The Thermodynamic Substance

8/10/2019 The Thermodynamic Substance

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2 The Thermodynamic Substance

The thermodynamic systems comprise of fluids, i.e. liquids and gases and a

thermodynamic process can be defined in terms of the fluid properties within a system. The

operational parts are excluded from thermodynamic analysis through a proper choice of theboundaries. A study of the behaviour of the working fluid (fluid properties) is, therefore

necessary for a thermodynamic analysis of energy and matter transformations. e consider

working substances at equilibrium state in form of fluid phases.

There are two points of view from which the behaviour of matter can be studied! the

microscopicand the macroscopic.

"rom the microscopicpoint of view, matter is composed of an extremely large number

of molecules, which are themselves built from atoms and are of complicated structure. "or

the simplest case of gases, it may be assumed that each molecule at a given instant has a

certain position, velocity, and energy, and for each molecule these change very frequently, asa result of collisions. The behaviour of the gas is described by summing up the behaviour of

each molecule. #uch a study is made in statistical thermodynamics.

$n classical thermodynamics, which is only concerned with the effects of the action of

many molecules, a macroscopic approachis adopted. %learly speaking, a certain quantity of

matter (water in a boiler or gas in a combustion engine) is considered without the events

occurring at the molecular level being taken into account. "or example, the macroscopic

quantity, pressure, is the average rate of change of momentum due to all the molecular

collisions which occur on a unit area. The effects of pressure can be felt and measured by

using, for example a pressure gauge. "or thermodynamic analysis the behaviour of matter

will be described in terms of macroscopic observable properties. &ngineeringthermodynamics uses the classical model. 'owever, sometimes we will use molecular picture

(microscopic model) for a better understanding of some phenomenon.

2.1 Quantity of Matter

The thermodynamic studies normally include the quantity of matter, e.g. the quantity

of water in a boiler or the amount of gas in the cylinder of a combustion engine. The quantity

of matter is characteried by mass m and number of molesn. The amount of matter of a

system is given by the mass enclosed within its boundary.

$n common language the massof a substance is also called its weight. ut they are

different conceptionally. eight is actually the force * with which a body of mass m will be

attracted from the surface of earth at a particular place having acceleration due to gravity g!

* + m g (.-)

The acceleration due to gravity is not a universal constant. $t depends on the place of

observation. At earths surface g + /.0- m s1. As the acceleration due to gravity depends on

altitude, the weight of a particular mass m will also vary with altitude.

The mass is determined through weighing by comparing with a standard mass. Thekilogramis the #$ ( #$ from the french #ystem $nternational ) unit of mass2 it is equal to the

mass of the international prototype of the kilogram mass of a lump of platinum1iridium.

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An alternative way to represent the amount of matter is as the number n of moles.The

mole is the amount of substance of a system which contains as many elementary entities as

there are atoms in 3.3- kilogram of %arbon1-. hen the mole is used, the elementary

species must be specified. These may be atoms, molecules, ions, electrons, other particles, or

specified groups of such particles.

&xamples of the use of mole!

- mol of 'contains about 4.3 -35'molecules, or -.366 -3

5' atoms.

- mol of 'g%lhas a mass of 67.30 g.

- mol of 'g8has a mass of 63-.-0 g.

- mol of e1has a mass of 960.43 g.

2.2 The Thermal Variables

"or a quantitative description of phases and of thermodynamic systems certain statevariables (properties) are needed. These properties should be measurable or computable from

other measured properties. The most common properties for fluid phases are temperature,

pressure and volume. These are known as thermal variables.

2.2.1 Volume

The volume : is the space occupied by a substance and is measured in cubic meters

(m;). The meter is the basic unit of length. Thus we write : + -m; if the system occupies - m;

of space. s >, is theunit of time and is defined as the duration of /-/45-773 periods of the radiation

corresponding to the transition between the hyperfine levels of the ground state of %aesium

1-55 atom. #ometimes instead of absolute volume, the term specific volume is used. #pecific

volume is the space occupied by unit mass of a substance and is measured in m;=kg.

=

V

m ?unit is m;=kg@

#imilarly the volume of one mole of a substance is called molar specific volume or simply

the molar volume and is also denoted by small .

m

VM

n

V==

?unit is m;=mol@

where ?unit is g=mol@ denotes the molar mass (molecular weight) of the substance, i.e. the

mass of - mol of substance. + m / n.

2.2.2 Pressure

Bressure is the force exerted by a system (fluid) on a unit area. The #$ unit of pressure

is defined aspressure + force = area

+ C = mD


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This composite unit is called > Bascal > and denoted by the symbol Ba

-Ba + -C m1

The pressure of the atmosphere is of the order of -39Ba. This shows that Bascal is

rather a small unit of pressure. $t is, therefore convenient to describe the pressure in terms of

kBa or Ba.

bar >, which nearly equals the pressure of one

atmosphere. The unit > bar > is defined as

- bar + -39Ba + -39Cm1+ -33 kBa.

The atmospheric pressure varies from region to region and is not constant throughout.

A standard atmospheric pressure is defined as

- standard atmosphere + - atm + -3-.59 kBa + -.3-59 bar

"igure .- shows two blocks of matter which have the same mass. They exert the same force

on the surface on which they are standing, but the narrow block exerts a higher pressure

because it exerts the force on a smaller area than the fatter block.

"igure .- ! %omparison of pressure exerted by the same force.

The instrument to measure the pressure is known as manometer. $f atmosphericpressure (which varies with altitude and weather) is measured the instrument is called

barometer. ost instruments indicate pressure relative to the atmospheric pressure pE,

whereas the pressure of a system is its pressure above ero, or relative to a perfect vacuum.

The pressure relative to the atmospheric pressure is calledgauge pressure (p*). The pressure

relative to a perfect vacuum is called absolute pressurepwhich is given by!

Absolute pressure + *auge pressure 8 Atmospheric pressure

p+ p*8 pE (.)

"igure . shows a few pressure measuring devices (manometers). "igure (a) showsan open u1tube manometer indicating gauge pressure, and "igure (b) shows an open u1tube

indicating vacuum, i.e. pressure below the atmospheric pressure. "igure (c) shows a closed u1

tube indicating absolute pressure.

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Evacuated

a)

p

h

pu

Hg

b)

p

h

pu

Hg

c)

p

h

Hg

"igure . ! #ome E1Tube anometers.

$f h is the difference in the heights of the fluid (mercury) columns in the two limbs of

the u1tube ?cf. "ig. (a) and "ig. (b)@, the density of the fluid (here, mercury) and g theacceleration due to gravity, then the gauge pressure p*is given by

p FA

m gA

A h gA

h gG = = = =

(.5)

where " is the force of the mercury column on the gas over an area A. "or mercury with a

density + -5.4F-35kg=m; a 743 mm (3.743 meter) column of mercury is equivalent to

-.3-5F-39C=mD + -.3-5 bar + - atm.

#ometimes the pressure is also expressed in kgf=m ( kg force per square meter ) or, as this

unit is large, in kgf=cm( also known as ata, atmosphere technical absolute ).

- standard atmosphere + - atm + -.355 kgf=cm

+ -.355 ata

2.2.3 Temperature

The temperature is a very familiar concept in everyday life as it is a measure of the

>hotness> or >coldness> of a body or fluid. ut it is quite difficult to give the concept a precise

definition. Temperature is associated with the ability to distinguish hot from cold. hen two

bodies at different temperatures are brought into contact, energy (heat) flows from the higher

temperature obGect (hot body) to the lower temperature obGect (cold body). After some time

they attain a common temperature and are then said to be in thermal equilibrium. 'ence, the

temperature may be defined as the property whose value is identical in the two systems whichare in thermal equilibrium.

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The temperature of a fluid is one of the properties of that fluid, along with pressure

and specific volume. $t is necessary to have a precise value of temperature in order to define

the state of a fluid. The basis for the temperature measurement is provided by the zeroth law

of thermodynamics: >hen a body A is in thermal equilibrium with a body , and also

separately with a body %, then and % will be in thermal equilibrium with each other>.

$n order to obtain a quantitative measure of temperature, a reference body, saythermometer, is used. A column of liquid inside the thermometer expands or contracts with

changes of temperature. The thermometric property is the expansibility of the liquid. hen a

thermometer is brought into contact with a body at a temperature higher (greater) than itself,

then heat is transferred from that body to the glass of the thermometer and from there to the

liquid inside the thermometer raising its temperature. The heat transfer stops when a thermal

equilibrium is reached. Cow the thermometer has the same temperature as that of the body

outside. The increase in temperature of the liquid results in its expansion. As the expansibility

of liquids is much higher than that of the glass, the expansion of glass may be neglected in

comparison to that of liquids. The extent of expansion of a liquid in a glass column is hence a

direct measure of the increase in the temperature of the liuid.

