th d ithermodynamic properties from volumetric...

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Advanced Chemical Advanced Chemical Engineering ThermodynamicsEngineering Thermodynamics

Th d iTh d iThermodynamic Thermodynamic Properties from Properties from Volumetric DataVolumetric Data

For any substance, regardless of whether it is pure or a mixture, most thermodynamic properties of interest in phase equilibria can be calculated from thermal and volumetric measurements.For a given phase (solid, liquid, or gas), thermal g p ( , q , g ),measurements (heat capacities) give information on how some thermodynamic properties vary with temperaturewhereas volumetric measurements give information on how thermodynamic properties vary with pressure or d it t t t t tdensity at constant temperature.Whenever there is a change of phase (e.g., fusion or vaporization), additional thermal and volumetric measurements are required to characterize that change.

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It is useful to express a selected thermodynamic function of a substance relative to that which the same substance has as an ideal gas at the same temperature and composition and at some specified pressure or densitypressure or density.

This relative function is often called a residual function.

The fugacity is a relative function because its numerical value is always relative to that of an ideal ygas at unit fugacity

in other words, the standard-state fugacity fi0 in Eq. (2-38) is arbitrarily set equal to some fixed value, usually 1 bar.

The thermodynamic function of primary interest is the fugacity that is directly related to the chemical potential The chemical potential is directly related to the Gibbs energythat by definition is found from the enthalpy andthat, by definition, is found from the enthalpy and entropy. Therefore, a proper discussion of calculation of fugacities from volumetric properties must begin with the question of how enthalpy and entropy, at constant temperature and composition, are related to pressure. p p , pOn the other hand, the chemical potential may also be expressed in terms of the Helmholtz energyIn that event the first question must be how entropy and energy, at constant temperature and composition, are related to volume.

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The answers to these questions may readily be found from Maxwell's relations.

We can then obtain exact equations for the thermodynamic functions U H S A and Gthermodynamic functions U, H, S, A, and G

from these we can derive the chemical potential and, finally, the fugacity.

If we consider a homogeneous mixture at some fixed composition we must specify two additional variablescomposition, we must specify two additional variables.

In practical phase-equilibrium problems, the commonadditional variables are temperature and pressure

We will give equations for the thermodynamic properties with T and P as independent variables

However, volumetric data are most commonly expressed by an equation of state that uses temperature and volume as independent variables

therefore it is a matter of practical importance to have available equations for the thermodynamic properties in terms of T and V.

S 3 1 3The equations in Secs. 3.1 and 3.4 contain no simplifying assumptions;

they are exact and are not restricted to the gas phase but, in principle, apply equally to all phases.

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At constant temperature and composition, we can use one of Maxwell's relations to give the effect of pressure on enthalpy and entropy:

Thermodynamic Properties with Independent Variables P and T

(3-1)

(3-2)

These two relations form the basis of the derivation for the desired equations.

First, expressions for the enthalpy and entropy are found.


The other properties are then calculated from the definitions of enthalpy, Helmholtz energy, and Gibbs energy:

(3-3)(3 3)

(3-4)

(3-5)

(3-6)( )

(3-7)

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The results are given in Eqs. (3-8) to (3-14). All integrations are performed at constant temperature and constant composition.


(3 8)(3-8)

(3-9)

(3-10)

(3-11)


(3-12)

(3-13)(3-13)

(3-14)

Where , is the partial molar volume of i. j

i i T,P,n= V n v

The dimensionless ratio fi/yiP = φi is called the fugacity coefficient.

For a mixture of ideal gases, φi =1, as shown later.

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The symbols have the following meanings:

hi0 = molar enthalpy of pure i as an ideal gas at temperature T

si0 = molar entropy of pure i as an ideal gas at temperature T and

1 barμi

0 = hi0 - Tsi

0 and fi0 = 1 bar

ni = number of moles of inT = total number of molesyi = ni/nT = mole fraction of i

All extensive properties denoted by capital letters (V, U, H, S, A, and G) represent the total property for nT moles and therefore are not on a molar basis. Extensive properties on a molar basis are denoted by lowercase letters (v, u, h, s, a, and g). In Eqs. (3-10) to (3-13), pressure P is in bars.


