CHAPTER 2.

PROPERTIES OF PURE SUBSTANCES

2.1 Pure Substance

- A substance that has a fixed chemical

composition throughout (ex: water, nitrogen,

helium, and CO2 )

- It can exist in more than one phase but its

chemical composition is the same in all the

phases. A good example is water.

-

- A chemical analysis will show that in a mixture

of liquid water and steam, the hydrogen and

oxygen atoms are in the ratio of 2 : 1

regardless of which sample (steam or liquid

water) is analyzed.

Steam

Liquid water

- Sometimes a mixture of gases, such as air, is

considered a pure substance as long as there is

no change of phase.

- A mixture of liquid air and gaseous air is not a

pure substance, since the composition of the

liquid phase is different from that of the vapour

phase.

2.2 Vapour-Liquid-Solid Phase Equilibrium

- There are 3 principal phases for a working fluid

i.e solid, liquid and gas.

- Let us carry out an experiment

- Consider as a system 1 kg of water contained

in the piston-cylinder arrangement shown in

Figure 1(a).

- Suppose the piston and weight maintain a

pressure of 0.1 MPa (1 bar) in the cylinder, and

that the initial temperature is 20oC

- As heat is transferred to the water, the

temperature increases appreciably, the specific

volume increases slightly, and the pressure

remains constant

- When the temperature reaches 99.6oC,

additional heat transfer results in a change of

phase, as indicated in Figure 1(b)

- Some of the liquid becomes vapour, and during

this process both the temperature and pressure

remain constant, but the specifiv volume

increases considerably

- When the last drop of liquid has vapourized,

further transfer of heat results in an increase in

both temperature and specific volume of the

vapour as shown in Figure 1 (c).

- The term saturation temperature designates

the temperature at which vaporization takes

place at a given pressure

- This pressure is called the saturation pressure

for the given temperature

- Thus for water at 99.6oC, the saturation

pressure is 0.1 MPa, and for water at 0.1 MPa,

the saturation temperature is 99.6oC.

- For a pure substance, there is a definite

relation between saturation pressure and

saturation temperature.

- This relationship is in the form of a curve

known as the vapor-pressure curve

Pressure

Temperature

Vapor pressure curve

-

FIGURE 1 Constant-pressure change from liquid to vapor phase for a substance

FIGURE 2 Temperature-Volume diagram for water

- If a substance exists as liquid at the saturation

temperature and pressure, it is called saturated

liquid.

- If the temperature of the liquid is lower than

the saturation temperature for the existing

pressure, it is called either a subcooled liquid or

a compressed liquid

- A subcooled liquid - implying that the

temperature is lower than the saturation

temperature for the given pressure

- A compressed liquid - implying that the

pressure is greater than the saturation

pressure for the given temperature

- When a substance exists as part liquid and part

vapor at the saturation temperature, its

quality, x is defined as the ratio of the mass

of vapour to the total mass

- Thus in Figure 1(b), if the mass of vapour =

0.2 kg and the mass of the liquid = 0.8 kg, the

quality = 0.2/1.0 atau 20%

- Quality is an intensive property and is given the

symbol ‘ x ‘

- liquidvapour

vapour

mmm

X+

=

- X = 0 : saturated liquid

- X = 1 : saturated vapour

- Quality has meaning only when the substance

is in a saturated state i.e. at saturation

pressure and temperature

- If a substance exists as vapour at the

saturation temperature, it is called saturated

vapour (quality is 100 percent or x =1)

- The saturated-liquid and saturated-vapour

states have the same pressure and the same

temperature, but are definitely not the same

state

- In a saturation state, pressure and temperature

are not independent properties.

- Two independent properties such as pressure

and specific volume or pressure and quality are

required to specify a saturation state of a pure

substance

- When the vapour is at a temperature greater

than the saturation temperature, it is said to

exist as superheated vapour

- The pressure and temperature of superheated

vapour are independent properties, since the

temperature may increase while the pressure

remains constant.

- Gases are highly superheated vapours

- We can also sketch the pressure versus

temperature diagram for a substance such as

water. This diagram is shown in Figure 3.

Liquid phase

Pressure

Vapour phase

Solid Phase Triple point

Temperature

Figure 3 P-T Diagram

- This diagram shows how the solid, liquid &

vapour phases may exist together in

equilibrium.

