ASSIGNMENT THERMODYNAMICS LABTHERMODYNAMIC PROPERTIES, VARIABLES & MODELS

SUBMITTED TO: SIR FARHAN AHMADSUBMITTED BY: GROUP#02(2009-CH-12,20,42,46,54,58)

DEPARTMENT OF CHEMICAL ENGINEERING, UET LAHORE

CONTENTS

Thermodynamic Properties Extensive Properties Intensive Properties Thermodynamic Variables Extensive Variables Intensive Variables Thermodynamic Models Benson group increment theory Joback method Non-random two-liquid mode (NRTL Equation) Klincewicz method Lydersen method Equation of state (EOS)

Ideal gas Law Cubic equation of state Non-cubic equation of state

References

THERMODYNAMIC PROPERTIES

Any characteristic of a system is called a property. A quantity which is either an attribute of an entire system or is a function of position which is continuous and does not vary rapidly

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over microscopic distances, except possibly for abrupt changes at boundaries between phases of the system is called thermodynamic property.

Thermodynamic properties are divided into two broad types.

Extensive Properties Intensive Properties

Extensive Properties

An extensive property is any property that depends on the size (or extent) of the system under consideration. For example; volume and mass. If the length of all edges of a solid cube is doubled, the volume increases by a factor of eight. The same cube will undergo an eight-fold increase in mass when the length of the edges is doubled.

Other examples of extensive properties include:

Energy,Entropy,Gibbs energy,Particle number,Momentum,Number of moles and Area

Intensive Properties

An intensive property is any property which does not depends upon the system size or independent of the amount of substance present in a system. Temperature, pressure and density are good examples.

An intensive property can exist at a point in space; e.g. the density of the atmosphere is different from point to point, with air nearest the ground having the highest density and air far above the earth's surface having the lowest.

Other examples of intensive properties include:

Chemical potential,Viscosity,Concentration,Specific energy and Specific heat capacity

A particularly important type of intensive property is the specific property, which is always given on a unit mass basis. An example is specific volume, which has units of volume/mass, typically expressed as cubic feet per pound or cubic meters per kilogram. Specific properties are intensive because they exist at a point. For instance, specific volume is simply the reciprocal of density.

Extensive VS Intensive Properties

When considering physical systems, it is often very useful to stay aware of whether the property being considered is intensive or extensive. An easy way to determine whether a

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property is intensive or extensive is to divide the system into two equal parts with an imaginary partition, as shown in Fig. Each part will have the same value of intensive properties as the original system, but half the value of the extensive properties.

Note:

A ratio of extensive variables will yield an intensive variable! (For example, mass/volume --two extensive variables -- gives density, an intensive variable) this is one way to understand why intensive variables "tell us more" about system.

Common Thermodynamic Properties of Single-Phase Pure Substances

Property Symbolic Designation

Units Classification

Extensive Intensive

Extensive Intensive

Mass M — kg — Fundamental property appearing in equation of state

Number of moles N — kmol — Fundamental property appearing in equation of state

Volume and specific volume

V v m3 m3/kg Fundamental property appearing in equation of state

Pressure — P — Pa or N/m2

Fundamental property appearing in equation of state

Temperature — T — K Fundamental property appearing in equation of state

Density — r — kg/m3 Fundamental property appearing in equation of state

Internal energy U u J J/kg Based on first law and calculated from calorific equation of state

Enthalpy H h J J/kg Based on first law and calculated from calorific equation of state

Constant-volume specific heat

— cv — J/kg_K Appears in calorific equation of state

Constant-pressure specific heat

— cp — J/kg_K Appears in calorific equation of state

Specific-heat ratio — g — Dimensionless

—

Entropy S s J/K J/kg_K Based on second law of thermodynamics

Gibbs free energy (or Gibbs function)

G g J J/kg Based on second law of thermodynamics

Helmholtz free A a J J/kg Based on second law of

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energy (or Helmholtz function)

thermodynamics

THERMODYNANIC VARIABLES

Thermodynamic variables are the quantities like pressure, volume and temperature, which help us to study the behavior of a thermodynamic system. There are some other thermodynamic variables such as entropy, internal energy, etc., but these thermodynamic variables can be expressed in terms of pressure, volume and temperature.

Thermodynamic state variables are of two kinds:

Extensive variables Intensive variables

Extensive variables

An extensive quantity or extensive variable is a physical property of a system that depends on the system size or the amount of material in the system.

Examples

Entropy, Gibbs energy, Mass, Number of moles, Area and Volume

Entropy

Entropy is a thermodynamic property that can be used to determine the energy not available for work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when converting energy to work. During this work, entropy accumulates in the system, which then dissipates in the form of waste heat.

