Pure Component System by Karthiga Nagarajan

Fugacity coefficient

karthigan1 — 2020-10-01 03:06:19 UTC

For a pure Ideal gas fugacity coefficient is one.

Definition

karthigan1 — 2020-10-01 03:13:11 UTC

Describes the tendency of a species to move to a stable phase.

Gas/Vapour: Lee/Kesler & 2nd Virial Coefficient

karthigan1 — 2020-10-01 03:13:13 UTC

Z is a fn of P_r, T_r and acentric factor
Both methods use Lee/Kesler chart/correlation table

2nd Virial:
Uses an analytical expression obtained from Lee/Kesler correlation table. (Pitzer Correlation)
Applicable for low to moderate pressure (Pr<1.5) and (Tr>0.7)

Refer to Figure 3.13 in Smith Van Ness Textbook on the validity of 2nd Virial Coefficients method.

Definition

karthigan1 — 2020-10-01 03:34:15 UTC

Change in Total G w.r.t change in composition (P,T constant)

Definition

karthigan1 — 2020-10-01 03:35:53 UTC

Deviation from Ideal Gas Behaviour
Fugacity (real state, T, P)divided by Fugacity (Ideal gas state, T, P)

Chemical Potential

karthigan1 — 2020-10-01 03:38:39 UTC

Pure species: Molar Gibbs Energy
Ideal Gas state: Negative infinity

Fugacity

karthigan1 — 2020-10-01 03:39:54 UTC

Pure species: In terms of residual molar Gibbs energy
Ideal Gas: System Pressure

Liquid: Poynting Equation

karthigan1 — 2020-10-01 08:59:42 UTC

Requires Saturated conditions data
PCF is one when P-Psat <10 bar
Reference state will be saturated conditions (Vapour & Liquid Phase in Equilibrium)

Liquid: Alternate method

karthigan1 — 2020-10-01 09:00:25 UTC

Governing eqn similar to Poynting Eqn.
Reference state will be a liquid state (not saturated condition) at same T but different P

Rackett Equation

karthigan1 — 2020-10-01 09:02:54 UTC

An empirical equation used to evaluate molar volume of saturated liquid.
Zc, Vc for chemical species is available in Appendix B.1

Saturated Liquid Volume

karthigan1 — 2020-10-01 09:03:47 UTC

Molar volume of liquid will be essentially the same as saturated liquid at the same

Gas/Vapour: PVT Experimental Data

karthigan1 — 2020-10-01 09:05:55 UTC

Methods:
a) Numerical Integration
b) Graphical method
c) Fit an eqn to the curve and integrate the equation

Reference state: For gas/vapour phase Ideal gas state

Special Cases

karthigan1 — 2020-10-09 08:49:03 UTC

For gases at Low P: Fugacity will be system P
For liquids at Low P: Fugacity will be saturation pressure

Derivations: Are they important?

karthigan1 — 2021-09-29 01:23:16 UTC

Yes. Understanding the derivation will help you to identify the constraints/assumptions/ limitations of the equation.

You will be able to judge whether an equation can be applied to a system in problem solving based on the limitations/constraints/ assumptions used to derive the equation.

For instance,

Poynting equation is applicable for pure liquids only. We will require property values at saturated conditions (VLE) to evaluate fugacity/fugacity coefficient of a pure liquids

Saturated Molar Volume Vs Liquid Molar Volume

karthigan1 — 2021-09-30 05:59:12 UTC

For liquids, molar volume is independent of system pressure. Molar volume at saturated condition (T, P_sat) will be same as liquid molar volume (T, P)

Integration of defining equation for fugacity

karthigan1 — 2021-09-30 06:01:52 UTC

Gases: Reference state will be ideal gas and final state will be system condition

Liquids: Reference state will be saturated condition (VLE) and final state will be system condition

Is Graphical/Numerical Integration required (PVT method)?

karthigan1 — 2021-09-30 06:04:33 UTC

For gases/vapour the effect of P on molar volume is significant and is well understood from the PVT data.
We should not directly (mathematically) integrate the equation and apply limits. It is mandatory to perform numerical/grahical integration

Ideal Gas approximation

karthigan1 — 2021-09-30 06:05:39 UTC

Gas/Vapour phase can be modeled as an ideal gas if system P<<10 bar or compressibility factor is closer to one.

For qn1b and c, why does Vi (sat) = Vi (L) when not at VLE

2022-09-29 08:25:21 UTC

Molar volume of liquid at system T, P is calculated as saturated liquid volume at system T (Psat). Effect of P on liquid properties is significantly small. We therefore use Vi sat instead of Vi L in part (c).

2022-09-29 08:38:46 UTC

Is the expression for fugacity used for both Vapor and liquid phase?
f = Pexp(P^3a/3R)
This expression is derived using EOS for pure A at vapour state. It's applicable only for the given vapour.

2022-10-05 14:03:08 UTC

How do you define 1 mole of a mixture, given that it doesn't have a fixed composition?
A mixture should have a well defined %composition of species to state the mole numbers.
Say a binary mixture, should have certain % of two species. % can be based on mole or mass. When based on mass, one can calculate the mole numbers of individual species using their mol wt. Sum of the moles of the individual species will be the mole number of the mixture.
Alternatively, molecular mass of mixtures based on %constituent species can be defined to calculate the mole number

According to the key learning points showed during the tutorial, at pure liquids at low pressure (below 10 bars), the PCF value is 1. For tutorial 1 question 1b, the PCF value I calculated is 1.45, thus the value of fi (L) = 1.45 P(sat). Thus are we still allowed to use the assumption mentioned above (PCF = 1) for P = 10 bars?