- Along the sublimation line, the solid & vapour

phases are in equilibrium

- Along the fusion line, the solid & liquid phases

are in equilibrium

- Along the vaporization line, the liquid & vapor

phases are in equilibrium

- The only point at which all three phases may

exist in equilibrium is the triple point

- The vaporization line ends at the critical point

because there is no distinct change from the

liquid phase to the vapour phase above the

critical point. The coordinates of this point are

known as the critical temperature Tc, critical

pressure Pc & critical specific volume vc

- Although we have made these comments with

specific reference to water, all pure substance

exhibit the same general behaviour

(qualitatively)

- Quantitatively, the triple point temperature and

critical temperature vary greatly from one

substance to another

2.3 Independent properties of a pure

substance

- The state of a simple compressible pure

substance is defined by two independent

properties

- If the pressure and temperature of

superheated steam are specified, the state of

the steam is determined. This also applies to a

compressed liquid.

- However, in a saturation state, pressure and

temperature are not independent properties

- The saturated-liquid and saturated-vapour

states have the same pressure and the same

temperature, but are definitely not the same

state

- Two independent properties such as pressure

and specific volume or pressure and quality are

required to specify a saturation state of a pure

substance

- The state of air, which is a mixture of gases of

definite composition, is determined by

specifying two properties as long as it remains

in the gaseous phase

- In this regard, air can be considered as a pure

substance

2.4 Property Tables

- Some thermodynamics properties can be

measured easily, but others cannot and are

calculated by using the relations between them

and measurable properties

- The results of these measurements and

calculations are presented in tables in a

convenient format

- The steam tables will be used to demonstrate

the use of thermodynamic property tables

- Property tables of other substances are used in

the same manner

- For each substance, the thermodynamic

properties are listed in more than one table

- A separate table is prepared for each region :

superheated vapour, compressed liquid and

saturated(mixture) regions

- Total energy of a system can be catogorized in

two groups: macroscopic and microscopic

- The macroscopic forms of energy are those a

system possesses as a whole with respect to

some reference frame

- Examples of macroscopic forms of energy are

kinetic and potential energies

- The microscopic forms of energy are those

related to the molecular structure of a system

and the degree of the molecular activity

- The sum of all the microscopic forms of energy

is called the internal energy of a system and is

denoted by U

- Enthalpy is a combination of the properties

U + PV.

- Enthalpy is also a property and is given the

symbol H :

H = U + PV (kJ)

Or per unit mass,

h = u + Pv (kJ/kg)

- Enthalpy is often encountered in the analysis of

power plants and refrigeration systems

2.4.1 Saturated States

- The properties of saturated liquid and

saturated vapour for water are listed in Tables

A-4(temperature table) and A-5(pressure table)

- Both tables give the same information

- The subscript f is used to denote properties of

a saturated liquid

- The subscript g is used to denote the

properties of saturated vapour

- Subscript fg denotes the difference between

the saturated vapour and saturated liquid

values of the same property

- For example :

vf = specific volume of saturated liquid

vg = specific volume of saturated vapour

vfg = difference between vg and vf

vfg = vg – vf

- The quantity hfg is called the enthalpy of

vaporization ( or latent heat of vaporization )

- hfg represents the amount of energy needed to

vaporize a unit mass of saturated liquid at a

given temperature or pressure

- hfg decreases as the temperature or pressure

increases, and becomes zero at the critical

point

2.4.2 Saturated Liquid-Vapour Mixture

- During a vaporization process, a substance

exists as part liquid and part vapour ( mixture )

- To analyze this mixture, we need to know the

proportions of the liquid and vapour phases in

the mixture

- This is done by defining a new property called

the quality, x as the ratio of the mass of

vapour to the total mass of the mixture

x = mg 1 – x = mf

mf + mg mf + mg

where mg = mass of vapour

mf = mass of liquid

m = total mass = mf + mg

- Consider a container that contains a saturated

liquid-vapour mixture with mass m and a

quality , x

- Volume of this mass is equal to the volume of

the saturated liquid and the saturated vapour

V = Vliquid + Vvapour

or

mv = mliquidvf + mvapourvg

Dividing the above equation by the total mass

and introducing the quality, x, we have:

