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5. Calculating Fugacity of Pure Gases

- To calculate the fugacity of a pure gas requires

a knowledge of the P,V,T behaviour of the

substance. This can take many forms, and our

choice is often governed by the required

precision, and the availability of

data/correlations. - In all cases, we can apply the following

relation - Section 11.7 of the text presents a generalized

method of calculating fi for pure gases that are

non-polar or slightly polar. - Lee-Kesler Correlation
- ? (?o)(?1)? 11.67
- where ?o and ?1 are tabulated functions of

reduced P and T and ? is the acentric factor of

the substance

Calculating Fugacity of Pure Gases

- Virial Equation
- We have already used another correlation in an

example. In cases where the simplest form of the

virial equation of state applies, we can

calculate fugacity from - 11.68
- where
- 3.65
- 3.66
- See the previously worked out example for a

demonstration of this approach.

Applicability of Simple Correlations

- It is very important to understand under what

conditions the simple correlations apply.

Pure Component VLE in Terms of Fugacity

- Consider a pure component at its vapour pressure
- Phase rule tells us, F2-21 1 degree of

freedom - Therefore, at a given T, there can only be a

single pressure, Psat for which a vapour and a

liquid are stable - Along the phase boundary, the chemical potentials

are equal - How do the fugacities of the liquid and gas

relate?

P liquid gas T

Pure Component VLE in Terms of Fugacity

- For the non-ideal, pure gas we can write
- 11.38a
- For a non-ideal liquid, we can define an

analogous expression - 11.38b
- At equilibrium, we apply the criterion on the

basis of chemical potential to give us - or
- 11.39
- In terms of fugacity coefficients
- 11.41
- All of these equations apply to a pure substance

at Pisat

Review of Chemical Equilibrium Criteria

- We now have several different forms of the

criterion for chemical equilibrium. While they

stem from the same theory, they differ in

practical applicability. - A system at equilibrium has the following

properties - the total Gibbs energy of the system is

minimized, meaning that no change in the number

of phases or their composition could lower the

Gibbs energy further - the chemical potential of each component, i, is

the same in every phase within the system - in p phases
- the fugacity of each component, i, is equal in

every phase of the system - in p phases

Calculating the Fugacity of Pure Liquids

- The derivation of the fugacity of a pure liquid

at a given T, P is comprised of four steps - Step 1. Calculate the fugacity of a vapour at

Pisat - Step 2. Calculate the change in Gibbs energy

between Pisat and the given pressure P

using the fundamental equation - dG VdP - SdT (constant T)
- which after integration yields
- Given that liquids are nearly incompressible

(Viliq is not a strong function of P) the

integral is easily equated to - (A)

Calculating the Fugacity of Pure Liquids

- 3. Using the definitions of fugacity
- we can take the difference
- (B)
- 4. Substituting A into B
- or
- or
- 11.44

Calculating the Fugacity of Pure Liquids

- We can now calculate the fugacity of any pure

liquid using two equations - 11.44
- and
- 11.35
- The exponential within Equation 11.44 accounts

for the change in Gibbs energy as we compress the

liquid from Pisat to the specified pressure, P.

This is known as the Poynting factor. - This contribution to fugacity is slight at all

pressures near Pisat, and is often assumed to be

unity.

Example Problem

Calculate the change in the chemical potential

of pure n-heptane during an isothermal expansion

at T186 oC from its saturation pressure

(P1Psat) to P22 bar. Under these conditions,

the ideal gas assumption for n-heptane is not

valid. Note the numerical solution to this

problem will count for only 5/50. Make good use

of your time by simply demonstrating to me the

solution procedure (as if you were describing the

solution algorithm). Specify the Appendices and

Tables that you use. Midterm Exam Question,

Fall 2005

Example Problem

Calculate the fugacity of n-pentane at T 25 oC

and P101 kPa. The saturation pressure of

n-pentane at 25 oC is Pisat 67.5 kPa.

What do we need to know so far

(In addition to the bullet items presented in

Lecture 3, Slide 15) How to write the

fundamental relations for auxialiary properties,

as they apply to open and closed (single, or

multi-component) systems How to calculate S,

H, and G changes of mixing for - Ideal gas

mixtures - Perfect solutions How to express

the chemical potential of pure (ideal and

non-ideal) compounds (gases and liquids), as well

as that of components in ideal mixtures How to

perform P,T-flash calculations (ideal mixtures,

only) How to calculate the fugacity and

fugacity coefficients of pure (ideal and

non-ideal) substances, existing as gases or

liquids

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