of an ideal gas using the equation

? ?I œ 8 - .X œ 8 - XEF EFE

F

@ @( (4.8)

since the heat capacity for the ideal gas is a constant. But the can be process, and is process any notE p Frestricted to a constant volume process!

Note: Although this equation is independent, the of does depend upon the process value kind-@of ideal gas we have (monatomic, diatomic, etc.). For polyatomic gases each type of motion

(vibration or rotation) has a “turn-on” temperature arising from quantum considerations. This

last equation, then, is valid over the whole range of temperatures only for monatomic ideal

gases. For polyatomic ideal gases it is valid only over limited temperature regions where can-@be considered constant.

THERMODYNAMIC POTENTIALS AND THE MAXWELL RELATIONSHIPS

The combined first and second law

$U œ X.W œ .I � T .Z (4.9)

provide some very useful relationships between the various system parameters of pure a substance. This equation

contains system properties (state variables), but, as pointed out in the first chapter, only of these arefive two

required to completely specify the macroscopic state of the system. This means that we can choose two of theany

system parameters (state variables) as independent, and the other three must be functions of these two parametersÞThis will allow us to derive a large number of relationships between the various system parameters. Some of

these relationships will prove to be particularly important in the development of statistical thermodynamics. In

addition, we can form of some of these system parameters to define system parameters whichcombinations new

are often quite useful.

In the material that follows we will be developing relationships between different system parameters.

These will be organized by examining how the change in a particular varies with otherThermodynamic potential

parameters of the system. The first of these is what we have already defined as the internal energy ofpotentials

the system.

Changes in The Internal Energy

Beginning with our statement of the combined first and second law, the change in the internal energy of a

system can be expressed directly in terms of a change in the entropy of the system and the change in the volume

of that system as

.I œ X .W � T .Z (4.10)

This equation is very useful when we are dealing with a constant volume process. In that case we can equate the

cnange in the internal energy directly to the change in entropy of the system.

Since the right-hand side of this last equation is written in terms of a change in entropy and a change in

volume, we will choose these two system parameters are our independent variables and write

.I œ .W � .Z œ X .W � T .Z`I `I

`W `ZŒ � Œ �

Z W

(4.11)

Chapter 4: General Relationships between State Variables 3

from which we can immediately see that

Œ �`I

`Wœ X

Z

(4.12)

and

Œ �`I

`Zœ � T

W

(4.13)

These last two relationships are significant because they give us new insight into the meaning of the

temperature and the pressure of a gas. We see that the of a gas is just a measure of the rate at which thepressure

internal energy of an system (constant entropy implies adiabatic processes) changes with a change inisolated

volume. Obviously the internal energy decreases as the volume increases. A high pressure system is one in which

the internal energy decreases very rapidly with a volume change, whereas a low pressure system is one in which

the internal energy varies slowly with volume.

Likewise, the is a measure of the rate at which the internal energy changes with attemperature entropy

constant volume. We can also express this equation for the temperature, using the reciprocity theorem, in the

form

Œ �`W "

`I Xœ

Z

(4.14)

This equation demonstrates that the reciprocal of the temperature is a measure of the rate at which the entropy of a

system changes with respect to internal energy when the volume is held constant. This particular equation will

turn out to be very significant in our study of statistical thermodynamics. We will see later on, in our study of

statistical thermodynamics, that it is fairly easy to define the entropy of a system relative to the internal energy

based upon a statistical model of the thermodynamic system. The definition of temperature, however, is not quite

so straight forward. This last relationship will allow us to define the temperature in a precise way when we work

with our statistical models.

The partial derivatives of the internal energy which we found in Equ. 4.12 and 4.13 can be utilized to

find yet another relationship between the system parameters by remembering that the second partials of exact

differentials must be the same, no matter what the order of differentiation. Thus

Œ � Œ � Œ � Œ �` `I ` `I

`Z `W `W `Zœ

W Z Z W

(4.15)

which gives

Œ � Œ �`X `T

`Z `Wœ �

W Z

(4.16)

This last equation is one of the so-called “Maxwell relations.”

Changes in the Enthalpy

We indicated that the combined first law in the form

.I œ X .W � T .Z (4.17)

turned out to give a very simple relationship between and for the case of constant volume processes..I .WHowever, this form of the equation is not as useful when we want to consider internal energy changes for constant

pressure processes, or if we simply want to calculate the change in internal energy for processes in which the

pressure is varying. We can, however, express this last equation in terms of a change in pressure as well an

entropy. The differential of the quantity can be written asTZ

. TZ œ T .Z � Z .T Ê T .Z œ . TZ � Z .T� � � � (4.18)

Chapter 4: General Relationships between State Variables 4

enabling us to rewrite the combined first and second law in the form

.I œ X .W � . TZ � Z .Tc d� � (4.19)

Rearranging this last equation, we have

. I � TZ œ X .W � Z .T� � (4.20)

We now the combination of system parameters to be a system parameter which we call thedefine new I � TZ Lenthalpy. The mentioned earlier, enthalpy is one of a group of combinations of thermodynamic parameters called

thermodynamic potentials (note that they all have the dimensions of energy). These thermodynamic potentials are

often useful in special situations or for special processes. For example, writing the combined first and second law

in terms of the enthalpy

.L œ X .W � Z .T (4.21)

we see that the change in enthralpy for a process (as is often the case in the laboratory) is justconstant pressure

the heat ( ) added to, or taken from, the system. So just as for constant volume processes,$ $U œ X.W U œ .I$U œ .L for constant pressure processes. Similarly, for a constant pressure process, we have

.L œ X.W œ UT T T$

from which we can see that

G ´ œU `L

.X `XT

T TŒ � Œ �$

In most engineering applications, changes in enthalpy are considered, rather than changes in internal energy,

because the enthalpy is related to the heat added to a system at constant pressure, whereas the internal energy is

related to the heat added to a system at constant volume - and working in the laboratory, we typically work under

constant pressure conditions.

Obviously the enthalpy of a system is a state variable since , , and are state variables, so we canI T Zwrite

.L œ .W � .T œ X .W � Z .T`L `L

`W `TŒ � Œ �

T W

(4.22)

which allows us to identify the partial derivatives of the enthalpy in terms of measurable system parameters,

giving

Œ �`L

`Wœ X

T

(4.23)

Œ �`L

`Tœ Z

W

(4.24)

Again, we can use the fact that the second partial of is the same, no matter what the order of differentiation,Land obtain another Maxwell relation

Œ � Œ �`X `Z

`T `Wœ

W T

(4.25)

Changes in the Helmholtz Free Energy

If we wish to write the combined first law

.I œ X .W � T .Z (4.26)

in a form which is more suitable to changes in temperature and volume than changes in entropy and volume, we

can write

.I œ . XW � W .X � T .Z� � (4.27)

Chapter 4: General Relationships between State Variables 5

or

. I � XW œ �W .X � T .Z� � (4.28)

We define a new thermodynamic potential called the Helmholtz free energy to giveJ œ I � XW

.J œ �W .X � T .Z (4.29)

from which we obtain

Œ � Œ �`J `J

`X `Z.X � .Z œ � W .X � T .Z

Z X

(4.30)

and the relationships

Œ �`J

`Xœ �W

Z

(4.31)

Œ �`J

`Zœ �T

X

(4.32)

Again you should note that is particularly useful when working at constant temperature, in which case.J

.J œ � T .ZX (4.33)

which is the amount of external work done on the system.