mliquid = mf mvapour = mg m = mf + mg mv = mliquidvf + mvapourvg

m m m

v = (1 –x)vf + xvg

vfg = vg - vf

v = vf + xvfg

similarly,

h = hf + xhfg

u = uf + xufg

s = sf + xsfg

- Quality has significance for saturated mixtures

only

- It has no meaning in the compressed liquid or

superheated regions

- Its value is between 0 and 1

- The quality of a system that consists of

saturated liquid is 0 or 0 percent

- The quality of a system that consists of

saturated vapour is 1 or 100 percent

- In saturated mixtures, quality can serve as one

of the two independent intensive properties

needed to describe a state

2.4.3 Superheated vapour

- A substance exists as superheated vapour

in the region to the right of the saturated

vapour line and at temperatures above the

critical point temperature

- Superheated region is a single-phase

region

- Temperature and pressure can be used as

two independent properties to determine a

state

2.4.4 Compressed Liquid

- Compressed liquid tables are not

commonly available

- Table A-7 is the only compressed liquid

table in the text

- Compressed liquid properties depend on

temperature much more strongly than

they do on pressure

- In the absence of compressed liquid data,

a general approximation is to treat

compressed liquid as saturated liquid at

the given temperature

- Thus,

y = yf@T

where y is v, u, or h

2.4.5 Reference state and reference

value

- The values of u, h and s cannot be

measured directly

- They are calculated from measurable

properties using the relations between

properties

- These relations give the changes in

properties, not the values of properties at

specified states

- We need to choose a reference state and

assign a value of zero for a

property/properties at that state

- For water, the state of saturated liquid at

0.01oC is taken as the reference state, and

the internal energy and entropy are

assigned zero values at that state

2.5 The P-v-T Surface

- The state of a simple compressible

substance is fixed by any two

independent, intensive properties

- Once the two appropriate properties are

fixed, all the other properties become

dependent properties

- Any equation with two independent

variables in the form z = z(x,y) represents

a surface in space

- Thus, we can represent the P-v-T

behaviour of a substance as a surface in

space

- T and v may be viewed as the

independent variables (the base) and P as

the dependent variable (the height)

- All the points on the surface represent

equilibrium states

- Single-phase regions appear as curved

surfaces

- Two-phase regions are surfaces

perpendicular to the P-T plane

- All the 2-D diagrams are the projections of

this 3-D surface onto the appropriate

planes

- A P-v diagram is a projection of the P-v-T

surface on the P-v plane

Figure 4. P-v-T surfac

2.6 Equation of State

- Property tables provide very accurate

information about properties

- A more practical approach would be to have

some simple relations/equations among the

properties that are sufficiently accurate

- Any equations that relates the pressure,

temperature, and specific volume is called an

equation of state

- Property relations that involves other properties

of a substance at equilibrium states are also

known as equations of state

- There are several equations of state, some

simple and others very complex

- The simplest and best-known equations of

state for substances in the gas phase is the

ideal-gas equation of state

- This equation predicts the P-v-T behaviour of a

gas accurately within some selected region

- The vapour phase of a substance is called a

gas when it is above the critical temperature

- Vapour usually implies a gas that is not far

away from a state of condensation

- From experimental observations it has been

established that the P-v-T behaviour of gases

at low density is closely given by the ideal-gas

equation of state :-

Pv = RuT (2.1)

where v = V/N and N is the mole

number

Ru is the universal gas constant

Ru = 8.3144 kJ/kmol K

- Dividing equation by the molecular weight, M,

we obtain:

Pv = RT

M M

v = V/N

v/M = V/NM

NM = jisim = m

v/M = V/m = v

R = Ru/M

R is the gas constant and is different for each gas

Equation 2.1 then becomes

Pv = RT (2.2)

This equation can also be written in terms of

the total volume

PV = mRT

And for a fixed mass, the properties of

an ideal gas at two different states are related

by :

P1V1 = P2V2

T1 T2

- At very low density all gases and vapours

approach ideal gas behaviour

- At higher densities, the behaviour may deviate

substantially from the ideal gas equation of

state

- Two questions arises :

1) over what range of density will the ideal

gas equation of state hold with accuracy ?

2) how much does an actual gas at a given

pressure and temperature deviate from

ideal gas behaviour ?

- To answer both questions, lets introduce

the compressibility factor, Z

Z = Pv

RT

or Pv = ZRT

- For an ideal gas, Z = 1

- The farther away Z is from unity, the more the

gas deviates from ideal-gas behaviour

- The pressure or temperature of a substance is

high or low relative to its critical temperature

or pressure

- Gases behave differently at a given

temperature and pressure

- Gases behave very much the same at

temperatures and pressures normalized with

respect to their critical temperatures and

pressures

- The normalization is done as :

Reduced pressure, PR = P/Pcr

Reduced temperature, TR = T/Tcr

Reduced specific volume, vR = vactual

RTcr/Pcr

- The Z factor for all gases is approximately the

same at the same reduced pressure and

temperature. This is known as the principle of

corresponding states

- The experimentally determined Z values are

plotted against PR and TR for several gases in

Figure 4

- By curve-fitting all the data, we obtain the

generalized compressibility chart that can be

used for all gases (Fig. A -30)