$t is necessary to have a common temperature scale for measurement. The most

common temperature scale is the %elsius scale. This uses two arbitrary fixed points, the

freeing point and the boiling point of water under standard atmospheric conditions, to define

3 3% and -33 3% respectively. "or a mercury thermometer, the difference in the length of

mercury column is divided in -33 equal parts. &ach part then represents - 3%.

This temperature scale is not satisfactory in many aspects. As each substance has a

different coefficient of thermal expansion this temperature scale will depend on the nature of

the liquid in the thermometer. &ven if the thermometers made of mercury and alcohol, with

linear scales, are made to agree at two temperatures, they will not agree at some intermediatetemperature. #econdly, the division in -33 equal parts is only applicable to a linear expansion

with temperature. #trictly, the coefficient of thermal expansion is not a linear function of

temperature. #uch a thermometer for temperature measurements is therefore not accurate.

Also the %elsius temperature scale shows negative values for temperatures which are below

the freeing point of water, e.g. the boiling point of liquid oxygen is 1-05H%. This seems

physically unsound as the other properties of state, namely pressure and volume have only

positive values. $t is therefore very much desirable to look for another scale for temperature.

An empirical temperature scale, which is also an absolute scale, is that of an ideal gas

thermometer which is described here in detail.

A schematic diagram of a constant volume gas thermometer is shown in "igure .5. A

small amount of gas is enclosed in a bulb which is connected via a capillary tube with one

limb of the mercury manometer. The other limb of the mercury manometer is open to the

atmosphere and can be moved vertically to adGust the mercury levels so that the mercury Gust

touches the reference mark on the capillary. The sensor is hereby the gas filled in the constant

volume bulb. The pressure in the bulb is measured through a height of the mercury column.

The thermometric property of this constant volume gas thermometer is the pressure of the

gas. $t changes in a characteristic way with the temperature of the bulb and is given by the

difference in level of the mercury column!

p+ pE8 g (.6)

where pEis the atmospheric pressure and the density of mercury.

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pu

V= const.

Mark

Hg tubin

z=p^

"igure .5 ! Iiagram showing the working principle of an ideal gas thermometer.

$f the pressure in the bulb is equal to the atmospheric pressure then + 3. #ince the

volume of the bulb is constant, a definite amount of gas (e.g. n mol) in the bulb under

thermal equilibrium conditions at a certain empirical temperature will result in a definite

value for the product (p), where = :=n. "or a definite temperature the value of (p) depends on

the nature of the gas. The value of can be varied by filling the bulb (of constant volume :)

with different amounts n of a gas. "igure .6 shows a plot of the values of (p) at a known

temperature ( + const.) against -= for different gases.

water gas

air

'elium

-=v

(pv)ig

pv

+ const.

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"igure .6 ! (p) versus -= plot for various gases.

$t is seen from the graph that although the readings of a constant volume gas

thermometer depend upon the nature of the gas, all gases indicate the same temperature, i.e.

the product (p), as -= is lowered and made to approach ero. This empirical experimental

finding is of great significance for defining a temperature scale which is based on physical

grounds. $t holds!

.const!p"!p"lim ig ==

3

-

for +const.

This relation describes an ideal gas. The instrument is called an ideal gas

thermometer. &ach ideal gas thermometer, irrespective of the gas filling indicates the same

value for (p)igand hence the same empirical temperature , if it is in thermal equilibrium

with a system. The value of (p)igcan be used to define an empirical temperature igof theideal gas thermometer

ig + % (p)ig (.9)

$t may be pointed out that the linear relation in &quation (.9) is chosen for the sake of

simplicity. Any other relationship betweenpand , e.g. quadratic could also be taken.

This definition has many advantages. The first is, that the temperature can not attain

negative values aspand have positive values only. #econdly, we need only one fixed point to

determine the constant % and hence to define the temperature scale. Thirdly, this temperaturescale does not depend on the nature of the gas (filling medium).

The scale is established by arbitrarily selecting a fixed point. The temperature of a

mixture of pure water, water vapor, and ice in thermal equilibrium is selected for this

purpose. #uch a mixture exists at only one temperature (called the triple point), and the

temperature of this mixture provides an easily reproducible standard. The gas thermometer is

brought in thermal equilibrium with pure water at its triple point, the corresponding value of

(p)igis determined. The value of (p)igis determined by measuring the values of (p) at different(i.e. with different fillings of a gas) and then extrapolating these to -= + 3. $t holds then

mol

#m

$p ig %&trig %&tr .7-

-)( ,., ==

(.4)

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$t is agreed that the triple point of water shall be 75.-433 J. 'ere J is used for

>Jelvin>, which is the unit of temperature in this scale (of ideal gas thermometer). The value

of % follows

% + 3.-37

mol'

#m (.7)

and the unit of temperature, Jelvin!

-

75-4

' t r & %

ig

= ,

. (.0)

'ence - Jelvin is one 75.-4 th part of the empirical temperature of ideal gas thermometer at

the triple point of water.

y an international agreement in -/96 the value of 75.-4 was selected to make the

unit of the new scale the same sie as that of the %elsius (formally %entigrade) scale. #ince

the ordinary ice point (freeing point of water) is 3.3- J lower than the triple point of water itfollows that 3H%+75.-9 J.

To measure the temperature of a system the gas thermometer is brought in thermal

equilibrium with the system and the value of (p)igis determined. The ideal gas thermometer

temperature of the system is then

mol

#mp

#m

mol'p$ igigig )(-37.3)( ==

This empirical ideal gas thermometer temperature satisfies the conditions of a

thermodynamic state property. Kater on in the course of our studies of entropy an absolute

thermodynamic temperaturescale will be defined. "ortunately it proves to be identical with

the ideal gas scale. $t will be shown that the point which we have defined as the absolute ero

of temperature is not an arbitrary point. At present we take it for granted that the

thermodynamic scale of temperature is identical with the ideal gas thermometer temperature

scale. e write!

ig + T (./)

The thermodynamic temperature T is therefore a measurable quantity. The &quation

(.9) may then be written as

pig + Lig + LT (.-3)

where L + -=% + -=3.-37+0.5-9 Cm=(molJ) and is called the molar gas constant (universal

gas constant).

$n everyday life the temperature is expressed in %elcius scale. The temperature in

%elcius (H%) is simply related to the thermodynamic temperature (via

t( in H% ) + (( in J ) 1 T3 (.--)

where T3 + 75.-9 J represents the ordinary ice point. The %elcius temperature scale

employs a degree of the same magnitude as that of the thermodynamic temperature, but itsero point is shifted.

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A second absolute temperature scale, used with the old &nglish system, is the )ankine

scale, defined by

((Lankins) + /=9 ((Jelvins)

This makes the water triple1point temperature correspond to 6/-.4/ L. The corresponding

relative temperature scale is "ahrenheit scale. The temperature differences are identical indegrees "ahrenheit (H") and Lankins (L), but a level of 3H" corresponds to 69/.47 L.

$t follows, that the relationship between the %elsius and "ahrenheit scale is

H" + H% F (/=9)85 (.-)

The temperature of the ice pointis about 5H" and that of thesteam point about -H".

The measurement of temperature with the help of ideal gas thermometer is a tedious

process ?extrapolation of many measured (p) to (p)ig@. The temperatures are, therefore

generally measured with other types of thermometers such as thermocouple and platinum

resistance thermometers. Caturally, these have to be calibrated against absolute temperature(measured with ideal gas thermometers). "or this purpose a practical temperature scale

($nternational Temperature #cale1$T#) was adopted first of all in -/7 and then revised from

time to time (latest revision -//3). The $T# is defined by!

1- Assigning *alues to certain accurately reproducible temperatures such as boiling and melting points.

2- +pecifying the type of thermometer to be used in each range of the scale.

3- +pecifying the interpolation formula to be used for each thermometer between the assigned *alues.

This scale is a practical scale and it is based on a number of fixed and easily reproducible

points which are assigned definite numerical values of temperature, and on specified

formulae which relate temperature to the readings on certain temperature measuringinstruments. The scale was so defined that it conforms closely to the ideal gas temperature

scale. The temperature interval from the oxygen point to the gold point is divided into three

main parts, as given below in $nternational Temperature #cale of -/40 ($T#140).

Table .-! Temperatures of "ixed Boints

Temperature H%

Cormal boiling point of oxygen 1-0./7

Triple point of water ( #tandard ) 8 3.3-

Cormal boiling point of water -33.33

Cormal boiling point of sulphur 666.43

(Cormal melting point of inc1suggested as analternative to the sulphur point) 6-/.93

Cormal melting point of antimony 453.93

Cormal melting point of silver /43.03

Cormal melting point of gold -345.33

(a) "rom 3 to 443H%!

A platinum resistance thermometer with a platinum wire whose diameter must lie between3.39 and 3.3 mm is used, and the temperature is given by equation

) , )-(-8At8t)

/


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where the constants)-,A0 andare computed by measurements at ice point, steam point and

sulphur point. The temperature t is in H%.