Units

Thermodynamic entropy has the dimension of energy divided by temperature, and a unit of joules per kelvin (J/K) in the International System of Units.

Gibbs Energy

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In thermodynamics the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system. Just as in mechanics, where potential energy is defined as capacity to do work.

The Gibbs free energy is a thermodynamic quantity which can be used to determine if a reaction is spontaneous or not. The definition of the Gibbs free energy is

∆G = ∆H- T∆S……………….(1)

Where ∆G is the free energy, ∆H is the enthalpy and ∆S is the entropy. If we consider G for a reaction where the temperature does not change, we have

G = H – TS…………………(2)

The sign of ∆G determines if a reaction is spontaneous or not.

∆G < 0: the reaction is spontaneous ∆G > 0: the reaction is not spontaneous ∆G = 0: the reaction is at equilibrium

Mass

Inertial mass can be defined as a quantitative measure of an object's resistance to the change of its speed.

Gravitational mass can be described as a measure of magnitude of the gravitational force which is

exerted by an object (active gravitational mass), or experienced by an object (passive gravitational force)

Units

The SI unit of mass is the kilogram (kg)

Number of moles

An amount of a substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 12 grams of pure carbon-12 (12C), the isotope of carbon with atomic weight 12

. This corresponds to a value of 6.02214179(30) ×1023 elementary entities of the substance.

Number of moles= givvenmassmolar mass

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Units

The numbers of moles are calculated in moles has the unit symbol mol.

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.

Units

In the International System of Units (SI), the standard unit of area is the square metre (m2), which is the area of a square whose sides are one metre long.

Intensive Variables

An intensive quantity or intensive variable is a physical property of a system that does not depends on the system size or the amount of material in the system.

Examples:

Temperature, chemical potential, density (or specific gravity), viscosity, concentration, specific energy and specific heat capacity

Temperature:

Temperature is a physical property of matter that quantitatively expresses the degree of hot and cold.

The temperature of a substance typically varies with the average speed of the particles that it contains, raised to the second power; that is, it is proportional to the mean kinetic energy of its constituent particles.

Units:

Celsius,Kelvin and Fahrenheit

Density:

The mass density or density of a material is defined as its mass per unit volume.

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Less dense fluids float on more dense fluids if they do not mix. This concept can be extended, with some care, to less dense solids floating on more dense fluids. If the average density (including any air below the waterline) of an object is less than water (1000 kg/m3) it will float in water and if it is more than water's it will sink in water.

Density=massvolume

The mass density of a material varies with temperature and pressure. (The variance is typically small for solids and liquids and much greater for gasses.) Increasing the pressure on an object decreases the volume of the object and therefore increases its density. Increasing the temperature of a substance (with some exceptions) decreases its density by increasing the volume of that substance.

Units

SI unit iskgm3 .

Viscosity:

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress.

Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity.

Newtonian: fluids, such as water and most gases which have a constant viscosity. Shear thickening: viscosity increases with the rate of shear. Shear thinning: viscosity decreases with the rate of shear. Shear thinning liquids are very

commonly, but misleadingly, described as thyrotrophic. Thixotropic: materials which become less viscous over time when shaken, agitated, or

otherwise stressed. Rheopectic: materials which become more viscous over time when shaken, agitated, or

otherwise stressed. A Bingham plastic is a material that behaves as a solid at low stresses but flows as a

viscous fluid at high stresses.

Units

The SI physical unit of dynamic viscosity is the Pascal-second (Pa·s), (equivalent to N·s/m2, or kg/ (m·s)).

The cgs physical unit for kinematic viscosity is the stokes (St). It is sometimes expressed in terms of centistokes (cSt). In U.S. usage, stoke is sometimes used as the singular form.

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1 St = 1 cm2·s−1 = 10−4 m2·s−1.1 cSt = 1 mm2·s−1 = 10−6m2·s−1.

Specific heat capacity

The specific heat capacity of a solid or liquid is defined as the heat required to raise unit mass of substance by one degree of temperature. The units for the specific heat capacity are

THERMODYNAMIC MODELS

Thermodynamic models are used to predict chemical and physical equilibria. Here we will discuss these models in detail i.e. their introduction, origin, applicability and their limitations. Benson group increment theory, Joback method, Non-random two-liquid mode (NRTL Equation) , Klincewicz method, Lee-Kesler method, Lydersen method, and equation of state (EOS).