We can again use the fact that the Helmholtz free energy is an exact differential to obtain the Maxwell

relationship

Œ � Œ �`W `T

`Z `Xœ

X Z

(4.34)

Changes in the Gibbs Free Energy

In a similar manner we define the Gibbs free energy as

K œ I � XW � TZ œ L � XW (4.35)

which gives

.K œ � W.X � Z .T

from which we obtain the relationships

Œ �`K

`Xœ �W

T

(4.36)

Œ �`K

`Tœ Z

X

(4.37)

and the final Maxwell relation

� œ`W `Z

`T `XŒ � Œ �

X T

(4.38)

The Gibbs free energy is often useful when we are dealing with systems with chemically active

constituents. A differential change in the Gibbs free energy can be written as

.K œ .I � X.W � W.X � T.Z � Z .T œ .I � ÐX.W � T.Z Ñ � W.X � Z .T (4.39)

For systems with chemically active constituents, the combined first and second law is given by

.I œ X.W � T.Z � .R. (4.40)

Chapter 4: General Relationships between State Variables 6

where is the chemical potential. The Gibbs free energy can, therefore, be expressed by the equation.

.K œ .R � W .X � Z .T. (4.41)

Thus, for constant pressure and temperature processes, the chemical potential is just equal to the change in the

Gibbs free energy with respect to the change in the number of particles in the system!

The Potential Memory Box (or Hints for Remembering Various Useful Relationships)

The relationships between the system parameters , , and , and the Thermodynamic Potentials ,Z X T W IJ K Lß, , and as well as the Maxwell relations turn out to be very useful relationships between the various system

parameters. One way of remembering these relationships is illustrated in the use of the “Memory Box” shown

below. This “Memory Box” can be remembered with the acronmy "Very Tall Physics Students Eat For Good

Health" Þ

Z X

W T

J Ð� Ñ

I K

L Ð� Ñ

The bold symbols , represent the Thermodynamic potentials, and are ordered alphabeticallyIß J ß Kß Lclockwise around our square “Memory Box”. The more conventional system parameters are positioned on the

four corners of the box. To determine an expression for the temperature , for example, you take the partial ofXthe potential with respect to the variable which is diagonally across from theon the opposite side of the box

temperature, holding the variable constant. For example:on the opposite side of the potential function

X œ œ`L `I

`W `WŒ � Œ �

T Z

The sign on the right indicates that the partials from which you obtain the temperature and the volume are

multiplied by a plus sign, whereas the partials from which you obtain the pressure and entropy require a minus

sign:

T œ � œ �`J `I

`Z `ZŒ � Œ �

X W

Remembering that the order of differentiation is unimportant when taking the second partials, we can use

these relationships to quickly derive the Maxwell equations. For example, we have just shown that

Œ �`I

`Zœ � T

W

and

Œ �`I

`Wœ X

Z

Taking the partial of the first of these equations with respect to (holding constant) and the partial of theW Zsecond of these equations with respect to (holding constant) we obtain the Maxwell relationshipZ W

� œ`T `X

`W `ZŒ � Œ �

Z W

Now, this last equation can also be obtained rather quickly simply by using our “Memory Box”. Notice

that the system parameters , , and in the partial derivative on the left side of this equation are all on T W Z adjacent

corners of our “Memory Box”, . If you goin the same order as indicated in that partial derivative, /� �`T `W Z

one more step around the box , you come to the system parameter , and if you now reversein the same direction X

Chapter 4: General Relationships between State Variables 7

directions, you have all the parameters (including ) found in the partial derivative on the right side of theXequation / . Now, since your starting point forin the same order as indicated in that partial derivative, � �`X `Z W

the partial on the left is on the of the box, while the starting point of the partial on the right is on the ofbottom top

the box, you introduce a sign. Whereas, if your starting point for the partials on each side of themust negative

equation were on the top, or on the bottom of our box, we have to introduce a negative sign.both both do not

This process will work, starting at any corner of the “Memory Box” and going in either direction to give all the

four of the Maxwell relations which are summarized below:

Œ � Œ �Œ � Œ �Œ � Œ �Œ � Œ �

`X `T

`Z `Wœ �

`X `Z

`T `Wœ

`W `T

`Z `Xœ

`W `Z

`T `Xœ �

W Z

W T

X Z

X T

CALCULATING GENERAL PROPERTIES FOR HOMOGENEOUS SUBSTANCES

As we mentioned earlier, the combined first and second law, in its original form

X .W œ .I � T .Z

contains five system properties, only two of which are independent. Typcially, we find it easier to measure things

like pressure, temperature and volume, or the gradients of these quantities than to measure the internal energy, or

entropy directly. Thus, we often write this equation and its various parameters as functions of these more easily

measured quantities. We now wish to examine some of the relationships that we can derive by considering two

special cases: 1) where and are the independent variables, and 2) where and are in the independentX Z X Tvariables.

. The first case we wish to consider is one in which assume that theIndependent Variables � �X ß Zindependent parameters for our system are the temperature and the volume . This means that we can expressX Zthe entropy and the internal energy as functions of temperature and volume, i.e., andW I W œ W X ß Z� �I œ I Xß Z T œ T X ß Z� � � �, and we can also express the pressure as a function of temperature and volume, . The

change in any one of these parameters can be expressed in a form similar to the equationdependent

.W œ .X � .Z`W `W

`X `ZŒ � Œ �

Z X

(4.42)

Thus, the combined first and second law can be written

X .X � .Z œ .X � .Z � T.Z`W `W `I `I

`X `Z `X `Z

X .X � .Z œ .X � � T .Z`W `W `I `I

`X `Z `X `Z

” • ” •Œ � Œ � Œ � Œ �” • Œ � ” •Œ � Œ � Œ �

Z X Z X

Z X Z X

(4.43)

from which we obtain

Œ � Œ �Œ � ” •Œ �

`W " `I

`X X `Xœ

`W " `I

`Z X `Zœ � T

Z Z

X X

(4.44)

Chapter 4: General Relationships between State Variables 8

The heat added to a system during a reversible process can be written$U

$U œ X .W œ X .X � .Z`W `W

`X `Z” •Œ � Œ �

Z X

(4.45)

If this process is a constant volume process, then and we have.Z œ !

$U œ X .W œ X .X`W

`XZ Z

ZŒ � (4.46)

We define the heat capacity at constant volume as the heat added to the system at constant volume divided by the

temperature change, or

G ´ œ X œU `W `I

.X `X `XZ

Z

Z Z

$ Œ � Œ � (4.47)

where we have used the relationship between the partials derived above. As mentioned earlier in the chapter, this

equation relates partial derivatives which are independent of any given processes to a process dependent variable,

GZ , which is a measurable quantity. The partials with respect to temperature, holding volume constant are,

therefore,

Œ �`I

`Xœ G

ZZ (4.48)

and

Œ �`W G

`X Xœ

Z

Z(4.49)

Now look back at the relationships between the partial derivatives of the energy and the entropy. Since

the entropy is a state variable, the second partial of the entropy will have the same value no matter what order the

partials are taken any ! (This last statement is true for state variable.) This means that

Œ � Œ � Œ � Œ �` `W ` `W

`Z `X `X `Zœ

X Z Z X

(4.50)

Œ � ” • Œ � œ �Œ � ” •Œ �` " `I ` " `I

`Z X `X `X X `Zœ � T

X Z Z X

(4.51)

" ` `I " `I " ` `I `T

X `Z `X X `Z X `X `Z `Xœ � � T � �Œ � Œ � ” • ” •Œ � Œ � Œ � Œ �

X Z X Z X Z#

(4.52)

But the internal energy is a state variable, so that the second partial of the energy will have the same value noalso

matter what order the partials are taken. This means that this last equation simplifies to give

" `I `T

X `Z `X� T œ” • Œ �Œ �

X Z

(4.53)

or

Œ � Œ �`I `T

`Z `Xœ X � T

X Z

(4.54)

(Notice that the quantity on the left hand-side of Equ. 4.53 is just equal to the S/ V . We will come back to� � � �` ` X

this later.)