(b) "rom 1-/3 to 3H%!

The same platinum resistance thermometer is used, and the temperature is given by

)+)-?-8At8t8 $(t1-33)t1@

where)-0 Aandare the same as before, and $is determined from a measurement at the

oxygen point.

(c) "rom 443 to -345H%!

A thermocouple, one wire of which is made of platinum and the other of an alloy of /3M

platinum and -3M rhodium, is used with one Gunction at 3H%. The temperature is given by the

formula

, a 2 bt 2 ct

where a0band care computed from measurements at the antimony point, silver point, and

gold point. The diameter of each wire of the thermocouple must lie between 3.59 and

3.49mm

An optical method is adopted for measuring temperatures higher than the gold point.

The intensity of radiation of any convenient wavelength is compared with the intensity of

radiation of the same wavelength emitted by a black body at the gold point. The temperature

is then determined with the help of Blancks law of radiation.

2.3 Pure Substances

The practical application of thermodynamics lies in the working of heat power plants

or refrigerating machineryetc., which operate with the help of some working fluidorsystem"thermodynamic system!. The processes carried out in these applications are analysed by

considering the effects produced on thesystems. $t is therefore, necessary to have information

on the properties of these working fluids or systems. $n many cases the working fluid is a

pure substancewhich may change its phase (A phase is any homogeneous part of a system

that is physically distinct). The important characteristic of a pure substance is that its

chemical composition is same throughout its mass. $t may exist in one or more phases. Thus,

a system consisting of a mixture of various phases of water, namely water and ice or water

and steam is a pure substance. A system consisting of oxygen as a vapor, a liquid, or a solid

or a combination of these is also a pure substance. #ometimes air, even though made up of

several gases, namely nitrogen, oxygen, carbon dioxide etc. and thus a gas mixture, is

considered and mathematically treated as a pure substance as long as there is no change of

phase. This is due to the fact that under normal conditions its composition remains constant

and that none of the constituent gases undergoes a chemical change during a thermodynamic

process. "or many applications it will be treated like a pure substance. #trictly speaking, this

is not true. e should rather say that air exhibits the characteristics of a pure substance.

Bure substances are used as the working medium in many energy transformation

devices. ater is used as the working fluid in steam power plants. $n heat pumps and

refrigerators the use of pure organic fluids is very common. ater vapor is used as energy

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carrier for the heating of devices in certain matter conversion processes. $t is, therefore very

much necessary to learn about the properties of pure substances.

2.3.1 The Thermal Properties and their Technical Use

&xperience has shown that there is an interrelation between the various properties of a

substance. Thus, there is a relationship between the pressure, specific volume, and

temperature, which may be expressed by

"(p,,()+3

This mathematical relationship between the thermal state variables pressure (p), specific

volume () and absolute temperature (() is conventionally known as the thermal euation of

state. $t may also be expressed as

p+p((,)+ ((,p)

(+ ((p,)

The relation between the three variables may be a simple one or very complicated

depending on the substance and the range of observation. $n general, the properties of every

pure substance may be represented by an equation of state.

2.3.2 Thermal Properties of Gases

e have seen in section ..5 that the products (p) of all real gases at giventemperature approach the same value, as the pressure approaches ero (-= approaches ero).

At the state of ero pressure , i.e. at vanishing density (3

-

) all real gases behave in a

similar manner. e can characterie this identical limiting behaviour of gases as ideal

beha*iour and call the state of ero pressure as an ideal gas state.

The behaviour of real gases at the ideal state (state of p 3) suggests the concept of an

ideal gas "perfect gas! 1 a hypothetical gas that will beha*e in an ideal manner at all

pressures. Cote that the ideal (perfect) gas is not real2 it is an imaginary gas that displays at

all pressures the same behaviour which the real gases display only at the state of p 3. "or aperfect (ideal) gas holds

(p)ig+ LT (.-5)

where L + 0.5-9 Cm=(molJ), is the universal gas constant and , the molar volume. #o in the

range of low pressures the molar density (

-

) increases linearly with pressure at a constant

temperature. These constant temperature curves are called isotherm ( isotherm + same

temperature ).

The isotherms appear as hyperbola in apdiagram. "igure .9 shows the p1 diagram

for a perfect gas.

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bar

vm /mol30 0.05 0.10 0.20

T= 273.1 !373.1 !

73.1 !

"73.1 !

1.5

1.0

0.5

0.1

"igure .9 !p1 Iiagram of an ideal gas.

$f is used for representing the specific volume in &quation (.-5) then it becomes

(p)ig+ (

)

M )T (.-6)

where is the molecular weight of the substance. The p1 diagram is not universal but

individual (different) for every gas. This difference is not due to the fact that the function is

different for every gas. $t is because the factor (

)

M ) is different for every gas and hence the

individuality. This factor ()M ) is called the specific gas constant.

A study of thep1 diagram explains why the power plants use gases as the working

medium. The productphas the dimensions of energy, namely Cm=mol or Cm=kg. y heating

at a constant volume the temperature and the pressure (c.f. "ig. .9) will increase. Thus the

energy supplied in the form of heat will be stored in gas (in form of a higher pressure). This

can be converted partly into work through the relaxation of pressure (decreasing the pressure)

and may be used, e.g. to push back a piston. This forms the basis for a spark ignition internal

combustion engine (


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"igure .4 ! $dealied


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"igure .7 ! %ondensation of a pure substance ( schematic t1: Iiagram ).

This process can be reversed if we place the cylinder on a hot plate . The temperature of the

liquid rises from state 9 to 6 from 3 o% to -33 o% with a small increase of volume. At 6 the

water begins to boil ( building of vapor bubbles 1 evaporation ). $f more heat is supplied the

process goes on further to state , where the last drop of liquid disappears. $f the supply of

heat continues the temperature as well as the volume of water vapor increase and we arrive at

state -.

At state 6 a liquid reaches the saturation temperature and is called saturated liuid.

#imilarly, at state where all the saturated liquid is converted to vapor, the vapor at saturationtemperature is called thesaturated *apor. etween the saturated liquid and saturated vapor

states ( between and 6 ), the fluidwill be at saturated temperature but will consist of a

proportion of liquid and a proportion of vapor in equilibrium. This condition is known as wet

*apor.

The specific volume in wet vapor range, i.e. in the coexistence range of liquid and

gas, depends not only on the saturation temperature=pressure but also on the relative masses

of vapor and liquid. The relative mass of vapor and liquid is expressed in terms of *apor

content3( mass fraction of the *apor also called *apor uality or dryness fraction !!

3 m

m m=

+

= =

= = =(.-9)

where m=is the mass of the liquid and m==is the mass of the vapor at saturation. The total mass

m+ m=8 m==.


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The volume per unit mass ( specific volume ) is

=======

==

)-( 33m

m

m

m

m

V+=+==

+=

83(//

4/

) (.-4)

&quation (.-4) states that the a*erage specific *olume of the two phase mi3ture0 0 is a

linear function of 3, and equals the specific volume of the liquid plus the volume increase

upon vaporiation (//4 /)times the mass fraction of the system vaporied. The vapor content ,

3, can be expressed explicitly by

===

=

=3

(.-7)

The temperature between the states and 6, i.e. the temperature at which both liquidand vapor coexist depends on pressure. $t has higher values at higher pressures and smaller

values at smaller pressures. This temperature is calledsaturation temperatureor boiling point

temperatureand is defined as that temperature at which vaporiation takes place at a given

pressure. The corresponding pressure is called the saturation "*apor! pressure. "or a pure

substance there is a definite relation between the saturated temperature and saturated

pressure. This relation between temperature and pressure during evaporation and

condensation is given by the so called *apor pressure cur*e( vapor pressure line ). "igure .0

shows the vapor pressure curve for water and some other pure substances in p1t diagram

above 3 o%.

The dependence of boiling temperature or condensation temperature on pressure, i.e.the shape of vapor pressure curve is very important from technical point of view and forms

the basis of the steam power plants, heat pumps and refrigerators. e will have a brief look

into these. A detailed study is left for next semester.

bar

200

150

100

50

0

0 100 200 300#$t

H %2

&H3

$%2'12

-9


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"igure .0 ! :apor pressure curves for some pure substances.

A simple steam power plant working on the vapor power cycle is schematically

shown in "igure ./. $n its simplest version the steam power plant consists of four elements1 a

boiler, a turbine, a condenser, and a pump.

'eat 5.

is tranferred from an external source ( furnace, where fuel is continuously burnt ) to

water in the boiler to produce steam having a pressure of 33 bar and at nearly 993 o%

(superheated steam2 the boiling point of water at 33 bar is 549.0- o%). Iuring this process

the water takes up energy which is stored in the form of high pressure and high temperature.

'ence the steam leaving the boiler has the potential (capacity) for doing work and reGecting

(giving) heat.