1. Benson Group Increment Theory

Benson Group Increment Theory (BGIT), or Group Increment Theory, uses the experimentally calculated heat of formation for individual groups of atoms to calculate the entire heat of formation for a molecule under investigation. This can be a quick and convenient way to determine theoretical heats of formation without conducting tedious experimentation.

Heats of formations are intimately related to bond dissociation energies and thus are important in understanding chemical structure and reactivity.[1] Furthermore, although the theory is old, it still is practically useful as one of the best group additivity methods aside from computational methods such as molecular mechanics. However, the BGIT has its limitations, and thus cannot always predict the precise heat of formation.

Origin

Benson and Buss originated the BGIT in a 1958 paper. Within this manuscript, Benson and Buss proposed four approximations:

1. A Limiting Law for Additivity Rules.2. Zero-Order Approximation. Additivity of Atomic Properties3. First Order Approximation. Additivity of Bond Properties4. Second Order Approximation. Additivity of Group Properties.

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These approximations account for the atomic, bond, and group contributions to heat capacity (Cp), enthalpy (ΔH°), and entropy (ΔS°). The most important of these approximations to the group increment theory is the Second Order Approximation, because this approximation "leads to the direct method of writing the properties of a compound as the sum of the properties of its group."

The "disproportionation" reactions that Benson and Buss refer to are termed loosely as "radical disproportionation" reactions. From this they termed a "group" as a polyvalent atom connected together with its ligands. However, they noted that under all approximations ringed systems and unsaturated centers do not follow additivity rules due to their preservation under disproprotionation reactions. One can understand this as you must break a ring at more than one site to actually undergo a disproportionation reaction. This holds true with double and triple bonds, as you must break them multiple times to break their structure. They concluded that these atoms must be considered as distinct entities. Hence we see Cd and CB groups which take into account these groups as being individual entities. Furthermore, this leaves error for ring strain as we will see in its limitations.

Applications

Simple Benson Model of isobutylbenzene

As stated above, BGIT can be used to calculate heats of formation which are important in understanding the strengths of bonds and entire molecules. Furthermore, the method can be used to quickly estimate whether a reaction is endothermic or exothermic. These values are for gas phase thermodynamics and typically at 298K.

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Limitations

As powerful as it is, BGIT does have several limitations which restrict its usage.

Inaccuracy

There is an overall 2-3 Kcal/mol error using the Benson Group Increment Theory to calculate the △Hf. The value of each group is estimated on the base of the average △△Hf

0shown above and there will be a dispersion around the average △△Hf0. Also, it can only

be as accurate as the experimental accuracy. That's the derivation of the error and there is nearly no way to make it more accurate.

Group Availability

The BGIT is based on empirical data and heat of formation. Some groups are too hard to measure, so not all the existing groups are available in the Table above.Sometimes approximation should be made when we meet those unavailable groups. For example, we need to approximate C as Ct and N as NI in C≡N,which clearly cause more inaccuracy, which is another drawback.

Ring Strain, Intermolecular and Intramolecular Interactions

In the BGIT, we assumed that a CH2 always makes a constant contribution to △Hf0 for a

molecule. However, a small ring such as cyclobutane leads to a substantial failure for the BGIT, because of its strain energy. A series of correction terms for common ring systems has been developed, with the goal of obtaining accurate △Hf

0 values for cyclic system. Representative values are given in the Table shown below. Note that these are not identically equal to the accepted strain energies for the parent ring system, although they are quite close. The group increment correction for a cyclobutane is based on △Hf

0 values for a number of structures, and represents an average value that gives the best agreement with the range of experimental data. In contrast, the strain energy of cyclobutane is specific to the parent compound, with their new corrections, it is now possible to predict △Hf

0 values for strained ring system, by first adding up all the basic group increments and then adding appropriate ring strain correction values.

2. Joback Method

The Joback method (often named Joback/Reid method) predicts eleven important and commonly used pure component thermodynamic properties from molecular structure only.

Basic Principles

Group Contribution MethodPrinciple of a Group Contribution Method

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The Joback method is a group contribution method. These kind of methods use basic structural information of a chemical molecule like a list of simple functional groups, adds parameters to these functional groups, and calculates thermophysical and transport properties as a function of the sum of group parameters.

Joback assumes that there are no interactions between the groups and therefore only uses additive contributions and no contributions for interactions between groups. Other group contribution methods, especially methods like UNIFAC, which estimate mixture properties like activity coefficients, use both simple additive group parameters and group interaction parameters. The big advantage of using only simple group parameters is the small number of needed parameters. The number of needed group interaction parameters gets very high for an increasing number of groups (1 for two groups, 3 for three groups, 6 for four groups, 45 for ten groups and twice as much if the interactions are not symmetric.).