The relationship in Equ. 4.54 tells us how the internal energy of the system changes with respect to� �volume if the temperature is held constant. You may be tempted to say that this partial has to be , and this iszero

Chapter 4: General Relationships between State Variables 9

indeed true the system is an , where . However, the equation which we derived from theif ideal gas TZ œ 8VXcombined first and second law ! This means that ais valid for any system no matter what the equation of state

measurement of the change in pressure with respect to temperature in a constant volume process can be used,

along with the measurement of the temperature and pressure of the system to determine how the internal energy

changes with volume.

You will remember from the first chapter that there are two typical quantities that are measured for many

systems - the coefficient of isothermal compressibility and the coefficient of thermal expansion . We pointed5 "out in that chapter that we can often rewrite the partials in terms of other partials using the cyclic and reciprocity

relationships. Using these in this case, we find that

Œ � Œ � Œ �`T `X `Z

`X `Z `Tœ � "

Z T X

(4.55)

or

Œ � � �� �

`T `Z `X

`X `Z `Tœ œ

Z

T

X

/

/(4.56)

"

,

so that our expression for the change in internal energy with respect to volume can be expressed in terms of

experimentally measurable quantities as

Œ �`I X

`Zœ � T

X

"

,(4.57)

Using this relationship for the change in internal energy with respect to volume, we can also obtain an expression

of the change in entropy with respect to volume from the equation

Œ � ” •Œ �`W " `I

`Z X `Zœ � T

X X

(4.58)

which gives

Œ � ” •œ �`W " X

`Z Xœ � T � T œ Î

X

"

," , (4.59)

We have been able to express the partial derivatives of , with respect to and in terms of theW I Z Xexperimentally measurable quantities , , , , and . This means that we can write an expression for theG T XZ , "heat added to the system,

$"

,U œ X .W œ X .X � X .Z œ G .X � .Z

`W `W X

`X `ZŒ � Œ �

Z XZ (4.60)

the entropy change in the system,

.W œ .X � .Z œ .X � .Z`W `W G

`X `Z XŒ � Œ �

Z X

Z "

,(4.61)

and the change in internal energy of the system

.I œ .X � .Z œ G .X � � T .Z`I `I X

`X `ZŒ � Œ � ” •

Z XZ

"

,(4.62)

in terms of experimentally measurable quantities.

. For an ideal gas and , and weEvaluation of Partials for Real and Ideal Gases " ,œ "ÎX œ "ÎTobtain the expected result

Œ �`I

`Zœ !

X

ideal gas (4.63)

Chapter 4: General Relationships between State Variables 10

that the internal energy of an ideal gas is independent of the volume. On the other hand, the partial derivative of

entropy with volume for an ideal gas is expressed as

Œ �`W T 8V V

`Z X Z @œ œ œ

X

ideal gas (4.64)

Notice that the change in entropy with volume (the slope of the entropy-volume curve) depends only upon the

specific volume of an ideal gas, and decreases as the volume becomes larger, going to zero as . AlsoZ p ∞notice that this partial is an intensive quantity, independent of the size of the system. Since the volume of the gas

is an extensive variable, the entropy must also be an extensive variable - depending upon the size of the system.

For a gas, however, where there is an interaction between the molecules, we expect that the internalreal

energy will be a function of the pressure and volume. As an example of a real gas, let's consider a gas which

obeys the Van der Waal equation. We found an expression for the coefficient of thermal expansion, , in the first"chapter for a Van der Waal's gas expressed in terms of the volumespecific

" œ œ" `@ V @ @ � ,

@ `X V X @ � # + @ � ,Œ � � �

� �T

#

$ # (4.65)

In terms of the volume, this isabsolute

" œ œ" `Z VZ Z � 8,

Z `X V X Z � #8+ Z � 8,Œ � � �

� ‘� �T

#

$ # (4.66)

The isothermal compressibility can be expressed as,

, œ � œ � œ �" `@ " `Z " "

@ `T Z `T Z `TÎ`ZŒ � Œ � � �X X X

(4.67)

For a Van der Waal's gas, the pressure can be expressed as

T œ � œ �VX + 8VX +8

@ � , @ Z � 8, Z� � � �# #

#

(4.68)

The partial is easily evaluated to give

Œ � � � � �� �`T 8VX #+8 #+8 Z � 8, � 8VXZ

`Z Zœ � � œ

Z � 8, Z Z � 8,X# #

# # $

$

#

$(4.69)

from which we can derive the isothermal compressibility

5 œ � œ" Z Z � 8, Z Z � 8,

Z #+8 Z � 8, � 8VXZ 8VXZ � #+8 Z � 8,

$ ## #

# $ $ ## #

� � � �� � � � (4.70)

We can now determine how the internal energy of a Van der Waal gas varies with volume

Œ �Œ �

Œ � ” •� � � �� ‘� � � � � �Œ � � �

`I X

`Zœ � T

`I "

`Zœ X † † � T

`I VZ Z � 8, 8VXZ � #+8 Z � 8, 8VX +8

`Z Z � 8, Zœ X † † � �

V X Z � #8+ Z � 8, Z Z � 8,

`I 8VX

`Z Z � 8,œ �

X

X

X

# $ # #

$ ## #

#

#

X

"

,

",

” •� �8VX +8 +8 +

Z � 8, Z Z @� œ œ

# #

# # #

(4.71)

Thus, for a “real” gas the internal energy depend upon the volume of the gas. Remember that the constant does +in the Van der Waal gas is related to the intermolecular forces acting on the particles. If this intermolecular force

is zero, then the internal energy is a function of the volume of the gas. But even for a real gas where there isnot

Chapter 4: General Relationships between State Variables 11

an intermolecular attraction, the dependence of the internal energy on the volume gets smaller as the specific

volume of the gas becomes larger, i.e., as the average distance between the molecules becomes larger. Thus, for

large specific volumes, the Van der Waal gas approaches an ideal gas.

Similarly, the partial of the entropy with respect to volume can also be evaluated for a Van der Waal gas

from the relation

Œ � ” • ” •Œ � � � � �`W " `I " +8 8VX +8 8V V

`Z X `Z X Z Z � 8, Z Z � 8, @ � ,œ � T œ � � œ œ

X X

# #

# #(4.72)

Notice that the entropy does not depend upon the intermolecular forces. The change in entropy with volume is

only a function of the specific volume accessible to the gas.

. Choosing and as independent variables, we write the combinedIndependent Variables � �X ßT X Tfirst and second law in the form:

X .X � .T œ .X � .T � T .X � .T`W `W `I `I `Z `Z

`X `T `X `T `X `T

X .X � .T œ � T .X � � T .`W `W `I `Z `I `Z

`X `T `X `X `T `T

” • Œ � Œ � ” •Œ � Œ � Œ � Œ �” • ” • ” •Œ � Œ � Œ � Œ � Œ � Œ �

T X T X T X

T X T T X X

T

(4.73)

or

Œ � ” •Œ � Œ �Œ � ” •Œ � Œ �

`W " `I `Z

`X X `X `Xœ � T

`W " `I `Z

`T X `T `Tœ � T

T T T

X X X

(4.74)

Just as in the previous case, the second partials of the entropy and the internal energy are equal, no

matter what the order of the partials. Taking the partial of the first equation above with respect to pressure and

the second equation with respect to temperature, we have

Œ � Œ � ” •Œ � Œ � Œ � Œ � Œ �` `W " ` `I `Z ` `Z

`T `X X `T `X `X `T `Xœ � � T

X T X T T X T

(4.75)

and

Œ � Œ � ” •Œ � Œ �” •Œ � Œ � Œ � Œ �

` `W " `I `Z

`X `T X `T `Tœ � � T

� � T" ` `I ` `Z

X `X `T `X `T

T X X X#

T X T X

(4.76)

If we subtract the second equation from the first, we obtain

! œ � T �" `I `Z " `Z

X `T `T X `X#X X T

” • Œ �Œ � Œ � (4.77)

Solving for the change in internal energy with respect to pressure (holding constant), we obtainX

Œ � Œ � Œ �`I `Z `Z

`T `X `Tœ �X � T

X T X

(4.78)

And this can be written in terms of and to give, "

Œ � � �`I

`Tœ �X Z � T � Z œ T Z � X Z

X

" , , " (4.79)

This equation tells us how the internal energy varies with pressure.