Equations (3-8) to (3-14) enable us to compute all the desired thermodynamic properties for any substance relative to the ideal-gas state at 1 bar and at the same temperature and composition, provided that we have information on volumetric behavior in the formform

V = F(T, P, n1, n2, ...) (3-15)

To evaluate the integrals, the volumetric information required in function F must be available not only for pressure P, where the thermodynamic properties are desired, but for the entire pressure

0 Prange 0 to P.In Eqs. (3-8) and (3-11), the quantity V appearing in the PV product is the total volume at the system pressure P and at the temperature and composition used throughout. This volume V is found from the equation of state, Eq. (3-15).

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For a pure component, , and Eq. (3-14) simplifies toi i=v v

(3-16)

Where vi is the molar volume of pure i. Equation (3-16) is frequently expressed in the equivalent form

(3-17)

where z, the compressibility factor, is defined by

(3-18)

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First, we consider the fugacity of a component i in a mixture of ideal gases. In that case, the equation of state is:

(3-19)(3-19)

and the partial molar volume of i is

(3-20)

Substituting in Eq. (3-14) gives:fi = yiP (3-21)

For a mixture of ideal gases, then, the fugacity of i is equal to its partial pressure, as expected.


Next, let us assume that the gas mixture follows Amagat's law at all pressures up to the pressure of interest.Amagat's law states that at fixed temperature and pressure, the volume of the mixture is a linear function of the mole numbers

(3-22)

where vi is the molar volume of pure i at the same temperature and pressure and in the same phase.Another way to state Amagat's law is to say that at constant

d h i i i ll itemperature and pressure, the components mix isometrically, i.e., with no change in total volume.If there is no volume change, then the partial molar volume of each component must be equal to its molar volume in the pure state.

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It is this equality which is asserted by Amagat's law.Differentiating Eq. (3-22), we have

(3-23)

Substitution in Eq. (3-14) yields

(3-24)

U i E (3 24) i h E (3 16) b iUpon comparing Eq. (3-24) with Eq. (3-16), we obtain

(3-25)

The Lewis fugacity rule.

Figure 3-1 Compressibility factors for nitrogen/butane mixtures at 171°C(Evans and Watson, 1956).( , )

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Fugacity of a Component in a Mixture at Moderate Pressures

To illustrate the use of Eq. (3-14) with a realistic examplewe compute now the fugacity of a component in a binary mixture at moderate pressures. In this illustrative calculation, we use, for simplicity, a form of h d W l i lid l dthe van der Waals equation valid only to moderate pressures:

(3-26)

where a and b are the van der Waals constants for the mixture. To calculate the fugacity, we must first find an expression for the partial molar volume; V = nTv

(3-27)


Differentiating Eq. (3-27) with respect to n1 gives

(3-28)

We must now specify a mixing ruleA relation that states how constants a and b for the mixture depend on the composition.We use the mixing rules originally proposed by van der Waals:

(3 29)

(3-30)

(3-29)

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(3-31)

(3-32)

In performing the differentiation, it is important to remember that n2 is held constant and that therefore nT cannot also be constant.Algebraic rearrangement and subsequent substitution into Eq. (3-

(3-33)

Algebraic rearrangement and subsequent substitution into Eq. (314) gives the desired result:

(3-34)


Equation (3-34) contains two exponential factors to correct for nonideality. We can therefore rewrite Eq. (3-34) by utilizing the boundary condition

(3 3 )(3-35)

(3-36)

Figure 3-2 presents fugacity coefficients for several g p g yhydrocarbons in binary mixtures with nitrogen. For comparison, we also show the fugacity coefficient of butane according to the Lewis fugacity rulein that calculation, the second exponential [in Eq. (3-34)] was neglected.

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We see that the Lewis rule is poor for butane for two reasons:

First, the mole fraction of butane is small (hence y22 near , ( y2

unity)

Second, the difference in intermolecular forces between butane and nitrogen (as measured by |a1

1/2-a21/2|) is large.