Model Strengths and Weaknesses

Strengths

The popularity and success of the Joback method mainly originates from the single group list for all properties. This allows to get all eleven supported properties from a single analysis of the molecular structure.

The Joback method additionally uses a very simple and easy to assign group scheme which makes the method usable also for people with only basic chemical knowledge.

WeaknessesSystematic Errors of the Joback Method (Normal Boiling Point)

Newer developments of estimation methods have shown that the quality of the Joback method is limited. The original authors already stated themselves in the original paper: “High accuracy is not claimed, but the proposed method are often as or more accurate than techniques in common use today.”

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The list of groups don't cover many common molecules sufficiently. Especially aromatic compounds are not differentiated from normal ring containing components. This is a severe problem because aromatic and aliphatic components differ strongly.

The data base Joback and Reid used for obtaining the group parameters was rather small and covered only a limited number of different molecules. The best coverage has been achieved for normal boiling points (438 components) and the worst for heat of fusion (155 components). Current developments which can use data banks like the Dortmund Data Bank or the DIPPR data base have a much broader coverage.

The formula used for the prediction of the normal boiling point shows another problem. Joback assumed a constant contribution of added groups in homologous series like the alkanes. This doesn't describe the real behavior of the normal boiling points correctly. Instead of the constant contribution a decrease of the contribution with increasing number of groups must be applied. The chosen formula of the Joback method leads to high deviations for large and small molecules and an acceptable good estimation only for mid-sized components.

3. Non-Random Two-Liquid Model

The non-random two-liquid model[1] (short NRTL equation) is an activity coefficient model that correlates the activity coefficients γi of a compound i with its mole fractions xi in the liquid phase concerned. It is frequently applied in the field of chemical engineering to calculate phase equilibria. The concept of NRTL is based on the hypothesis of Wilson that the local concentration around a molecule is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind Uii and that with the molecules of the other kind Uij. The energy difference also introduces a non-randomness at the local molecular level. The NRTL model belongs to the so-called local-composition models. Other models of this type are the Wilson model, the UNIQUAC model, and the group contribution model UNIFAC. These local-composition models are not thermodynamically consistent due to the assumption that the local composition around molecule i is independent of the local composition around molecule j. This assumption is not true, as was shown by Flemmer in 1976.

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General equations

The general equation for ln(γi) for species i in a mixture of n components is:

with

There are several different equations forms for αij and τij, the most general of which are shown above.

Parameter determination

The NRTL parameters are fitted to activity coefficients that have been derived from experimentally determined phase equilibrium data (vapor–liquid, liquid–liquid, solid–liquid) as well as from heats of mixing. The source of the experimental data are often factual data banks like the Dortmund Data Bank. Other options are direct experimental work and predicted activity coefficients with UNIFAC and similar models. Noteworthy is that for the same liquid mixture several NRTL parameter sets might exist. The NRTL parameter set to use depends on the kind of phase equilibrium (i.e. solid–liquid, liquid–liquid, vapor–liquid). In the case of the description of a vapor–liquid equilibria it is necessary to know which saturated vapor pressure of the pure components was used and whether the gas phase was treated as an ideal or a real gas. Accurate saturated vapor pressure values are important in the determination or the description of an azeotrope. The gas fugacity coefficients are mostly set to unity (ideal gas assumption), but for vapor-liquid equilibria at high pressures (i.e. > 10 bar) an equation of state is needed to calculate the gas fugacity coefficient for a real gas description.

4. Klincewicz Method

In thermodynamic theory, the Klincewicz method[1] is a predictive method based both on group contributions and on a correlation with some basic molecular properties. The method estimates the critical temperature, the critical pressure, and the critical volume of pure components.

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Model description

As a group contribution method the Klincewicz method correlates some structural information of a chemical molecule with the critical data. The used structural information are small functional groups which are assumed to have no interactions. This assumption makes it possible to calculate the thermodynamic properties directly from the sums of the group contributions. The correlation method does not even use these functional groups, only the molecular weight and the number of atoms are used as molecular descriptors.

The prediction of the critical temperature relies on the knowledge of the normal boiling point because the method only predicts the relation of the normal boiling point and the critical temperature and not directly the critical temperature. The critical volume and pressure however are directly predicted.

Model Quality

The quality of the Klincewicz method is not superior to older methods, especially the method of Ambrose gives somewhat better results as stated by the original authors and by Reid et al.The advantage of the Klincewicz method is that it is less complex.