Chapter 4: General Relationships between State Variables 12

An alternate way of determining / is to realize that when the temperature is held constant, we� �`I `T X

can choose the independent variables to be the volume and the pressure, and a change in one of these will

necessarily imply a change in the other. Thus, from Equ. 4 57� �Þ

Œ � Œ � Œ � Œ � � �Œ � � �

`I `I `Z `I

`T `Z `T `Zœ œ † � Z

œ � T † � Z œ �X Z � T ZX

X X X X

,

"

,, " ,

(4.80)

The partial of the entropy with respect to the pressure can now be evaluated, using this last relationship to

give

Œ � ” • ” •Œ � Œ � Œ �`W " `I `Z " `I

`T X `T `T X `Tœ � T œ � T Z œ � Z

X X X X

, " (4.81)

The heat added to a system during a reversible process can be written$U

$U œ X .W œ X .X � .T`W `W

`X `T” •Œ � Œ �

T X

(4.82)

If this process is a constant pressure process, then and we have.T œ !

$U œ X .W œ X .X`W

`XT T

TŒ � (4.83)

We define the heat capacity at constant pressure as the heat added to the system at constant pressure divided by

the temperature change, or

G ´ œ X œ � TU `W `I `Z

.X `X `X `XT

T

T T T

$ Œ � ” •Œ � Œ � (4.84)

where we have used the relationship between the partials derived above. As mentioned earlier in the chapter, this

equation relates partial derivatives which are independent of any given processes to a process dependent variable,

GT , which is a measurable quantity. The partials with respect to temperature, holding volume constant are,

therefore,

Œ �`W G

`X Xœ

T

T(4.85)

Œ � Œ �`I `Z

`X `Xœ G � T œ G � T Z

T TT T " (4.86)

Again we have been able to express the partial derivatives of , with respect to and in terms ofW I T Xthe experimentally measurable quantities , , , , and . This means that we can write an expression forG T Z XT , "the heat added to the system,

$ "U œ X .W œ X .X � X .T œ G .X � X Z .T`W `W

`X `TŒ � Œ �

T XT (4.87)

the entropy change in the system,

.W œ .X � .T œ .X � Z .T`W `W G

`X `T XŒ � Œ �

T X

T" (4.88)

and the change in internal energy of the system

Chapter 4: General Relationships between State Variables 13

.I œ .X � .T œ G � T Z .X � T Z � X Z .T`I `I

`X `TŒ � Œ � c d c d

T XT " , " (4.89)

in terms of experimentally measurable quantities.

For an ideal gas and , whicnEvaluation of Partials for Real and Ideal Gases. " ,œ "ÎX œ "ÎTgives the expected result

Œ �`I

`Tœ !

X

ideal gas (4.90)

The change in entropy with respect to pressure for an ideal gas is given by

Œ �`W Z 8V

`T X Tœ � Z œ � œ �

X

" ideal gas (4.91)

Notice that this partial is an extensive quantity, since the pressure is an intensive variable and the entropy is an

extensive variable.

To see how the internal energy of a real gas varies with pressure, we again use the relationships and " ,derived for the Van der Wall gas. To obtain the desired result, let's use Equ. 4 80� �Þ

Œ � Œ � � � � �� �

`I `I +8 Z Z � 8,

`T `Z Zœ † � Z œ † � † Z

8VXZ � #+8 Z � 8,X X

# #

#

#

$ # #, (4.92)

Œ � � �� �

`I +8 Z Z � 8,

`Tœ �

8VXZ � #+8 Z � 8,X

# #

$ # # (4.93)

and

Œ � � � � �� ‘ � ‘� � � �

`W VZ Z � 8, VZ Z � 8,

`Tœ � Z œ � † Z œ �

VX Z � #8+ Z � 8, V X Z � #8+ Z � 8,X

# $

$ $# #" (4.94)

for the Van der Waal gasÞ In this section we will start againThe Difference in Heat Capacities for Homogeneous Substances.

with the combined first and second law written in terms of the independent variables and :Z X

$U œ .I � T .Z œ .X � .Z � T .Z`I `I

`X `ZŒ � Œ �

Z X

(4.95)

We have identified the first partial as the heat capacity at constant volume, , so we can write this last equationGZas

$U œ G .X � � T .Z`I

`ZZ

X” •Œ � (4.96)

We have already found an expression for , which gives us� �`IÎ`Z X

Œ �`I X

`Zœ � T

X

"

,(4.97)

so we obtain

$"

,U œ G .X � .Z

XZ (4.98)

To find an expression for the heat capacity at constant pressure, we will write the change in volume in terms of a

change in temperature and a change in pressure to obtion

Chapter 4: General Relationships between State Variables 14

$"

,U œ G .X � .X � .T

X `Z `Z

`X `TZ

T X” •Œ � Œ � (4.99)

or

$ , "" " "

, , ,U œ G � .X � � Z .T œ G � .X � X Z .T

X Z X X Z” • ” •c dZ Z

# #

(4.100)

During a constant pressure process, , giving.T œ !

$"

,U œ G � .X

X ZT Z

#” • (4.101)

But the heat capacity at constant pressure is just

G œ œ G �U X Z

.XT Z

T#$ "

,(4.102)

so the difference in the heat capacities is given by

G � G œ � T � T Z œX X Z

T Z

#” •Œ �" "

, ," (4.103)

Now, , , , and are all quantities, so we see that for all homogeneous substances ! For anX Z G / G" ,#T Zpositive

ideal gas, this difference is just given by

G � G œ œ 8VTZ

XT Z (4.104)

but for a real gas (a Van der Waal gas) we find

G � G œ8 V XZ

8VXZ � #+8 Z � 8,T Z

# # $

$ # #� � (4.105)

which reduces to the ideal gas result if .+ œ !

ENTROPY CHANGES FOR AN IDEAL GAS

One of the most obvious uses of the combined first and second law

$U œ X .W œ .I � T .Z (4.106)

is to find an equation for the change in entropy of a system Solving for we obtainÞ .W

.W œ � .Z.I T

X X(4.107)

Thus, the change in entropy of a system can be determined simply by integrating from the initial to the final states

of the system

?W œ .W œ � .Z.I T

X XEF

E E E

F F F

( ( ( (4.108)

For an ideal gas, where and , and where we can assume the is a constant, we can.I œ 8- .X TZ œ 8VX -@ @

write this as

?W œ .W œ 8- � 8V.X .Z

X ZEF

E E E

F F F

@( ( ( (4.109)

Chapter 4: General Relationships between State Variables 15

Thus, the change in entropy of an ideal gas as it goes from state to state (characterized by , , andE F T Z XE E E

T Z XF F F, , , respectively) is given by

?W œ 8- � 8VX Z

X ZEF @

F F

E Eln ln (4.110)Œ � Œ �

This same equation can be expressed in terms of changes in pressure by using the ideal gas law to obtain

?W œ 8- � 8V œ 8- � 8V � 8VX X T X X T

X X T X X TEF @ @

F F F F

E E E E

E E

F Fln ln ln ln ln (4.111)Œ � Œ � Œ � Œ � Œ �

or

?W œ 8 - � V � 8V œ 8- � 8VX T X T

X T X TEF @ :

F F

E E

E E

F F� � Œ � Œ � Œ � Œ �ln ln ln ln (4.112)

Likewise, one can use the ideal gas law to elliminate the temperature and write the change in entropy in terms of

the volume and pressure. Thus, we can easily calculate the changes in the entropy of an ideal gas, provided we

know the temperature and volume, the temperature and pressure, or the volume and pressure for the initial and

final states:

?W œ 8- � 8VX Z

X Z

œ 8- � 8VX T

X T

œ 8- � 8-T Z

T Z

EF @F F

E E

:F

E

E

F

@ :F F

E E

ln ln

ln ln

ln ln

(4.113)Œ � Œ �Œ � Œ �Œ � Œ �

In what follows we will find expressions from which we can calculate the changes in entropy for system, notany

just an ideal gas.