- The following observations can be made from

the generalized compressibility chart:

o At very low pressures (PR < < 1), the

gases behave as an ideal gas regardless of

temperature

o At high temperatures ( TR > 2 ), ideal gas

behaviour can be assumed with good

accuracy regardless of pressure (except

when PR > > 1)

o The deviation of a gas from ideal-gas

behaviour is greatest in the vicinity of the

critical point. In this region it is preferable

to use the property tables

2.7 Other Equations of State

- The ideal-gas equation of state is vey simple,

but its range of applicability is limited

- It is desirable to have equations of state that is

accurate over a larger region

- Such equations are naturally more complicated

- Examples of such equations are: Equation Formula Van der Waals ( ( )( ) RTbvvaP 2 =−+

Beattie-Bridgeman ( ) 232 1vABv

vTc

vTR

P u −+⎟⎠⎞

⎜⎝⎛ −=

Benedict-Webb-Rubin

Virial Equation

2.8 SPECIFIC HEATS

It takes different amounts of energy to raise the temperature of identical masses of different substances by one degree

We need about 4.5 kJ of energy to raise the temperature of 1 kg of iron to from 20 to 30oC

It takes about 41.8 kJ of energy to raise the temperature of 1 kg liquid water from 20 to 30oC

It is desirable to have a property that will enable us to compare the energy storage capabilities of various substances

This property is the specific heat

The specific heat is defined as the energy required to raise the temperature of a unit mass of a substance by one degree

In thermodynamics, we are interested in two kinds of specific heats:

o Specific heat at constant volume, Cv o Specific heat at constant pressure, Cp

Cv is defined as :

vv T

uC ⎟⎠⎞

⎜⎝⎛∂∂

=

Cp is defined as

pp T

hC ⎟⎠⎞

⎜⎝⎛∂∂

=

These equations are property relations and are independent of the type of processes

They are valid for any substance undergoing any process

Like any other property, the specific heats depend on the state of a substance

The only relevance Cv has to a constant-volume process is that Cv happens to be the energy transferred to a system during a constant-volume process per unit degree rise in temperature

This is how the values of Cv are determined

This is also how the name specific heat at constant volume originated

The same explanation applies to Cp Cv is related to the changes in internal energy. It is a measure of the variation of internal energy of a substance with temperature

Cp is related to the changes in enthalpy. It is a measure of the variation of enthalpy of a substance with temperature

A common unit for specific heats is kJ/kg.oC or kJ/kg.K

2.9 INTERNAL ENERGY, ENTHALPY, AND SPECIFIC HEATS OF IDEAL GASES

The behaviour of an ideal gas is given by the relationship :

Pv = RT

It has been demonstrated mathematically and experimentally that for an ideal gas, the internal energy is a function of temperature only, i.e :

u = u(T)

Using the definition of enthalpy and the equation of state of an ideal gas

h =u + Pv

Pv = RT h = u + RT Since R is constant and u = u(T), it follows that the enthalpy of an ideal gas is a also a function of temperature only:

h = h(T)

Since u and h depend only on temperature for an ideal gas, the specific heats Cv and Cp also depend on temperature only Thus, the partial derivatives in the definition for Cv and Cp can be replaced by ordinary derivatives

The differential changes in the internal energy and enthalpy of an ideal gas is given by:

du = Cv (T) dT

dh = Cp (T) dT

The change in internal energy or enthalpy for an ideal gas during a process from state 1 to state 2 is obtained by integrating the above equations

There are 3 ways to determine the internal energy and enthalpy changes of ideal gases

1. By using the tabulated u and h data (

Table A-17). This is the easiest and most accurate

2. By using the Cv and Cp relations as a function of temperature and performing the integrations (Table A-2c). This is inconvenient for hand calculations but desirable for computerized calculations

3. By using average specific heats (Table A-2b). This is very simple and convenient when property tables are not available. They are reasonably accurate if the temperature interval is not very large

The ∆u and ∆h of ideal gases are expressed as

)()(

)()(

12

2

1,12

12

2

1,12

TTCdTTChhh

TTCdTTCuuu

avpp

avvv

−≅=−=∆

−≅=−=∆

∫

∫

Cv,av and Cp,av are evaluated from Table A-2b at the average temperature (T1 + T2)/2

A special relationship between Cv and Cp for ideal gases can be obtained by differentiating the relation h = u + RT, which yields

dh = du + RdT

Replacing dh by CpdT and du by CvdT and dividing the results by dT, we obtain

Cp = Cv + R

This is an important relationship since it enables us to determine Cv from a knowledge of Cp and the gas constant R

The specific heat ratio k is defined as

k = Cp/Cv

For incompressible substances (liquids and solids), both the are identical and denoted by C:

Cv = Cp = C (refer Table A-3) The ∆u and ∆h of incompressible substances are given by

Pvuh

TTCdTTCu av

∆+∆=∆

−≅=∆ ∫ )()( 12

2

1