If the gas in excess were hydrogen or helium instead ofIf the gas in excess were hydrogen or helium instead of nitrogen, the deviations from the Lewis rule for butane would be even larger.

Figure 3-2 Fugacity coefficients of light hydrocarbons in binary mixtures with nitrogen at 343 K. Calculations based on simplified form of van der Waals' equation.

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Fugacity of a Pure Liquid or Solid


The first term on the right-hand side gives the fugacity of


(3-37)

g g g ythe saturated vapor, equal to that of the saturated condensed phase. Equation (3-37) becomes

(3-38)

(3-39)

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Figure 3-3 Fugacity coefficients from vapor-phase volumetric data for four saturated liquids.

Because the liquids are at saturation conditions, no Poyntingcorrection is required.

Table 3-1 gives some numerical values of the Poyntingcorrection for an incompressible component with vi

c=100 cm3/mol and T= 300 K.


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Figure 3-4 Fugacity of liquid water at three temperatures from saturation pressure to 414 bar. The critical temperature of water is 374°C.

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At constant temperature and composition, the effect of volume on energy and entropy:

Thermodynamic Properties with Independent Variables V and T

(3-40)(3 40)

(3-41)

These two equations form the basis of the derivation for the desired equations.First, expressions are found for the energy and entropy.The other properties are then calculated from their definitions:


(3-42)

(3-43)

(3-45)

(3-44)

(3-46)

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In Eqs. (3-49) to (3-52), the units of V/niRT are bar-1

No additional terms for change of phase (e.g„ enthalpy of vaporization) need be added to these equations when they are applied to a condensed phase.


(3-47)

(3-48)

(3-49)

(3 48)


(3-50)

(3-51)(3-51)

(3-52)

(3-53)

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For a pure component Eq. (3-53) becomes


(3-54)

Fugacity of a Component in a Mixture According to van der Waals' Equation

To illustrate the applicability of Eq. (3-53), we consider a mixture whose volumetric properties are described by van derWaals' equation:

(3-60)

v is the molar volume of the mixture a and b are constants that depend on composition.

(3-61)

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We want to calculate the fugacity of component i in the mixture at some given temperature, pressure, and composition. Differentiating Eq. (3-61) with respect to ni, we have

Substituting into Eq. (3-53) and integrating, we obtain

(3-62)

(3-63)


At the upper limit of integration, as V→ ∞

and Eq. (3-63) becomes

(3-64)

(3-65)

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At the upper limit of integration, as V→ ∞

and Eq. (3-63) becomes

(3-64)

(3-65)


The composition dependence of constants a and b:

(3-66)

(3-67)

(3-68)

where aij is a measure of the strength of attraction between a molecule i and a molecule j. If i and j are the same chemical species, then aij is the van derWaals a for that substance.

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If i and j are chemically not identical and if we have no experimental data for the i-j mixture, we then need to express aijin terms of ai and aj. j

For i≠j, it was suggested many years ago by Berthelot, on strictly empirical grounds, that

(3-69)

This relation, often called the geometric-mean assumption.


If we adopt the mixing rules given by Eqs. (3-67), (3-68), and (3-69), the fugacity for component i, given by Eq. (3-65), becomes

(3-70)

where v is the molar volume and z is the compressibility factorwhere v is the molar volume and z is the compressibility factor of the mixture.

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For a numerical example, consider the fugacity of hydrogen in a ternary mixture at 50°C and 303 bar containing 20 mol % hydrogen, 50 mol % methane, and 30 mol % ethane. Using Eq (3 70) we find that the fugacity of hydrogen is 114 5Using Eq. (3-70), we find that the fugacity of hydrogen is 114.5 bar.Constants a and b for each component were found from critical properties.

Th l l f h i l l d f dThe molar volume of the mixture, as calculated from van derWaals' equation, is 62.43 cm3/mol.

From the ideal-gas law, the fugacity is 60.8 bar, while the Lewis fugacity rule gives 72.3 bar.


To illustrate the applicability of Eq. (3-53), we consider a mixture whose volumetric properties are described by van derWaals' equation:

v is the molar volume of the mixture a and b are constants that depend on composition.

th d ithermodynamic properties from volumetric...

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