Appendix 4.1

The and Equations:$U X .WA Systematic Derivation of Relationships

Between Parameters of Arbitrary Systems

In this appendix we will follow a very systematic approach to develop many equations which relate various

partial derivatives to experimentally measurable quantities. To simplify the expressions and the math, we

consider only systems where 0. The first law simplifies toclosed .R œ

.I œ X.W � T.Z (A4.1.1)

and is a function only of and , where only two of these variables are independent. Since the entropyI Wß X ß T ß Zcannot usually be measured directly, we wish to express the energy in terms of the more easily measured variables

T ß Z ß X and . This means that we have a choice of expressing the energy as

I I Z X I I T X I I T Z = ( , ) or = ( , ) or = ( , ). (A4.1.2)

Since we can often measure the amount of heat which is added to a system during various processes, we will

rearrange the energy equation to give

$U X .W œ .I T .Z = + (A4.1.3)

By considering the energy as a function of any two of the three easily measurable parameters and , weI Tß Z ß Xcan derive the three so-called equations.$U

I. I œ Z ßX� � Here we have

.I œ .X � .Z`I `I

`X `ZŒ � Œ �

Z X

(A4.1.4)

which gives for the equation$U

$U œ X.W œ .X � .Z � T.Z`I `I

`X `ZŒ � Œ �

Z X

(A4.1.5)

or

$U œ X.W œ .X � � T .Z`I `I

`X `ZŒ � ” •Œ �

Z X

(A4.1.6)

which is the first equation.$U

II. I œ IÐT ßXÑ The second equation arises from the assumption that = ( , ), giving$U I I T X

$U œ X.W œ .X � .T � T.Z`I `I

`X `TŒ � Œ �

T X

(A4.1.7)

but we must remember that d is also an exact differential, with = ( , ), so that this equation becomesZ Z Z T X

$U œ X.W œ .X � .T � T .X � .T`I `I `Z `Z

`X `T `X `TŒ � Œ � ” •Œ � Œ �

T X T X

(A4.1.8)

Chapter 4: General Relationships between State Variables 17

or

$U œ X.W œ � T .X � � T .T`I `Z `I `Z

`X `X `T `T” • ” •Œ � Œ � Œ � Œ �

T T X X

(A4.1.9)

which is the second equation.$U

III. I œ IÐT ßZ Ñ The last equation is found by taking = ( , ) so that$U I I T Z

$U œ X.W œ .T � .Z � T.Z`I `I

`T `ZŒ � Œ �

Z T

(A4.1.10)

or

$U œ X.W œ .T � � T .Z`I `I

`T `ZŒ � ” •Œ �

Z T

(A4.1.11)

which is the third equation.$U In this appendix we will carefully examine some of the consequences of these three equations. In

particular we want to determine expressions for the change in the internal energy and the entropy of a system in

terms of experimentally measureable quantities. Once we determine expressions for these partial derivatives, we

will be able to write out expressions for the heat flowing into a system or the change in entropy of the system, or

the change in internal energy of the system in terms of experimentally measureable quantities. Some of the

relationships derived in the main part of the chapter will be repeated here for completeness, but there are many

more relationships derived in this appendix than in the chapter itself.

I. Consequences of the First equation: and $U I œ IÐX ß Z Ñ W œ WÐX ß Z Ñ The First equation is$U

$U œ X.W œ .X � � T .Z`I `I

`X `ZŒ � ” •Œ �

Z X

(A4.1.12)

In addition, since the entropy can also be written as a function of and we can writeX Z

X.W œ X .X � X .Z`W `W

`X `ZŒ � Œ �

Z X

(A4.1.13)

Examining these two equations, we immediately find some relationships between the energy partials and the

entropy partials:

X œ`W `I

`X `XŒ � Œ �

Z Z

(A4.1.14)

and

X œ � T`W `I

`Z `ZŒ � ” •Œ �

X X

(A4.1.15)

This last equation is very useful in studying the variation of the internal energy of a gas as a function of volume.

Rearranging this equation, one can write

Œ � Œ �`I `W

`Z `Zœ X � T

X X

(A4.1.16)

Chapter 4: General Relationships between State Variables 18

Now we can use the Maxwell relation (derived in the chapter)

Œ � Œ �`W `T

`Z `Xœ œ

X Z

"

,(A4.1.17)

to write

Œ �`I X

`Zœ � T

X

"

,(A4.1.18)

We will now use the equation to find expressions for these partial derivatives in terms of measurable$Uquantities, such as the heat capacities and , the isothermal compressibility and the coefficient to thermalG GZ T ,expansion , and the easily measureable parameters and ." T ß Z ß X

A. Isochoric Processes. We will first consider a system undergoing an isochoric process ( = 0), so that the.Zfirst equation simplifies to give$U

$U œ X.W œ .X`I

`XZ Z Z

ZŒ � (A4.1.19)

where the subscripts indicate the variable which must remain constant in the process. We the heat capacitydefine

at constant volume as the of the amount of heat added to the system, , to the rise in temperature,G UZ Zratio $.XZ , of the system for a given process.

G œ œ X œU `W `I

.X `X `XZ

Z

Z Z Z

$ (A4.1.20)Œ � Œ �

[Note: Remember that the ratio of to is simply a ratio since is not an exact differential, while the$ $U .X UZ Z

ratio of to is equivalent to the partial derivative of with respect to when is held constant, because.W .X W X ZZ Z

the entropy is a state variable.]

We see from this equation that the heat capacity at constant volume is directly related to the how the energy

of the system changes with temperature if we hold the volume constant. Likewise, it is directly related to the

change in entropy of the system during this isochoric change in temperature. These partials can therefore be

written in terms of an experimentally measureable quantity according to:

Œ � Œ �`I `W G

`X `X Xœ G œ

Z ZZ

Z and (A4.1.21)

[Note: We have written these equations in terms of variables, which means that theseextensive

relationships depend upon the of the system. The heat capacity, therefore, also depends uponsize

the size of the system. To get the equations in terms of variables, we must divide by theintensive

mass or the number of moles in the system. But this must be done for all the variables. For

example, the combined first and second law would have to be written as

$ %; X.= œ . T .@ = + (A4.1.22)

where / , / , etc. Under these circumstances the heat capacity is given by; œ U 8 = œ W 8 -specific

- œ œG ;

8 .X

$(A4.1.23)

All the equations which we derive can easily be changed to form by dividing theintensive

extensive ratiosvariables by the number of moles. Notice that and are defined in terms of of" ,the volume so that their definition is independent of size!]