"igure ./ ! #chematic diagram of a simple steam power plant.

The high pressure, high temperature steam leaving the boiler expands in the turbine ( to a

pressure 3.364 bar ) to produce shaft work B Twhich in turn produces electric current via a

generator. The low end pressure at the turbine corresponds to a condensation temperature of

53o

% according to the vapor pressure curve of water. #o the vapor can reGect heat5

.

3when itis brought in contact with a cold surrounding ( condenser, where cooling water at -9 o%

circulates ). 'ereby it is condensed completely. Then the liquid water is pumped back to the

boiler (pressure level 33 bar) with the help of a condensate pump by supplying energy B#Bin

the form of work. 'ere one should note that the working of a steam power plant is possible

only because of the characteristic ( namely steep ) vapor pressure curve of water.

The steam power plant works only because the different pressures at the entry and exit

of the turbine correspond to technically attainable temperatures (993 H% and 53 H%) for the

heat supply and the heat reGection. A flat vapor pressure curve in comparison to that of water

would be unsuitable for the technical realiation of the power plant cycle. ecause then after

the expansion in the turbine, the vapor will attain a saturation Temperature of 53 H% at a muchhigher pressure. As the expansion only up to a relatively higher value for pressure would be

possible, the obtainable work will be smaller (lower work production). Also in the case of a

flat vapor pressure curve the saturation temperature corresponding to desired high pressure at

BTB

#B

5

3

-

turbinepump

condenser

boiler

-4


17/51

the entrance of the turbine would also be very high and difficult to attain. The quantitative p1t

behaviour of water is therefore technically very much important for the functioning of steam

power plants.


18/51

a suitable temperature. A refrigerator and a heat pump are thus distinguished only by the

desired effect0 i.e. cooling or heating.

2.3.! Thermal Properties in the entire state re"ion

Till now our interest has been confined to fluid region (liquid 1 vapor region) for the

simple reason that the working substance in power cycles etc. is a fluid. e saw the

importance of vapor pressure curve of water for its use as working fluid in steam power plant.

e now look at the behaviour of pure substances over the entire range of its existence. e

study the properties of the most common substance water. %onsider a unit mass of ice (water

in solid state) at 13 H% and - atm (-.3-59 bar) contained in a cylinder and piston. Ket the

ice be heated slowly so that its temperature is always uniform. $f the pressure is held constant

(isobaric process) and heat is supplied continuously, the following changes in the mass of

water are observed!

"igure .--! 'eating of '< at constant pressure

(a) The temperature of ice rises and approaches 3 H% as a limit. There will be smallincrease in volume.

(b)


19/51

$f the heating of ice at 13 H% to steam at 93 H% were done at a constant pressure of bar

similar regimes of heating would have been obtained with similar saturation states.

$f the initial pressure of the ice at 13 H% is 3.43 kBa, heat transfer to the ice first results in

an increase in temperature to 1-3 H%. At this point, however, the ice would pass directly from

the solid phase to the vapor phase in the process known as sublimation. "urther heat transfer

would result in superheating of the vapor.

$f we start with ice with an initial pressure of 3.4--5 kBa and a temperature of 13 H% and

heat it slowly. Then as a result of heat transfer the temperature will increase until it reaches

3.3- H%. At this point if further heat is supplied some of the ice will become vapor and some

will become liquid, because at this point it is possible to have the three phases in equilibrium

(triple point).

2.3.!.1 Pressure#temperature $ia"ram

The state changes of pure water, upon slow heating at different constant pressures, are

plotted on p1(coordinates in "igure .-. This diagram shows how the solid, liquid and

vapor phases may exist together in equilibrium.

$ritical (oint

ri(le (oint* +

$ ,

E -

H

em(erature

i0uid

(ase

olid(ase

a(or(ase

"igure .- ! Bressure1temperature diagram for a substance like water.

The curve passing through the melting points at different pressures is called thefusion

cur*e " fusion line !, and the curve passing the vaporiation or condensation temperatures at

different pressure is called the *aporization cur*e "line!.Thesublimation cur*e "line!gives

the vapor pressure of solid at different temperatures. The fusion curve, the vaporiation

curve, and the sublimation curve meet at the triple point "triple phase state!.

Along the fusion curve, solid and liquid phases are in equilibrium, along the

sublimation curve solid and vapor phases are in equilibrium and along the vaporiation curve

the liquid and the vapor phases are in equilibrium. The intersection of these three curves at

the triple point in a p1T diagram represents the conditions under which three phases can

-/


20/51

coexist in equilibrium. These conditions are represented by a point only on a pressure4

temperature diagram6 on other property diagrams they are represented by a line or an area .

$f three phases of a pure substance exist together in a system, only one value of pressure and

only one value of temperature are possible. Thestate of each phasepresent in the system is

fixed. 'owever, more information is required (namely, the proportions of the phases present)

to determine thestate of the system(mi3ture of liuid 2 *apor) at the triple point temperature

and pressure. "or example, the specific volume of the system with - per cent vapor content ismuch less than that of the system with -3 per cent vapor content.

The triple point of water is at 3.4--5 kBa and 75.-4 J. The vaporiation curve ends

at the critical point because there is no distinct change from the liquid phase to the vapor

phase above the critical point. The liquid and vapor are indistinguishable from each other.

There is no sudden change from the liquid state to the vapor state. The critical pressure and

the critical temperature are the highest pressure and temperature at which distinguisable

liquid and vapor phases can coexist together in equilibrium.

A look at the "igure .- shows that a solid at the state A will pass directly to vapor

at state if it is heated at constant pressure (which is less than the triple point pressure). At a

constant pressure above the triple point pressure ( from state & ) the substance first passes

from the solid to the liquid state at a definite temperature and then passes to vapor state " at a

higher temperature. The constant pressure line %1I passes through the triple point. At a

pressure above the point the change from the state * to ' is not distinguishable.

ater is one of a very few substances that expand on freeing. ost substances

contract on freeing and the freeing temperature increases as the pressure increases, so that

thesolid4liuid saturation cur*e or fusion "melting! lineappears as in "igure .-5, i.e. the

slope of the fusion line is positive.


21/51

"igure .-5 ! Bressure1temperature diagram for a substance that contracts on freeing.

2.3.!.2 Pressure#%olume $ia"ram

The pressure1specific volume diagram for a substance that contracts on freeing is

shown in "igure .-6. The line h4lis a plot of the specific volume of saturated liquid versuspressure and is called the saturated liuid line. As the pressure increases (and consequently

the saturation temperature increases), the specific volume of saturated liquid increases

slightly. The region immediately to the left of the saturated liquid line represents compressed

or subcooled liquid states. The line l4g is a plot of the specific volume of saturated vapor

versus pressure and is called thesaturated *apor line. As the pressure increases, the specific

volume of saturated vapor decreases. The region immediately to the right of the saturated

vapor line represents superheated vapor states. #olid, liquid , and vapor states are represented

by points in three different regions on a pressure1volume diagram Gust as they are on a

pressure1temperature diagram. 'owever, mixtures of two phases are represented by areas on

ap1 diagram while they are represented by lines on a p1(diagram. ixtures of three phases

are represented by a line on ap1diagram (by a point on ap1(diagram).

-


22/51

"igure .-6 ! Bressure1volume diagram for a substance that contracts on freeing.

The area beneath the dome formed by the saturated liquid line h4land the part of the

saturated vapor line l4gdown to the line h4irepresents mixtures of liquid and vapor. All the

states represented by this area on thep1 diagram are represented by the line on ap1(diagram

between the triple point and the critical point.

$n "igure .-6 saturated solid states at pressures higher than the triple phase pressure

are represented by points along the line r4s, called thesaturated solid line "cur*e!. The liquid

at the freeing temperature (saturated liquid) is represented by points along the line h4m,

called thefreezing liuid lineto avoid the confusion with thesaturated liuid line(liquid at

the boiling point). ixtures of solid and liquid are represented by points in the region

bounded by lines r4s0 h4mand r4h. All points within this region lie on the fusion line on thep1

(diagram. Kine u4wrepresents states of minimum specific volume for the solid at various

pressures, that is, the solid can not be compressed further. elow the triple phase pressure,

liquid can not exist. This area beneath the line u4r4h4i4grepresents solid vapor mixtures. The

states in this region ofp1diagram lie on the sublimation line of ap1(diagram. The line u4r

represents saturated solid states and the line i4grepresents saturated vapor states. The line r4h4ion thep1 diagram represents the states in which solid, liquid, and vapor can exist together.

