PETE 310PETE 310

Lectures # 9 & 10Lectures # 9 & 10

Ideal and Real GasesIdeal and Real Gases

Equations of State Equations of State The Ideal GasThe Ideal Gas

Ideal gas propertiesIdeal gas properties• Volume of gas molecules is Volume of gas molecules is

negligible compared with gas volumenegligible compared with gas volume• Forces of attraction or repulsion Forces of attraction or repulsion

between molecules or walls of between molecules or walls of container are zerocontainer are zero

• No loss of internal energy due to No loss of internal energy due to collisionscollisions

VV

PP TT11

TT22PV

1

TV

At fixed TAt fixed T

At fixed PAt fixed P

Gay Lussac’s LawGay Lussac’s Law

Boyle’s LawBoyle’s Law

Avogadro’s LawAvogadro’s Law

At the same P and T equal Volumes of gas At the same P and T equal Volumes of gas contain the same # of molecules contain the same # of molecules Na = 2. 73Na = 2. 73XX10102626 molecules/lb mole molecules/lb mole

Internet Lesson on Ideal Gas Internet Lesson on Ideal Gas BehaviorBehavior

Experimental Experimental instructionsinstructions

Problems with Problems with solutionssolutions

http://jersey.uoregon.edu/vlab/Piston/

http://dbhs.wvusd.k12.ca.us/GasLaw/WS-Ideal.html

Hi, I am Boyle

Ideal Equation of State Ideal Equation of State ConstructionConstruction

TT22

TT11

PP11

PP22

VV11 VVxx VV22

2

2

1

211

T

V

T

V

VPVP

x

x

Step 2Step 2

Step 1Step 1

2

22

1

11

T

VP

T

VP

CombiningCombining

Boyle & Charles lawsBoyle & Charles laws

http://dbhs.wvusd.k12.ca.us/GasLaw/KMT-Gas-Laws.html

a cool website with lots of info…a cool website with lots of info…

BoyleBoyle CharlesCharles

Boyle & Charles lawsBoyle & Charles laws

TPV

1 PTV

)/(

nVV

RT

PV

T

VP

T

VP

M

M

2

22

1

11

R = gas constant, depends upon units used R = gas constant, depends upon units used See page 95 McCain bookSee page 95 McCain book

Ideal Gas MixtureIdeal Gas Mixture

The pressure in a vessel containing The pressure in a vessel containing an ideal gas mixture (an ideal gas mixture (nn) or a single ) or a single gas component (gas component (nnkk) is ) is

t

kk

t

V

RTnP

V

nRTP

http://home.a-city.de/walter.fendt/phe/gaslaw.htm

Partial Pressure Partial Pressure

PPkk is the partial is the partial

pressure of pressure of component component k, k, and by and by definitiondefinition

kkk y

n

n

P

P

cN

ik PP

1

P pk

T1

T1

n1

,n2

, nk…, n

k

Density of ideal gasDensity of ideal gas

RT

P

VM

m

V

nRTP

twt

Watch out the units!Watch out the units!

Mixtures of Ideal GasesMixtures of Ideal Gases

Dalton’s law of partial pressuresDalton’s law of partial pressures

Amagat’s law of partial volumesAmagat’s law of partial volumes

Specific gravity of a gasSpecific gravity of a gas

Apparent Molecular Weight of Apparent Molecular Weight of a Gas Mixturea Gas Mixture

Determine the MDetermine the Mwawa and density of a and density of a

mixture of 30%Cmixture of 30%C11, 40%C, 40%C22, and 30%C, and 30%C33

at T=200at T=200ooF and P=4000 psiaF and P=4000 psia

Behavior of Real GasesBehavior of Real Gases

1RT

PVM

ZRT

PVM

Equations of State for GasesEquations of State for Gases

Ideal gasIdeal gas

Real gas Real gas

ZZ

PV

RT

RT

PVid

M

M

1

Z is the ratio of the “real molar volume” Z is the ratio of the “real molar volume”

over the “ideal molar volume” over the “ideal molar volume”

of a substance measured at the same of a substance measured at the same

pressure and temperature.pressure and temperature.

Compressibility Factor ChartsCompressibility Factor Charts

T r

P r

Z

1

The Principle of Corresponding The Principle of Corresponding StatesStates

““All fluids when compared at All fluids when compared at the same reduced temperature the same reduced temperature and reduced pressure, have and reduced pressure, have approximately the same approximately the same compressibility factor, and all compressibility factor, and all deviate from ideal gas behavior deviate from ideal gas behavior to about the same degree” to about the same degree”

The Principle of Corresponding states (POC) The Principle of Corresponding states (POC) originated with single component fluids. originated with single component fluids.