B. Isobaric Processes. Next we will consider the system undergoing an isobaric process ( = 0). In this case.Tthe first equation gives us$U

Chapter 4: General Relationships between State Variables 19

$U œ X.W œ .X � � T .Z`I `I

`X `ZT T T T

Z XŒ � ” •Œ � (A4.1.24)

If we now divide this equation by (a change in temperature at constant pressure), we get.XT

$U .W `I `I .Z

.X .X `X `Z .Xœ X œ � � T

T T T

T T TZ XŒ � ” •Œ � (A4.1.25)

The first term in this equation is, by definition, the heat capacity at constant pressure, . Again, it is aGT not

partial derivative because is not an exact differential. However, the other ratios which appear in this equation$Uare partial derivatives, so that this equation should be written

G œ œ X œ � � TU `W `I `I `Z

.X `X `X `Z `XT

T

T T Z X T

$ Œ � Œ � ” • Œ �Œ � (A4.1.26)

This last equation can be further simplified by using the identities:

G œ œ`I " `Z

`X Z `XZ

Z TŒ � Œ � and (A4.1.27)"

to give

G œ X œ G � � T Z`W `I

`X `ZT Z

T XŒ � ” •Œ � (A4.1.28)"

This equation can now be used to determine the isothermal change in internal energy as the volume of the system

changes. We simply use the experimentally determined values of and and obtainG ß G ß T ß ß ZT Z "

Œ �`I G � G

`Z Zœ � T

X

T Z

"(A4.1.29)

Likewise, the entropy partial is

Œ �`W G

`X Xœ

T

T(A4.1.30)

C. Isothermal Processes. For an isothermal process ( = 0) the first equation gives.X U$

$U œ X.W œ � T .Z`I

`ZX X

X X” •Œ � (A4.1.31)

This expression is interesting. It is a measure of the amount of heat which is added to a system during an

isothermal process, i.e., it is the amount of heat added to the system during a process in which the temperature of

the system does not change! Now the heat capactity is defined as = / , so that for an isothermal process,G U .X$the heat capacity of the system is essentially infinite! Thus the concept of heat capacity in the usual sense is not

very useful in this case. However, somewhat analogous to the definition of heat capacity, we define the latent

heat capacity the amount of heat added to a system (or taken away) as the volume(or latent heat) of the system as

of the system changes even when the temperature remains constant. We use the symbol to stand for the latentPZheat capacity of the system [Note that this is an exception to the way we normally viewduring a volume change.

subscripts Usually a subscript will mean a variable is held , but when we deal Þ fixed with latent heats, the

subscript will stand for the quantity which changes.] The latent heat for a volume change is therefore given by

the equation

P œ œ X œ X œ � TU .W `W `I

.Z .Z `Z `ZZ

X X

X X X X

$ Œ � ” •Œ � (A4.1.32)

From Equ. (13.25) we can write asPZ

Chapter 4: General Relationships between State Variables 20

P œG � G

ZZ

T Z

"(A4.1.33)

We have, however, introduced a new partial derivative of the entropy which can now be expressed in terms of

measureable parameters:

Œ �`W P G � G

`Z X X Zœ œ

X

Z T Z

"(A4.1.34)

Using the definition of the latent heat, the first equation can be expressed in the simple form:$U

$U œ X.W œ X Z C d + L d (A4.1.35)Z Z

D. Adiabatic ProcessesÞ U œ X.W UFor an adiabatic process ( = 0) we can write the first equation$ $

$U œ X.W œ G .X � P .Z œ !W Z W Z W (A4.1.36)

$"

U œ X.W œ G .X � .Z œ !G � G

ZW Z W W

T Z (A4.1.37)

Here we use the subscript to denote an adiabatic (isentropic) process. Dividing by we obtainW .ZW

.X `X P G � G G � G

.Z `Z G Z G Z Gœ œ � œ � œ

W Z T Z Z T

W Z Z ZWŒ �

" "(A4.1.38)

which gives the partial

Œ �`X G � G

`Z Z Gœ

W

Z T

Z"(A4.1.39)

However, this last equation could have been derived from previous results in this section by using the cyclic and

reciprocity rules:

Œ � � �� �

`X `W `Z G � G G � G

`Z `X `Z G Z Z Gœ � œ � œ � œ

G � G

X ZG

XW

X

W

: Z

Z

: Z Z :

Z Z

/

/(A4.1.40)

"

" "

This last equation can also be written

Œ �`X " �

`Z Zœ

W

#

"(A4.1.41)

where is the of specific heats ( / ).# #ratio œ - -: @

The last equation above is perfectly general, for any real substance. For an ideal gas, is a constant and#" œ "ÎX , so that this last equation can be integrated to obtain the equations relating the pressure, volume and

temperature for adiabatic processes in ideal gases.

Chapter 4: General Relationships between State Variables 21

Summary of Results for the First equation: $U I œ IÐZ ß XÑ

Summarizing the expressions which we have developed for the energy and entropy partials, we have, from

the first equation:$U

Œ � Œ � Œ �Œ � Œ �Œ � Œ �

`W G `W G `W P G � G

`X X `X X `Z X X Zœ œ œ œ

`I `I G � G X

`X `Z Zœ G œ � T œ � :

`X G � G `X : G � G

`Z Z G `Z G Z Gœ œ �

Z T X

Z T Z T Z

Z XZ

T Z

W I

Z T : Z

Z Z Z

"

" ,

"

" "

(A4.1.42)

where the last expression on the right is derived from the energy partials, using the cyclic and reciprocity rules.

The Maxwell relationships also gives us some additional information. For example

Œ � Œ �`W `T

`Z `Xœ œ

X Z

"

,(A4.1.43)

can be used to gain some additional information concerning the heat capacities. We obtain

Œ �`W P G � G

`Z X X Zœ œ œ

X

Z T Z

" ,

" (A4.1.44)

which gives

G � G œXZ

T Z

#"

,(A4.1.45)

for the difference in the heat capacities. This equation tells us that is greater than since , G G X ß Z ßT Z#always "

and are always greater than zero! For an ideal gas, where 1/ and = 1/ , this reduces to5 œ X :" ,

G � G œ 8V ÐT Z ideal gas) (A4.1.46)

or in terms of molar specific heat capacities

- � - œ V ÐT Z ideal gas) (A4.1.47)

We can use the first equation to calculate the amount of heat added to the system for a given temperature and$Uvolume change in terms of measureable quantities

$"

U œ X.W œ G .X � .ZG � G

ZZ

T Z” • (A4.1.48)

or, using the Maxwell relation,

$"

,U œ X.W œ G .X � .Z

XZ (A4.1.49)

II. Consequences of the Second equation: and $U I œ IÐX ßT Ñ W œ WÐX ßT Ñ The second equation is given by$U

$U œ X.W œ � T .X � � T .T`I `Z `I `Z

`X `X `T `T” • ” •Œ � Œ � Œ � Œ �

T T X X

(A4.1.50)

Likewise, in terms of the differential change in the entropy

X.W œ X .X � X .T`W `W

`X `TŒ � Œ �

T X

(A4.1.51)

giving

Chapter 4: General Relationships between State Variables 22

X œ � T`W `I `Z

`X `X `XŒ � Œ � Œ �

T T T

(A4.1.52)

and

X œ � T`W `I `Z

`T `T `TŒ � Œ � Œ �

X X X

(A4.1.53)

Again, we want to determine expressions for the change in the internal energy and the entropy of a system in

terms of experimentally measureable quantities. This will enable us to write out expressions for the heat flowing

into a system or the change in entropy of the system, or the change in internal energy of the system in terms of

experimentally measureable quantities.