The triple point state is thus represented by a line on a p1 diagram as against a point on ap1(

diagram. Boint hrepresents the saturated liquid at the triple phase pressure, point irepresents


23/51

saturated vapor and the point rrepresents saturated solid at the same pressure. The points

between rand hrepresent solid1liquid, solid1vapor or solid1liquid1vapor mixtures. The points

between hand irepresent solid1vapor, liquid1vapor or solid1liquid1vapor mixtures. The phase

transition from liquid to solid at constant pressure is accompanied by a decrease in specific

volume. $t is to be pointed out again that at constant pressure the phase transition from liquid

to solid water (ice) is accompanied by an increase of specific volume.

The isotherms (lines of constant temperature) are also shown in "igure .-6 as broken

lines. $n all two phase regions the constant temperature lines coincide with constant pressure

lines.

$t is to be kept in mind that the specific volume scale is greatly distorted in p1

diagram. "or all substances the change in specific volume between points hand i is many

times greater than between r and hand also the critical specific volume is only slightly

greater than at ror h. To emphasie this, only parts of the p1v diagrams of carbon dioxide

and water are shown in "igure .-9 with true linear scales. Esually the volume axis is plotted

logarithmically.

"igure .-9 !p1 diagrams to true linear scale for carbon dioxide and water.

2.3.!.3 p##TSurfaces

The relationship between pressure, specific volume and temperature can be

represented in a three1dimensional diagram. $t is conventional to assign thep11(surfaces, the

vertical ordinate as pressure axis while the volume and temperature axes lie in horiontal

plane. "igure .-4 shows the p11( diagram for water (a substance that expands on freeing)

and "igure .-7 for other substances (substances in which the specific volume decreases

during freeing). Thep1( andp1 proGections are also shown in these diagrams. Thep1( and

p1 diagrams have been discussed in detail.


24/51

The temperature, pressure and specific volume at this point are called the critical temperature

((c), critical pressure(pc) and critical *olume "c! respecti*ely.

$t will be noted that there are six distinct regions in each of the p11( surfaces.

#ingle Bhase Legion!

(a)solid! :olume is very small2 temperature is below triple point2 pressure has

any value.

(b) liuid: :olume is not much different from that of solids2 temperature is higher

than triple point2 pressure has any value.

(c) *apor: :olume is large2 temperature is below critical2 pressure is high or low.

7f the temperature is abo*e criticalthe state is referred to as gas.

Two Bhase Legions!

(a)solid 4 liuid These are regions (between two single phases) where

(b) liuid 4 *apor the two phases can exist simultaneously in quilibrium.

(c)solid 4 *apor

A p11( diagram provides a better picture of the relationship among pressure, specific

volume and temeperature because here each point represents only one euilibrium state.

:arious regions of a p11( diagram overlap on a p1 proGection. Any given point on a p1

diagram can represent a solid, a solid1liquid, a liquid or a liquid1vapor mixture depending on

the temperature

6


25/51

"igure .-4 ! Bressure1volume1temperature surface for a substance that expands on freeing

(e.g. water).

9


26/51

"igure .-7 ! Bressure1volume1temperature surface for a substance that contracts on freeing.

2.3.& luid Models

A quantitative description of the properties of fluids is necessary for quantitative

thermodynamic analysis. The properties of fluids may be described by using fluid models.The properties of pure substances are arranged in the form of tables as functions of pressure

and temperature. #eparate tables are provided to give the properties of substances in the

saturation states and in the liquid and vapor phases. The properties of water are arranged in

the so calledsteam tables. Tables .- and . give the properties of each phase in the vapor1

liquid saturation region, namely saturated liquid water and saturated steam. hen a liquid

and its vapor are in equilibrium at certain temperature and pressure, only the pressure or the

temperature is sufficient to identify the saturation state. $f the temperature is given, the

saturation pressure gets fixed, or if the pressure is given, the temperature gets fixed. "or

saturated liquid or the saturated vapor only one property is required to be known to fix up the

state. #team Tables .- and . give besides thermal properties temperature, pressure and

specific volume other properties internal energy, enthalpy and entropy too, which will be

introduced in a later chapter. "rom Table .- one can read the saturation pressure and the

specific volume of saturated liquid and saturated vapor for a particular temperature. #imilarly,

4


27/51

Table . may be used to read other saturation properties at a given pressure. $f data are

required for intermediate temperatures or pressures, linear interpolation is done.

Table .5 gives the values of the properties (volume, internal energy, enthalpy and

entropy) of superheated vapor for each tabulated pair of values of pressure and temperature,

both of which are now independent. Cow the interpolation or extrapolation is to be done for

pairs of values of pressure and temperature. Table .6 lists the properties of subcooled water(compressed liquid water ). Table .9 lists the ice1vapor saturation (sublimation) values.

#imilar tables for other technically important substances are also available.

'owever, now with the development of computer based calculation it is more

convenient to use analytical equations, e.g. equation of state in place of these table. The

model of ideal gas (perfect=ideal gas equation) is used for the calculation of thermal

properties of gases at low densities (pressures up to a few bars). This model is based on the

universal behaviour of gases irrespective of their nature at low densities.

(p)ig+ L (

or

(pV)ig+ nL ( (.-0)

(pV)ig+ m(

)

M ) ( (.-/)

This model is quite accurate and is used for describing the behaviour of gases at low

pressures. A real gas obeys ideal gas law at temperatures that are high relative to the critical

temperature and at pressures that are low relative to the critical pressure. The gas oxygen,

nitrogen, air or hydrogen may be treated as ideal gases at ordinary temperature and pressure.

The ideal gas equation of statep+ L(is valid only under the following assumptions!1 the volume occupied by the molecules themselves is negligibly small compared to

the volume of the gas.

1 there is no attraction between the molecules of the gas.

#o at low pressures and high temperatures, where these conditions are fulfilled the real gases

follow ideal gas equation. ut as pressure increases, the intermolecular forces of attraction

and repulsion increase, and also the volume of the molecules becomes appreciable compared

to the total gas volume. Then the real gases deviate considerably from the ideal gas equation.

To describe the behaviour of gases at higher pressures some other equation of states

may be used. A well known equation is :an der aals &quation, which is given below!

)(ba

p =+ ))((

(.3)

where NaO and NbO are the constants specific to the substance. The coefficient ais introduced to

account for the existence of mutual attraction between the molecules. The coefficient b

accounts for the finite sie of molecules.

Leal gases conform more closely with :an der aals equation of state than the ideal

gas equation of state, particularly at higher pressures. A widely used equation of state withgood accuracy is the Ledlich1Jwong &quation!

7


28/51

)(-

b(

a

b

)(p

+

=

(.-)

The constant aand bare evaluated from the critical data.

a ) (

p

c

c

=3679 9. .

(.)

b )(

p

c

c

=3 3044.

(.5)

Another widely used equation for describing the properties of gases is the virial

equation of state!

....;>D>>- ++++= p8p$p)(

p

(.6)

or an alternative expression

...;

-

++++=

8$

)(

p

(.9)

,%, , % etc. are called virial coefficients. and are called second virial coefficient, %

and % are called third virial coefficients, and so on. "or a given gas, these coefficients are

functions of temperature only. The ratio )(

p

is called the compressibility factor z. "or an

ideal gasz + -. The magnitude of zfor a certain gas at a particular pressure and temperature

gives an indication of the extent of deviation of gas from the ideal gas behaviour. The virial

equations (.6) and (.9) may also be written as!

z p $ p 8 p= + + + +- > > D > ; .... (.4)

or

...;- ++++=

8$z

(.7)

The terms...

;,,

8$

etc. arise on account of molecular interactions. $f no such interactions

exist ( e.g. at very low pressures ) then + 3, % + 3 etc. 'ence z+ -, i.e. p+ L(.

A similar and accurate model for liquid region does not exist. "or calculation the fluid

model of ideal liquid is used!

il+ const.

According to this model (ideal liquid model) the volume (molar or specific) of a

liquid is supposed to be constant. The small dependence of volume on temperature and

0


29/51

pressure is neglected. "igure .-0 shows the t1 diagram of water. e see that the specific

volume is almost independent of temperature and pressure up to 33 H%. The model of ideal

liquid will not describe the real behaviour of liquids exactly but may be used for a reasonable

and good approximation.

500

400

300

200

100

0

0 5 10cm /gv

#$

3

p= 244 bar

p= 144 bar

p= 2 bar

p= 1 bar

+ *

critical (oint

boiling li0uid

saturated va(or

"igure .-0 ! t1 diagram of water.

The vapor pressure curve of pure substances in a given temperature region may be

described by an equation of form!

ps(() + p3exp ?A(-1To=()@ (.0)

wherepsis the saturated vapor pressure at temperature (2 p3, T3and A are three substance

specific constants. According to this equation a log(p) versus -=(plot should be linear. $n

"igure .-/ log(p) is plotted against inverse Temperature ?see the logrithmic scale for p@ for

some substances and we see a linear relationship. The critical temperature and triple point

temperature for the substances are also indicated. There are of course some substances which

do not show linear behaviour.