The aspect where the Klincewicz method is unique and useful are the alternative equations where only very basic molecular data like the molecular weight and the atom count are used.

Deviation diagrams

The diagrams show estimated critical data of hydrocarbons together with experimental data. An estimation would be perfect if all data points would lie directly on the diagonal line. Only the simple correlation of the Klincewicz method with the moelcular weight and the atom count have been used in this example.

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Equations

Klincewicz published two sets of equations. The first uses contributions of 35 different groups. These group contribution based equations are giving somewhat better results than the very simple equations based only on correlations with the molecular weight and the atom count.

Group-contribution-based equations

Equations based on correlation with molecular weight and atom count only;

with

MW: Molecular weight in g/mol

Tb: Normal boiling point in K

A: Number of atoms

5. Lydersen Method

The Lydersen method is a group contribution method for the estimation of critical properties temperature (Tc), pressure (Pc) and volume (Vc). The Lydersen method is the prototype for and ancestor of many new models like Joback, Klincewicz, Ambrose, Gani-Constantinou and others.

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The Lydersen method is based in case of the critical temperature on the Guldberg rule which establishes a relation between the normal boiling point and the critical temperature.

Equations

Critical temperature

Guldberg has found that a rough estimate of the normal boiling point Tb, when expressed in kelvins (i.e., as an absolute temperature), is approximately two-thirds of the critical temperature Tc. Lydersen uses this basic idea but calculates more accurate values.

Critical pressure

Critical volume

M is the molar mass and Gi are the group contributions (different for all three properties) for functional groups of a molecule.

6. Equation of State

In physics and thermodynamics, an equation of state is a relation between state variables.[1]

More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.

Major equations of state

For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:

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In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.

= pressure (absolute)

= volume

= number of moles of a substance

= = molar volume, the volume of 1 mole of gas or liquid

= absolute temperature

= ideal gas constant (8.314472 J/(mol·K))

= pressure at the critical point

= molar volume at the critical point

= absolute temperature at the critical point

Classical ideal gas law

The classical ideal gas law may be written:

The ideal gas law may also be expressed as follows

where ρ is the density, γ = Cp / Cv is the adiabatic index (ratio of specific heats), e = CvT is the internal energy per unit mass (the "specific internal energy"), Cv is the specific heat at constant volume, and Cp is the specific heat at constant pressure.

Cubic equations of state

Cubic equations of state are called such because they can be rewritten as a cubic function of Vm.

Van der Waals equation of state

The Van der Waals equation of state may be written:

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where Vm is molar volume, and a and b are substance-specific constants. They can be calculated from the critical properties pc,Tc and Vc (noting that Vc is the molar volume at the critical point) as:

Also written as

The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons:

1. Molecules are thought as particles with volume, not material points. Thus V cannot be too little, less than some constant. So we get (V − b) instead of V.

2. While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (p + something) instead of p. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface.

With the reduced state variables, i.e. Vr=Vm/Vc, Pr=P/Pc and Tr=T/Tc, the reduced form of the Van der Waals equation can be formulated:

The benefit of this form is that for given Tr and Pr, the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:

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For Pr<1 and Tr<1, the system is in a state of vapor-liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.

Redlich–Kwong equation of state

Introduced in 1949 the Redlich–Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. The Redlich–Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):

Peng–Robinson equation of state

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In polynomial form:

where, ω is the acentric factor of the species, R is the universal gas constant and Z=PV/(RT) is compressibility factor.

Non-cubic equations of state

Dieterici equation of state

where a is associated with the interaction between molecules and b takes into account the finite size of the molecules, similarly to the Van der Waals equation.

The reduced coordinates are:

Virial equations of state

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the

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mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. In this case B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only.

It can also be used to work out the Boyle Temperature (the temperature at which B = 0 and ideal gas laws apply) from a and b from the Van der Waals equation of state, if you use the value for B shown below:

The BWR equation of state Benedict–Webb–Rubin equation

where

p = pressure

ρ = the molar density

REFERENCES

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http://www.cambridge.org/us/engineering/turns/assets/0521850436c02_p046-171.pdf http://www.aimath.org/WWN/phasetransition/Defs16.pdf http://www.eng.fsu.edu/~dommelen/quantum/style_a/qsbv.html http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470697261.html http://en.wikipedia.org/wiki/Category:Thermodynamic_models Book: Equilibrium stage separation operations in Chemical Engineering By Ernest J.

Henley & Sieder Book: Chemical Process Equipment By Stainley M. Walas Introduction to chemical Engineering Thermodynamics By J.M.Smith, H.C.Van Ness and

M.M.Abbott

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