A. Isobaric Processes. Consider the isobaric process ( = 0).T

$U œ X.W œ � T .X`I `Z

`X `XT T T

T T” •Œ � Œ � (A4.1.54)

Dividing by the temperature change at constant pressure, we obtain the heat capacity at constant pressure, ,GT

G œ œ X œ � T œ � T ZU `W `I `Z `I

.X `X `X `X `XT

T

T T T T T

$"Œ � Œ � Œ � Œ � (A4.1.55)

This equation is the same one we obtained earlier, we have a different energy partial. Here we determine hownot

the energy of the system varies with temperature during an isobaric process in terms of experimentally measurable

parameters from the partial

Œ �`I

`Xœ G � T Z

TT " (A4.1.56)

We do obtain again the previous result,

Œ �`W G

`X Xœ

T

T(A4.1.57)

Notice that the equation for can be writtenGT

G œ œ X œ � œ œU `W `I ` TZ ` I � TZ `L

.X `X `X `X `X `XT

T

T T T T T T

$ Œ � Œ � Œ � Œ � Œ �[ ] [ ](A4.1.58)

where the quantity is the of the system, as defined earlier. Here we see that L œ I � TZ enthalpy the enthalpy

of the system under constant pressure processes is similar to the internal energy of the system under constant

volume processes, since

G œ`L

`XT

TŒ � (A4.1.59)

and

G œ`I

`XZ

ZŒ � (A4.1.60)

B. Isothermal Processes. Now consider an isothermal process ( = 0).X

$U œ X.W œ � T .T`I `Z

`T `TX X X

X X” •Œ � Œ � (A4.1.61)

In the same way that we defined the latent heat before, we again define the latent heat of pressure change andPTfind from the equation above

Chapter 4: General Relationships between State Variables 23

P œ œ X œ � TU `W `I `Z

.T `T `T `TT

X

X X X X

$ Œ � ” •Œ � Œ � (A4.1.62)

Remember that the isothermal compressibility is given by

, œ �" `Z

Z `T (A4.1.63)Œ �

X

so that we can write

P œ œ X œ � TU `W `I

.T `T `TT

X

X X X

$,Œ � ” •Œ � v (A4.1.64)

The quantity is an experimentally measureable quantity, but it can also be determined in terms of other morePTfamiliar quantities. We will show in what follows that

P œ � P Z œ � Z œ G � GG � G

ZT Z Z T

T Z, ,

" "

,Œ � � � (A4.1.65)

This will mean that the energy and entropy partials can be expressed as

Œ �`I

`Tœ P � T Z

XT , (A4.1.66)

or

Œ � � �`I

`Tœ G � G � T Z

XZ T

,

", (A4.1.67)

and

Œ �`W P

`T Xœ

X

T(A4.1.68)

in terms of measurable quantities.

To see the relationship between and , we will consider the first and second equationsP P UZ T $

$U œ G .X � P .ZZ Z (A4.1.69)

and

$U œ G .X � P .TT T (A4.1.70)

There are actually two different ways we can understand the relationship between and , and we willP PT Z

examine both. First, consider three different processes which can be diagramed on a , graph: 1) = 0; 2)T Z .T.Z .X = 0; 3) = 0, as shown below.

For the case where = 0 we immediately have = C and we can calculate the amount of heat.T U .X$ T

added to the system during the temperature increase . Likewise for the case where = 0 we can write =.X .Z U$C . Now for the case where = 0, we have two expressions which we can useZ .X .X

$U .Z = L (A4.1.71)Z

or

$U .T = L (A4.1.72)T

Chapter 4: General Relationships between State Variables 24

Volume (cubic meters)

Pre

sure

(P

asc

als

)

0 2 4 6 8 10

2

4

6

8

T(A)

T(B)T(C)

dP = 0dV

= 0

dT = 0A

B

Now for an isothermal change ( = 0) the pressure and volume are related, so that if changes by then we.X T .Tknow exactly how much changes. Thus if is constant and changes from state we can calculate Z X T E p F U$from the second equation above. But during this process, changes also from the value it has for state to thatZ Efor state , so that we can in essence calculate the heat added to, or removed from, the system using Equ.F(A4.1.71) above. Now the process is the same in both cases and the end points are the same so that we must have

P .Z P .TZ X T X = (A4.1.73)

or

P `T `Z "

P `Z `T Zœ œ œ �

Z

T X X

�"

Œ � ” •Œ �,

(A4.1.74)

Now since we have already found an expression for in terms of easily measured parameters of the system, thisPZequation allows us to determine :PT

P œ � P Z œ � Z œ G � GG � G

ZT Z Z T

T Z, ,

" "

,Œ � � � (A4.1.75)

This same result can be obtained directly from the equations

$U .X P .Z = C + (A4.1.76)Z Z

and

$U .X P .T = C + (A4.1.77)T T

Since is a function of and , we can write the first equation asZ T X

$U .X .T � .X`Z `Z

`T `X = C + L (A4.1.78)Z Z

X T” •Œ � Œ �

or

$U œ G � P .X � P .T`Z `Z

`X `T” • Œ �Œ �Z Z Z

T X

(A4.1.79)

Chapter 4: General Relationships between State Variables 25

Now if we compare this equation with the second equation$U

$U .X .T = C + L (A4.1.80)T T

we see that

G œ G � P ZT Z Z " (A4.1.81)

and

P œ P � Z œ � P Z œ G � GT Z Z Z T (A4.1.82)� � � �, ,,

"

which is the result we obtained earlier.

C. Adiabatic Processes. We now examine adiabatic processes ( ), for which the second equation gives.W œ ! U$

$U œ X.W œ ! œ � T .X � � T .T`I `Z `I `Z

`X `X `T `T” • ” •Œ � Œ � Œ � Œ �

T T X X

(A4.1.83)

or

$U œ X.W œ ! œ G .X � P .TT W T W (A4.1.84)

from which we obtain

Œ � � � � �`X P G � G

`T G G Gœ � œ � œ �

G � G

W

T Z T

T T T

Z T,"

(A4.1.85)

,

"

As we demonstrated last time this equation can be obtained directly from the entropy partials by using the

reciprocal and cyclical properties of partials.

We can also use this last equation to derive a useful relationship between the pressure and the temperature

of an ideal gas which undergoes an adiabatic process. Deviding the top and bottom of the equation by andGZsubstituting for and , this last equation can be written, "

Œ � � � � �`X G � G X " �

`T G :œ � œ �

W

Z T

T

, #

" #

(A4.1.86)

Since is constant for an ideal gas, we can integrate this equation to obtain#

X: œ Ð"�## const. adiabatic process for ideal gas)A4.1.87)(

Summary of and I œ IÐX ßT Ñ W œ WÐX ßT Ñ Summarizing the expressions which we have developed for the energy and entropy partials using the second

$U equation, we have:

Œ � Œ � � �Œ � Œ � � �Œ � � �

`I `I

`X `Tœ G � T Z œ P � T Z œ G � G � T Z

`W G `W P

`X X `T X Xœ œ œ G � G

`X P G � G

`T G Gœ � œ �

T XT T Z T

T X

T TZ T

W

T Z T

T T

" , ,,

"

,

"

,

"

The heat added to the system, from the second equation is given by$U

$U œ X.W œ � T .X � � T .T`I `Z `I `Z

`X `X `T `T” • ” •Œ � Œ � Œ � Œ �

T T X X

(A4.1.88)

Chapter 4: General Relationships between State Variables 26

or

$,

"U œ X.W œ G .X � G � G .TT Z T� � (A4.1.89)

which can be further simplified to give

$ "U œ X.W œ G .X � X Z .TT (A4.1.90)

III. Consequences of the third equation: and $U I œ IÐT ßZ Ñ W œ WÐT ß Z Ñ The third equation is$U

$U œ X.W œ .T � � T .Z`I `I

`T `ZŒ � ” •Œ �

Z T

(A4.1.91)

and times the change in entropy is given byX

X.W œ X .T � X .Z`W `W

`T `ZŒ � Œ �

Z T

(A4.1.92)

so that we have

X œ`W `I

`T `TŒ � Œ �

Z Z

(A4.1.93)

and

X œ � T` `I

`Z `ZŒ � Œ �S

(A4.1.94)T T

Now we take the partial with respect to holding constant in the first equation and the partial with respect toZ TT Z I W holding constant in the second equation, and make use of the fact that and are exact differentials to

obtain:

Œ � Œ �`W `W

`T `Zœ Z � Z

Z T

, " (A4.1.95)

We now examine the third equation under certain processes to obtain expressions for the partials of energy$Uand entropy in terms of experimentally measurable quantities.