/


30/51

1/T

'12

H %

H$%

2&2

2

2

375#$

31#$

112#$6157#$

1000

100

10

1

0.1

0.01

0.001

0.0001

0 0.2 0.4 0.6 0.8

bar

1/!

6254#$

62#$

6214#$

4#$

68#$

"igure .-/ ! :apor pressure curve of some substances in logpversus -=(diagram.

2.! Mi'tures

Law materials are generally found in nature as mixtures. Bure substances as well

mixtures with desired properties are produced through appropriate material transformations

of substances available in nature. ixtures of desired properties may also be produced by

mixing the pure substances in appropriate ratios. "or example, the useful product ammonia is

produced by mixing hydrogen and nitrogen under suitable conditions. $n many other

engineering applications we encounter thermodynamic systems, which are mixtures of pure

substances. The mixtures are sometimes referred to as solutions, particularly a solid dissolved

in liquid. A knowledge of the qualitative and quantitative behaviour of mixtures is thus

necessary for the thermodynamic analysis of the material transformation.

A mixture is defined as any collection of molecules, ions, electrons. &ach of these is

called a species (or a constituent of the mixture) if it is distinguishable from the others by

virtue of its chemical structure. $n reacting mixtures the amount of each constituent can not

necessarily be varied independently. $n nonreacting mixtures each and every constituent can

53


31/51

be varied independently and every constituent is also a component. %omponents are those

constituents the amount of which can be independently varied. y varying the amount of

components the composition may be varied. The composition of a mixture is expressed in

various forms which are described below.

Mass fraction(The ratio of the mass of a constituent a to that of the total mixture is known as the

mass fraction( wa) of the constituent. "or instance, the mass fraction of a component a in a

mixture is!

w m

ma

a

(

=

where mT+ ma8 mb8 mc8 ....(./)

$f wais multiplied by -33 we get weight per cent. The sum of all ws is equal to -, i.e.

wi+ -.

Mole fraction(

The mole fraction 3aof a component a is the amount of a expressed as a fraction of

the total amount of molecules ( i.e. moles ). $t is the ratio of the number of moles of a

constituent to the total number of moles in the mixture.

(

aa

n

n3 =

nT+ total number of moles + na 8 nb 8 nc8.. (.53)

$f3ais multiplied by -33 we get mol per cent. The sum of all3s is equal to -, i.e.

3i+ -.

Volume fraction(

The ratio of the volume occupied by a constituent if theoretically assumed to be

separated (at the pressure and temperature of the mixture) to the total volume of the mixture

is known as *olume fraction a.

(

aa

V

V=

where VT+ Va 8 Vb8 ... (.5-)

$f ais multiplied by -33 we get volume per cent. The sum of all s is equal to -, i.e.

i+ -.

The mass fraction and the mole fraction are related to each other as under!

=

==

ii

ii

ii

ii

i

ii

3M

3M

nM

nM

m

mw

(.5)

5-


32/51

===

i

i

i

i

i

i

i

i

i

ii

M

w

M

w

M

m

M

m

n

n3

(.55)

2.!.1 The Thermal Properties and their Technical )pplications

The mixtures undergo changes when they are heated and cooled or when their

compositions are changed. Iuring heating or cooling phase changes may also take place. e

will first see, how the intensive phase properties (also called the degrees of freedom) are

related to the number of phases coexisting in equilibrium.

2.!.2 The Phase *ule

e have seen in our studies of the properties of pure substances that a system

consisting of a single component (for which we write %+-), when two phases are present

(e.g., liquid and vapor phases2 we write B+) only pressure or temperature could be changed

independently. "or example, for reading a property (say the molar volume) from the steam

table in a two phase system we needed only one state property (temperature or pressure). e

say that such a system has only one degree of freedom (variance "+-). $n saturated vapor1

liquid range we could change only one property (t or p) and the other was automatically

fixed. #imilarly, at the triple point of a pure substance where three phases exist

simultaneously, we could not change any property without disturbing the equilibrium

between the three phases. The three phase equilibrium is given by a single fixed value of

temperature and of pressure (Co degree of freedom for this system2 variance of the system is

ero). "or a single phase one component system both the temperature and the pressure could

be changed independently.

e write here the general relation between the degrees of freedom of a system

(variance, "), the number of components (%) and the number of phases at equilibrium (B) for

a system of any composition!

" + % 1 B 8 (.56)

This is the phase rule applicable to non reacting systems. $t was first formulated by *ibbs.

"or systems of two components, i.e. for a binary mixture, %+ and hence " + 6 1 B.

$n a single phase region this system has three degrees of freedom i.e. its temperature,

pressure and composition (e.g. mole fraction) can be varied independently. e therefore note

that the properties of a fluid mixture depend not only on the pressure and temperature but also

on its composition.

, (t, p, x) (.59)

where 3 is the mole fraction of say component no. . The dependence of on the mole

fraction of component - (3-) is already taken into account by considering 3 because the

relation3-+ (- 13) holds.

A similar relationship for specific volume is

5


33/51

, (t,p, w) (.54)

The thermal properties of fluid mixtures in the two phase region (liquid1vapor region)

are very much important because many operations performed in chemical, petroleum, and

related industries involve the contact between phases. The analysis or design of processes

such as distillation, liquid1liquid extraction requires an understanding of principles of phase

equilibrium and an availability of equilibrium data.

"or a binary system in a two phase region the number of degrees of freedom i.e. the

number of intensive phase properties is ( 1 8 + ). #o if the composition and pressure

are fixed, the temperature is automatically fixed.

Cow we look at the vapor1liquid equilibrium of a binary system by considering a

mixture of toluene (component -) and cyclohexane (component ).

120

110

100

90

80

70

0.0 0.5 0.65 1.0

*

$

,E+

u(ereatedva(or

ubcooledli0uid

aturatedva(or

aturatedli0uid

t

#$

92 92

0.5 0.65 1.00.0

140

120

100

80

60

40

*

$

+ E,

aturatedli0uid

aturatedva(or

ubcooledli0uid

u(ereatedva(or

p

k:a

$onstant p= 141.3 k:a $onstant t= 4 #$

"igure .3 ! Bhase diagrams for toluene (-) 8 cyclohexane () system.

oth toluene and cyclohexane exist as liquids at room temperature (vapor pressure of

toluene at 9 H% + 0.67 mm 'g + 5.7/4 kBa 2 vapor pressure of cyclohexane at 9 H%+

/7.77 mm 'g + -5.359 kBa). e consider a mixture of toluene and cyclohexane at --9 H% at

a constant pressure of - atm (-3-.5 kBa). At this temperature both toluene and cyclohexane

exist as superheated vapors (b.p. of toluene + --3.94 H% 2 b.p. of cyclohexane + 03.79 H%)

and form a gaseous mixture (only one phase, namely vapor phase). Applying the phase rule

we get the number of degrees of freedom " for this situation ! " + % 1 B 1 + 1-8 + 5. e

have fixed the temperature and pressure and hence there remains one more variable

(composition) which is necessary to completely describe the system. The composition can be

selected in the entire composition range ( xcyclohexane + 3.3 to -.3 ). e select the composition of

the gas phase as xcyclohexane + 3.49 and xtoluene + 3.59. $f we cool down this mixture then at acertain temperature the vapor will start to condense as liquid and will do so over a range of

temperature. The temperature range lies between the b.p. of cyclohexane and toluene

(03.79H% to --3.94H%). ithin this range the mixture will exist as a two phase system (one

55


34/51

vapor phase and one liquid phase) ?"ig. .3 shows a t1x diagram@. "or the two phase region

the number of degrees of freedom " + 18 + . #o if the pressure and temperature are given

(e.g.p+ -3-.5 kBa, t+ /3H%) then there is no more variable left. The composition of the

vapor phase and of the liquid phase is fixed automatically ( points and I ). The

compositions of liquid phase and of the vapor phase are not identical. The vapor phase is

richer in the more volatile component, namely cyclohexane (having lower boiling point and

higher vapor pressure) and the liquid phase is richer in toluene (higher b.p. and lower vaporpressure). This fact is the basis for separation of liquid mixtures by distillation. e will learn

more about it when we study ideal systems in the forthcoming chapter.

The above mentioned mixture (3+ 3.49 ) of toluene and cyclohexane at /3H% and at

63 kBa pressure (low pressure) will exist as a homogeneous gaseous mixture.


35/51

2.0

1.5

1.0

0.5

0

0 100 200 300 400 500

t

#$

mol

"igure .- ! %hemical equilibrium of ammonia synthesis ( $nitial composition! - mol C 8 5

mol ').

2.!.! Models for luid Mi'tures

"or a quantitative description of fluid mixtures we need suitable models. asically,

any state property of mixture depends on temperature, pressure and its composition. "orexample, the molar volume of a fluid mixture will depend on T, p and the mole fractions of

all the components and may be written as!