Material properties are usually Material properties are usually expressed in terms of reduced expressed in terms of reduced parameters such as:parameters such as:

• Reduced Temperature: Reduced Temperature:

Typical Reduced ParametersTypical Reduced Parameters

cr TTT /

• Reduced Pressure:Reduced Pressure:

• Reduced Molar Volume:Reduced Molar Volume:

cr PPP /

cMMrVVV /

Typical Reduced ParametersTypical Reduced Parameters

Reduced ParametersReduced Parameters

Usually Usually TTrr and and PPrr VVrr obtained as a obtained as a function of function of TTrr and and PPrr

These are called two-parameter These are called two-parameter Corresponding States modelsCorresponding States models

Three-parameter corresponding Three-parameter corresponding states models improve predictions states models improve predictions but third parameter is not but third parameter is not VVrr (not (not independent variable)independent variable)

This third parameter is called the This third parameter is called the acentric acentric factor.factor.

It takes into account the non-spherical It takes into account the non-spherical nature of moleculesnature of molecules

Peng Robinson and the Soave Redlich Peng Robinson and the Soave Redlich Kwong equations of state (EOS) are Kwong equations of state (EOS) are examples of three parameter examples of three parameter corresponding states models.corresponding states models.

Generalized Corresponding Generalized Corresponding States Three-Parameter States Three-Parameter

Compressibility Factor ChartsCompressibility Factor Charts

Following the POC only one Following the POC only one compressibility factor chart can be compressibility factor chart can be used to determine volumetric used to determine volumetric properties of any pure fluid properties of any pure fluid by using by using its reduced propertiesits reduced properties. The shape of . The shape of this chart is in general.this chart is in general.

Corresponding States Corresponding States Correlations & ModelsCorrelations & Models

The objective is then to find a model (models) to The objective is then to find a model (models) to predict the Z factor. predict the Z factor.

Ideal gas behavior is described from the ideal gas Ideal gas behavior is described from the ideal gas Equation of State (EOS) with a compressibility Equation of State (EOS) with a compressibility factor of 1.factor of 1.

Extension of Corresponding Extension of Corresponding States to MixturesStates to Mixtures

ZZ factor charts (all built from EOS) factor charts (all built from EOS) are also used for multicomponent are also used for multicomponent systems in this case the coordinates systems in this case the coordinates used are used are “pseudo-reduced “pseudo-reduced properties”properties”

For a mixture you can use the same For a mixture you can use the same charts as for a pure component.charts as for a pure component.

Pseudoreduced PropertiesPseudoreduced Properties

For mixtures the same type of charts For mixtures the same type of charts apply but using “pseudoreduced apply but using “pseudoreduced properties” which are defined similarly as properties” which are defined similarly as the ratio of pressure (or temperature) with the ratio of pressure (or temperature) with “pseudoreduced critical pressure" (or “pseudoreduced critical pressure" (or temperature). These pseudocritical temperature). These pseudocritical properties are an average of the critical properties are an average of the critical properties of the components in the properties of the components in the mixture. Charts for mixtures can also be mixture. Charts for mixtures can also be used for single component fluids.used for single component fluids.

Compressibility Compressibility factor Z as a factor Z as a function or function or

pseudoreduced pseudoreduced pressurepressure

Pseudocritical Properties of Pseudocritical Properties of Natural GasesNatural Gases

Pseudoreduced PressurePseudoreduced Pressure

Pseudoreduced TemperaturePseudoreduced Temperature

pcpr P

PP

pcpr T

TT

Pseudocritical Properties of Pseudocritical Properties of Natural GasesNatural Gases

If only the specific gravity and Mw of If only the specific gravity and Mw of of the gases is known then charts of the gases is known then charts are available to estimate these are available to estimate these pseudocritical properties (McCain pseudocritical properties (McCain figure 3-10 ). figure 3-10 ).

Pseudocritical Properties of Pseudocritical Properties of Natural GasesNatural Gases

Naturally the degree of accuracy is Naturally the degree of accuracy is reduced substantially. We well see reduced substantially. We well see methods when compositional methods when compositional information is available, in this case: information is available, in this case:

cii

N

ipc PyP

c

1

cii

N

ipc TyT

c

1

Pseudocritical Properties of Pseudocritical Properties of Natural GasesNatural Gases

Once Once ZZ is evaluated you can find the is evaluated you can find the gas density asgas density as

3/ ftlbmV

Mg

Z-factor Z-factor chart for chart for

low low reduced reduced

pressurespressures