A. Isochoric Processes For an isochoric process (d = 0) the third equation givesZ U$

$U œ X.W œ .T`I

`TZ Z Z

ZŒ � (A4.1.96)

Dividing by a small temperature change at constant volume we obtain

G œ œ X œU .W `I .T

.X .X `T .XZ

Z Z Z

Z Z ZZ

$ Œ � (A4.1.97)

which is the heat capacity at constant volume. The ratios of the exact differentials must be partial derivatives, so

we can write

G œ œ X œU `W `I `T

.X `X `T `XZ

Z

Z Z Z Z

$ Œ � Œ � Œ � (A4.1.98)

We can use the cyclic relationship to evaluate :� �`TÎ`X Z

Œ � � �� �

`T `Z `X Z

`X `Z `T � Zœ � œ � œ

Z

T

X

/

/(A4.1.99)

" "

, ,

Chapter 4: General Relationships between State Variables 27

so that we obtain for the heat capacity at constant volume:

G œ œ X œU `W `I

.X `X `TZ

Z

Z Z Z

$ "

,Œ � Œ � (A4.1.100)

This gives us another energy partial which we can evaluate experimentally

Œ �`I G

`Tœ

Z

Z,

"(A4.1.101)

B. Isobaric Processes For an isobaric process, the third equation becomes$U

$U œ X.W œ � T .Z`I

`ZT T T

T” •Œ � (A4.1.102)

and we can again divide by a small temperature change at constant pressure to give.XT

G œ œ X œ � TU `W `I `Z

.X `X `Z `XT

T

T T T T

$ Œ � ” • Œ �Œ � (A4.1.103)

or

G œ œ X œ � T ZU `W `I

.X `X `ZT

T

T T T

$"Œ � ” •Œ � (A4.1.104)

which gives us another energy partial

Œ �`I G

`Z Zœ � T

T

T

"(A4.1.105)

C. Adiabatic ProcessesÞ Now for an adiabatic process we can write

$U œ X.W œ ! œ .T � � T .Z`I `I

`T `ZŒ � ” •Œ �

Z TW W (A4.1.106)

where we have again used the subscript to stand for adiabatic (isentropic) changes. Now by using the energyWpartials derived above

Œ � Œ �`I G `I G

`Z Z `Tœ � T œ

T Z

T Z

" "

, (A4.1.107)

we can write

$,

" "U œ X.W œ ! œ .T � .Z

G G

ZZ T

W W (A4.1.108)

from which we can derive

Œ �`T G G

`Z Z G Z Gœ � ‚ œ �

W

T T

Z Z" , ,

" (A4.1.109)

Again, we can use this last equation to derive a useful relationship between the pressure and the volume of

an ideal gas which undergoes an adiabatic process. Writing the ratio of the specific heats as and substituting the#value of for an ideal gas, we have,

Œ �`T :

`Z Zœ �

W

#(A4.1.110)

Chapter 4: General Relationships between State Variables 28

which can be integrated to give

:Z œ# constant (A4.1.111)

We could also examine an isothermal process, but we would get no new results.

Summary of and I œ IÐZ ßT Ñ W œ WÐZ ßT Ñ We now summarize the partial derivative relationships which we have found in this subsection:

Œ � Œ �Œ �Œ � Œ �

`I G `I G

`T `Z Zœ œ � T

`T G G

`Z Z G Z Gœ � ‚ œ �

`W `W

`T `Zœ Z � Z

Z T

Z T

W

T T

Z Z

Z T

,

" "

" , ,

"

, "

(A4.1.112)

The heat added to the system using the third equation is given by$U

$U œ X.W œ .T � � T .Z`I `I

`T `ZŒ � ” •Œ �

Z T

(A4.1.113)

or

$,

" "U œ X.W œ .T � .Z

G G

ZZ T

(A4.1.114)

ADDITIONAL APPLICATION

The Internal Energy of an Ideal Gas (The Free Expansion of a Gas):

When we first introduced the concept of an ideal gas, we started with a model for that gas which assumed

that their was no interaction potential energy between the molecules making up the system, that is, the internal

energy of the system could not depend upon the location of the particles or their relative separation. This implies

that the internal energy of an ideal gas should not depend upon the volume of the gas. In fact, we showed that the

internal energy of an ideal gas is a function of the temperature only, not the volume or the pressure. From these

fundamental assumptions, we derived the equation of the ideal gas

TZ œ R5X œ 8VX (A4.1.115)

Let us approach this problem from a somewhat different perspective. Let's assume that the equation of state

has been determined by empirical observations (which, in fact, is the case historically). We can now use the

relationships which we obtained in the last chapter to investigate the internal energy of such a gas and how it

depends upon various parameters. Rather than start with arbitrary relationships to prove this point, however, we

will actually develop the equations we need.

If we assume a closed system, the internal energy can be a function of at most two independent parameters.

Let's make the choice , so that we can writeI œ IÐZ ß X Ñ

.I œ .Z � .X`I `I

`Z `XŒ � Œ �

X Z

(A4.1.116)

From the results we obtained above, we find

Œ �`I

`Xœ G

ZZ (A4.1.117)

and

Œ �`I X

`Zœ � T

X

"

,(A4.1.118)

Now for an ideal gas this equation becomes

Œ � � �� �

`I X "ÎX

`Z "ÎTœ � T œ !

X

(A4.1.119)

which tells us that any gas which obeys the ideal gas equation is one for which the internal energy of the gas

cannot depend upon the volume! We can use the chain rule

Œ � Œ � Œ �`I `I `Z

`T `Z `Tœ œ

X X X

0 (A4.1.120)

to see that the internal energy of an ideal gas must be independent of both volume and pressure changes, and thus

must be a function of the temperature of the gas only!

The fact that an ideal gas is not a function of the pressure or the volume of the gas implies that if we were to

cause an ideal gas to expand freely into a larger volume, the internal energy (and thus the temperature of the gas)

would not change. Such an experiment is called a free-expansion, and is performed by placing a gas in a chamber

with two sections separated by a valve. One side is evacuated and the other side is filled with a gas. Once the gas

reaches equilibrium, the valve is opened and the gas is allowed to freely expand into the adjacent volume. From

the first law we have

.I œ U� [ œ U$ $ $ (A4.1.121)

since no work is done in a free expansion (there are no walls which move). Thus, any change in internal energy

would cause heat to flow into or out of the system. Now if the system is isolated, so that the walls of our special

box are adiabatic, then the change in internal energy must be zero! Thus, for an adiabatic free-expansion we have

Chapter 4: General Relationships between State Variables 30

that . Now let's invision a process for which , but for which the process if reversible, or “quasi-.I œ ! .I œ !static". In this case, we would have

.I œ ! œ X.W � :.Z (A4.1.122)

or

X .X � .Z œ :.Z`W `W

`X `Z” •Œ � Œ �

Z X

(A4.1.123)

Now

G œ œ XU `W

.X `XZ

Z

Z Z

$ Œ � (A4.1.124)

and

Œ � Œ �`W `T G � G

`Z `X Z Xœ œ

X Z

T Z

"(A4.1.125)

so that we have

G .X œ T � X .Z`T

`XZ

Z” •Œ � (A4.1.126)

or

.X œ T � X .Z" `T

G `XZ Z” •Œ � (A4.1.127)

For an ideal gas, the quantity in brackets is zero, so that the temperature of the gas would not change for a

small volume change. This is reasonable if the internal energy of the gas does not depend upon the relative

distances between the particles. However, for a Van der Waal's gas, the change in temperature is zero. Thisnot

is because Van der Waal's model attempts to treat the gas from a more realistic point of view. As the relative

distance between molecule increases, the potential energy of the system changes - effectively changing the

location of the “bottom of the potential well". If the internal energy cannot change, and the effective potential

energy of the system “rises", then the thermal energy of the gas (the difference between the total energy and the

potential energy) must decrease. Thus the temperature of the gas must decrease. Notice that we can, in effect,

determine for a given change in volume for gas (even if we do not know the equation of state) provided.X any

we know , and for the gas.G ß G ß ß Z ß T XZ T "