(T, p, PxGQ) +xii (.57)

where PxGQ represents the mole fraction of all components. x irepresents the mole fraction of a

component i and i, the partial molar volume of the component. The partial molar volume of a

component i, iis the molar *olume of that component in the mi3ture.$n general, it is different

from the molar volume of the pure component i. $t is a complex function of temperature,

pressure and the composition.

i+ i( T, p, PxGQ) (.50)

Lelations similar to &quations (.57) and (.50) may be written for the corresponding

specific volumes.

The difference between the partial molar volume and the molar volume may be

understood from the following experiment!

At a given temperature and pressure (say room temperature and atmospheric pressure)

we add a small amount of water (say - mol) to a very large volume of pure water. The

increase in volume is found to be!-0 cm5 (+ the molar volume of water, ow)

'owever if we add the same amount of water (i.e. - mol) to a large volume of pure ethanol,

the increase in volume is not -0 cm; (i.e. ow) but only -6 cm; (the molar volume of water in

59


36/51

the mixture, w).


37/51

"or gases the model of ideal gas mixture is used. As in the case of pure gases, it is

assumed that the molecules themselves have negligible volumes and the molecule of type iis

not influenced by the presence of molecule of type. &ach component k behaves as if it were

alone in the available space (volume) and has the pressure (its partial pressure)!

V

)(np k

ig

k =(.63)

i.e. V

)(np ig -- =

, V

)(np ig =

, V

)(np ig 55 =

....

Adding these equations, we obtain

pV

)(n

V

)(n

V

)(nnnppp k

igigig===+++=+++ .....)(... 5-5-

(.6-)

&quation (.6-) is known as Ialtons law of partial pressures.

#ubstituting the value of Vfrom &q. (.6-) in &q. (.63) for pkigwe get

p3pn

np k

k

kig

k == (.6)

"or an ideal gas mixture, the partial pressure of a component is therefore equivalent to

the mole fraction.

"urther

),(),,(

Q)P,,(

p(nnp(Vp

)(nV

p

n)(

p

)(n

p

)(nnp(V

i

ig

oii

i

i

i

ig

oiiig

:

:

i

i:

ig

===

===

(.65)

Iivision through the total number of moles n yields

),(Q)P,,( p(33p( ig

oii:

ig

= (.66)


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2.!.!.2 Moist )ir

$n addition to mixtures consisting of ideal gases, we may frequently come across such

mixtures where one of the constituents is a *apor. The vapor in the mixtures is at such a lowpartial pressure that we can treat it as an ideal gas. 'ence the general rules applicable to ideal

gas mixtures apply to these mixtures as well.

A significant difference in the behavior of gas1vapor mixtures as compared to ideal

gas mixtures, however, lies in the fact that the composition of gas1vapor mixtures may

change on heating or cooling while that of ideal gas mixture does not change normally.

hen the temperature of the gas4*apormixture is lowered below a certain limit, it may

condense or solidify and thus the composition of the mixture may change. The vapor

condenses to liquid form if its partial pressure is above triple point pressure and solidifies

directly if the pressure is below the triple point pressure.

%ompletely dry air does not exist in nature. ater vapor in varying amounts is

diffused through it. e usually come across moist airwhich is actually a mixture of dry air

and water vapor (a gas1vapor mixture). A study of properties of moist air has become

important due to its wide application in air1conditioning, cooling towers, evaporative

condensers and weather forecasting etc.

e have seen in chapter .5.6 that every pure substance has a characteristic saturation

curve (see also "igure .5). At any given temperature, the equilibrium pressure of a pure

substance, in the vapor phase, can not exceed its saturation pressure. As the pressure reaches

the saturation pressure, the vapor condenses into liquid. The same rules hold for the

condensable component in a gaseous mixture, i.e. the partial pressure of a component(namely water vapor) in air can not exceed its saturation pressurefor the temperature of air.

Any attempt to increase the partial pressure of water vapor above saturation value will result

in its condensation.

p: ps:(() (.67)

wherep:is the partial pressure of water vapor and ps:is the saturation pressure of pure water

vapor at temperature (.

5liquid

dewpoint

saturation linevapor

-

p

T

50


39/51

"igure .5 ! (1pdiagram for moist air.

This may be seen by considering "igure .5. Assume that the moisture in the system is at

point -, inside the superheated vapor range. #tate - of water vapor represents a partial

pressurep-+ p:+pat a temperature (-. The pressure at point which is situated on the

saturation line at the same temperature ((+(-) gives the corresponding saturation pressure

ps:.

The condensation of water vapor can also occur if the gaseous mixture is cooled at

constant pressure up to the saturation temperature (point 5).

The properties of gas1vapor mixture (moist air) may be quantitatively modelled by

using the model of ideal gas4*apor mi3ture. $n this model the unsaturated gas1vapor mixture

is considered as an ideal gas mixture of components! gas (dry air) and vapor (water vapor).

Iry air which is a mixture of non1condensable gases is treated as a single component. $fwater vapor is mixed with an unsaturated gas1water vapor mixture it reaches saturated state at

a definite vapor content above which some water condenses. The ideal gas1vapor mixture

then consists of an ideal gas mixture of (dry air 8 water vapor) and a condensed phase of pure

vapor (liquid water). The vapor content in a gas1vapor mixture up to saturation point is

generally represented as the partial pressure of the vapor according to equation (.63)!

VM

)(m

V

)(np

V

VVV ==

This is the pressure exerted by the vapor on the container wall if it were alone in the

volume :. Thus in an ideal gas1vapor mixture the saturated state is defined so that the partial

pressure of vapor is equal to the saturation vapor pressure of the pure vapor at the

temperature of the mixture!

p:+ps:(() +ps(() (.60)

?psfor saturation pressure of pure water vapor@

The values ofps:+pscan be read from the steam tables. (he standard tables and diagrams

for water *apor are also applicable to the moisture0 pro*ided the partial pressure is used

instead of pressure. The saturation partial pressure gives the maximum possible vapor contentin the gas phase. The unsaturated ideal gas1vapor mixture which does not contain condensed

phase (liquid) is characteried by &q. (.67). The saturation sate may be arrived either by

supplying more water vapor to the gaseous mixture or by decreasing the temperature. The

temperature at which water vapor starts condensing is called the dew point temperature or

simplydew point, tdp, of the mixture (see "ig. .52 the temperature at point 5 represent the

dew point for state -). $t is equal to the saturation temperature of the vapor corresponding to

its partial pressure in the mixture. At dew point holds!

p: +ps:(tdp) (.6/)


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air is provided according to the model of Rideal gas = water vapor mixtureS through the vapor

pressure curve of pure water.

e now look at some other definitions relevant to moist air. +pecific humidity or

humidity ratio3, is defined as the mass of water vapor (or moisture) per unit mass of dry air

in a mixture of air and water vapor. $f m*is the mass of dry air in mixture and m:the mass of

water vapor then

G

V

m

m3 =

(.93)

#pecific humidity is a suitable property to describe the state of unsaturated and

saturated gas1vapor mixture. The advantage of describing the relative humidity in terms of

the mass of dry airinstead of mass of the mixture is that the mass of dry air does not change

when water condenses or evaporates into the air (mass of dry air remains constant whereas

the mass of moist air changes with the amount of moisture).

Another important property of moist air is the relative humidity, . The relativehumidity is defined as the ratio of the partial pressure of water vapor, p:, in a gas1vapor

mixture to the saturation pressure,ps:, of pure water, at the same temperature of the mixture!

!("p

p

sV

V=

(2.51)

The specific humidity may be related to the relative humidity and to the partial

pressure. "or unsaturated gas1vapor mixture using the ideal gas law we can write!

(M

)

Vpm

V

VV

)(

=

and

(M

)

Vpm

G

GG

)(

=

asp+p:8p*it follows!

)( VG

VV

GG

VV

G

V

ppM

pM

Mp

Mp

m

m3

===

(.9)

or !!("pp"

!("p.

!pp"

p.

!pp"

p

.3

sV

sV

V

V

V

V

=

=

= 4343

/40

-0

(.95)

"or a given specific humidity, the partial pressure of water vapor is

)4.3( 3

3ppV

+=

(.96)

63


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The saturated specific humidity xsis found when +-, i.e.

))((

)(4.3

))((

)(),(

(pp

(p

(pp

(p

M

Mp(3

sV

sV

sV

sV

G

Vs

=

=

(.99)

The volume of an ideal gas1vapor mixture is given by the addition of the volume of

gas and the volume of vapor

V+ V*8 V:

$n practice the mass of gas (air) remains constant in moist air and it is only the mass

of vapor (moisture, water vapor) which changes. $t is, therefore practice to define the specific

properties of gas1vapor mixtures in terms of the mass of the gas m*. "or example, the specific

volume of an unsaturated gas1vapor mixture or a Gust saturated gas1vapor mixtures is defined

as the volume per kg of gas!

p

(3

M

)

M

)

pM

)(3

pM

)(

3m

mm

m

V

VGVG

3

VG

G

VVGG

G

3

@?-

-

+=+=

+=+

==

+

+

(.94)

The subscript (-83) in -83denotes that the specific volume is for a mixture of - kg gas and 3

kg vapor.

"or the specific volume of saturated gas1vapor mixture the volume of the condensedphase (liquid) should also be considered. ut as the specific volume of the condensate is very

small as compared to specific volume of gas, it may be neglected and the above equation

(.94) may also be used for saturated ideal gas1vapor mixtures.

2.!.!.3 ,deal Solutions

The term solution is generally used for solids, liquids or gases dissolved in liquids. As

the thermodynamic system is a fluid or a fluid mixture we will be more interested in the

properties of liquid solutions and gas solutions. "irst we look into liquid solutions. The

behavior of real liquid mixtures (solutions) is complex. 'owever, to simplify thermodynamic

and mathematical treatment of real solutions the fluid model R$deal #olution = $deal Kiquid

#olutionS is used. The model R$deal Kiquid #olutionS serves as a first step into the analysis of

real liquid systems.

As in the case of gas mixtures we set the partial molar volume of a component i in

ideal solution equal to its molar volume as pure substance i.e.

6-


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),(Q)P,,( p(3p( loi:il

i = (.97)

where loi (T,p) is the molar volume of pure liquid i at the temperature and pressure of the

solution (liquid mixture). e write, therefore, for the molar volume of an ideal solution

(liquid mixture)

il( T, p, PxGQ) + xiiil, xioil( T, p) (.90)

#imilar relation holds for the partial specific volume of an ideal liquid mixture.

il( T, p, PwGQ) + wioil (T,p) (.9/)

ith3iand wiUwe denote the mole fraction and the weight fraction of any component i in the

liquid phase. P3Q and PwQ represent the mole fractions and weight fractions of all

components.

2.!.!.! ,deal Systems

e have described the properties of gas mixture with the help of a model of R$deal

gas mixtureS. The properties of liquid mixtures could be described with the help of R$deal

#olutionS model. Cow we consider the vapor1liquid phase equilibrium of a mixture of

volatile liquids. To describe this equilibrium we define an R$deal #ystemS. e denote it with

superscript RisS.

%onsider a solution composed of several volatile substances in a container which is

initially evacuated. #ince the components are volatile, some of the solution evaporates to fillthe space above the liquid with vapor. After the attainment of thermodynamic equilibrium at a

temperature (, the total pressure p within the container is the sum of the partial pressures of

all the components, i.e.

p+ p-8p8 ... (.43)

These partial pressures are measurable, as are the equilibrium mole fractions x-U, x

U,.... in the

liquid. Ket one of the components, i, be present in relatively large amount as compared to any

of the others. Then it is found experimentally that

pi+3i;ps3i (.4-)

whereps3iis the vapor pressure of the pure component i. This is Laoults law. $t states that the

partial pressure of a substance in a mixture is proportional to its mole fraction and its vapor

pressure when pure. 7t is a limiting law and it is followed in any solution as 3 i approaches

unity. Laoults law is obeyed very well over a wide range of concentrations, when the

components are similar in structure and chemical nature. An example is the mixture of n1

hexane and n1heptane. #imilarly, a mixture of benene and toluene (methylbenene) obeys

the Laoults law throughout the composition range and is termed as an ideal system. The

ideal system is defined as one in which each component obey )aoult


43/51

e now apply Laoults law to binary mixtures in which both components are volatile.

The partial vapor pressures of the two components of an ideal binary mixture are proportional

to the mole fractions of the components in the liquid. The total pressure of the vapor is the

sum of the two partial vapor pressures. #o we have in a binary liquid solution

3-U83

U+ -

p-+3-Ups3-+ (-13

U)ps3- (.4)

and also p+3Ups3 (.45)

$f the total pressure over the solution isp, then

p+p-8p+ (-13U)ps3-83

Ups3

or p+ps3-83U(ps3Vps3-) (.46)

&quation (.46) relates the total pressure over the mixture to the mole fraction of component

in the liquid at constant temperature. $t further shows that pis a linear function of3U. This

relationship is shown in "igure .6. The curve is called bubble4point cur*e. $t is clear from

"ig. .6 that the bubble1point curve in an ideal system is a straight line in p13diagram ( also

called vapor pressure diagram). The bubble1point curve connects the vapor pressures of the

pure components. The terms 3Ups3and (-13

U) ps3-are also plotted in "igure .6 and show

the Laoults law.

The expression for the dew4point cur*emay be derived as follows!

3W

+p=p (.49)

? e use Xdouble slashX in the superscript to represent the composition (here, mole fraction)

of the vapor phase. hen the liquid and the vapor phase exist together we will consistently

use Xsingle slashX in the superscript for liquid phase and Xdouble slashX in the superscript for

the vapor phase composition. "or overall composition no superscript will be used@

65


44/51

$onstantt= 74 #$

$onstantp= 141.3 k:a

150

100

50

00 0.5 1.0

k:a

120

100

90

80

70

bubble-pointcurve

dew-pointcurve

+

,p

41

p42

0 0.5 1.0

dew-pointcurve

bubble-pointcurve

#$t41

t42

p = xp2is

2 42

p = ;16x )p1is

2 41

2 2

Mole >raction benzenex ,x

+,

2 2

Mole >raction benzenex?x

"igure .6 !p13and (13Iiagrams for an ideal binary system.

Esing the values ofpandpfrom equations (.45) and (.46) we obtain

p

p3

!pp"3p

p33 s

=

ss

=

s

s

=== 3

3-33-

3 =

+=

(.44)

solving for3Uyields

3Ups3+3

Wps3-83U3

W (ps31ps3-)

or )( 3-3

>>

3

3-

>>

>

sss

s

pp3p

p33

=

(.47)

Butting the value of3Ufrom (.47) in (.46) we get

)(

)(

3-3

>>

3

3-33-

>>

3-

sss

ssss

pp3p

ppp3pp

+=

)( 3-3>>

3

33-

sss

ss

pp3p

ppp

=

(.40)

&quation (.40) expressespas a function of3W, the mole fraction of component in vapor.

This function is also plotted in "igure (.6). $t is called dew1point curve.


45/51

3

>>

3-

>>

--

ss p

3

p

3

p+=

(.4/)

&quation (.40) and (.4/) give :K1&quilibria of an ideal system at constant

temperature (that means under isothermalconditions).

The solution of equation (.46) for equilibrium temperatures at specified 3U is

inconvenient. 'ence the isobaric V>4?uilibria cannot be represented by such explicit

equations. y rearranging equation (.46) follows the expression for the bubble point curve

of a binary system

p

(p3

p

(p3 ss

)()()-(- 3>

3->

+=

(.73)

"or the dew1point curve on rearrangement of equation (.4/) we get!

)()()-(-

3

>>

3-

>>

(p

p3

(p

p3

ss

+=

(.7-)

&quations (.73) and (.7-) can only be solved iteratively as the vapor pressure is a function

of temperature.

The bubble1point curve ( (13U) is most readily constructed by solving (.46) for the

bubble1point compositions 3U at representative values of ( between the saturation

temperatures of the pure components

)()(

)(

3-3

3->

(p(p

(pp3

ss

s

=

(.7)

The corresponding equilibrium compositions on the dew1point curve are obtained by

substituting the value of3Ufrom equation (.7) in equation (.44).

)@()(?

)@(?)()(

3-3

3-33>

>>

(p(p

(pp

p

(p

p

(p33

ss

sss

==

(.75)

The :K& (vapor1liquid equilibrium, i.e. 3W vs.3

U) diagram of an ideal system at a given

temperature may be constructed most conveniently using equation (.44). The isobaric :K&

diagram may be constructed from the corresponding (13 diagram. "igure .9 shows

schematic :K& diagram for an ideal system. The :K& diagrams at constant temperature and

at constant pressure do not differ much from one another.


46/51

"igure .9 ! :K& Iiagram for an ideal system.

The enrichment of more volatile component in the vapor phase is represented by the

so called @relati*e *olatility@or @*olatility ratio@defined as

-

-

-

==

3333=

(.76)

"or a binary ideal system the relative volatility is then

)(

)(

=))((

=))((

3-

3

>

-3-

>

-

>

3

>

-(p

(p

3p

(p3

3p

(p3

s

s

s

s

is==

(.79)

&quation (.79) shows clearly that the relative volatility in a binary ideal system is the ratio

of vapor pressure of the two components, i.e. it depends only on temperature. $t does not

depend on the composition of the mixture. $ts temperature dependence is weak and for small

temperature ranges the relative volatility may be considered as constant.

There are only a few systems which may be considered as ideal systems. 'owever,

the model of ideal systems helps very much to understand the behavior of real solutions.

ecause now only the deviations from the ideal system behavior is to be considered. oth

The Thermodynamic Substance

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