2022-10-07 09:17:14 UTC

Please check the calculation and dimensions while calculating PCF. For tutorial 7 Q1b PCF should be 1.007

2022-10-09 03:39:05 UTC

For the red arrow, why do we have to integrate twice?

Shouldn't 1/P dP = lnP? Why do we need to integrate lnP again

Differential of ln x is 1/x*dx based on function of function rule. Using this dP/P in the integral is rewritten as d(lnP)
We aren't integrating twice.

Also, Z is a function of P. It's mathematically incorrect to write the integral as (Z-1) ln P.

e0775197 — 2022-10-16 13:56:16 UTC

Does this mean that residual properties can used to describe liquid phase as well? Yes
Also, why is there an ideal gas state for the liquid phase?

Ideal gas is a reference state (hypothetical state) where there are no molecular interactions. Concept of residual will exist for any system where the interaction among molecules is significant.

Part II-1-Single Component Systems-Problem Solving Slide 18/23

2022-10-18 00:56:23 UTC

why is VA(L) = 0.12m^3/mol?
From the PV data provided, it's the molar volume of liquid (saturated)

Question 2 Part a

2022-10-29 15:16:26 UTC

Hi Prof, I'm having trouble obtaining the residual volume values for the table. Here is one row of the data. Could I check if the conversion has been done correctly?

Volume is dependent on pressure. It is incorrect to integrate the equation. You have to plot (V-RT/P) Vs P and find area under the curve.

I understand, but i'm having trouble finding the value of V-RT/P. Can I check if the residual volume is correct for that particular row of data?

qn 2 part a

2022-10-29 15:40:14 UTC

Hi Prof, to clarify my question, could I ask if the working to obtain the residual property of molar volume for the first data point is correct?
Please check the units for R, 8.314 m3Pa/K-mol. Calculation seems fine.

Qn 2 part b

2022-10-30 06:31:32 UTC

Hi Prof,

I'm struggling at this question because i keep getting a math error. Could I ask if the terms used are correct? molar volume of liquid is in terms of cm3/mol and I used 83.14 cm3-bar/K-mol.

Please note volume is reported as m3/K-mol. Check the units of all terms and then substitute in the PCF.

2022-11-16 06:14:19 UTC

hi prof,

May i clarify what the highlighted portion mean? What is greatly bigger than Psat such that fugacity coefficient is equal to 1?

Phi i sat is one when Pi sat is less than 10 bar

criterion of exactness

2023-10-03 13:48:30 UTC

prof can u explain criterion of exactness in class

Why is fugacity needed when the compressibility factor already defines the deviation of a real fluid from the ideal gas state?

2023-10-04 04:47:14 UTC

For a control volume at constant temperature, molar volume (of the mixture) is constant, therefore V and T cancel and we are left with P1 = P2 (from the ideal gas law).

And as the pressure is constant too, we can conclude that P1 = P2 = Psat. How is fugacity relevant when the compressibility factor, Z, pretty much does the same thing?

2023-10-04 15:30:08 UTC

for this qn how do u know P>Psat ie how did u find Psat

2023-10-05 07:13:31 UTC

When deriving various property relations for equilibrium, we set temperature to be constant first. Is this because we are accounting for situation under equilibrium, where T and P are constant (or we say specific)?

2023-10-05 07:40:31 UTC

Dear Dr Karthiga , what is the diff between chemical potential and Gibbs free energy

2023-10-05 07:41:22 UTC

Dumb question - why is setting temperature constant easier than setting pressure constant?

2023-10-05 08:03:22 UTC

What is the physical significance of Pr < 1.5?

Hi Prof, can you please explain again why Vi^sat = Vi^L for question 1?

2023-10-05 08:10:27 UTC

2023-10-05 08:11:09 UTC

Posting queries results in the latest query being posted at the very bottom. Is it possible to change the Padlet so that the most recent queries are shown on top?

Why are the Fi(sat) values obtained from method 1 and 2 so different

2023-10-05 08:11:31 UTC

2023-10-05 08:12:50 UTC

Hi prof, using the 2 methods (generalised correlations and Virial coefficient), we obtained 2 different values for phi sat. How do we determine which one is more accurate?

2023-10-05 12:11:22 UTC

I did poorly for the first half of this semester, but I am motivated to do better after attending your lecture today. Your problem-solving lectures are very thorough and concise. Thank you, Prof Karthiga!

Poynting Equation

2023-10-12 18:18:03 UTC

Why is poynting correction =1 for low pressure liquid

Saturated condition fugacity at Pi(sat)<10 bar

2023-10-14 03:57:37 UTC

Hi Prof, could you please explain why fsat equals Psat at low pressure for saturated condition? is it because the system resembles an ideal gas state (even though it has liquid and is in VLE)?

Liquids at Low P

2023-10-14 04:01:57 UTC

Hi prof could u please explain why fugacityof liquid at P<10 bar equals Pi(sat) and not the system P? Because fugacity equals system pressure for gases at low pressure(P) and saturated condition at low pressure(system P=Psat)... Why is it different for liquids at low pressure?

Feedback on Lecture Quiz

2023-10-17 02:59:15 UTC

Hi Prof, while we appreciate the use of the lecture quiz at a 'concept check' at the end of each week, I think most students find it being graded quite daunting and stressful

One better alternative may be to give the marks based on participation - i.e. if you did poll your answer, you will be provided marks regardless of right or wrong

Perhaps the opinions of the students could be polled on this matter, since the lecture quiz is a new thing the teaching team is trialing and refining on :)

2023-10-17 12:23:01 UTC

Hi prof for this question shouldnt the denominator for fugacity coefficient of liquid be x1f1 instead of x1P?

Practice 1

2023-10-18 03:01:57 UTC

Hi prof, how do we determine the volume of liquid A for part (b)? I was thinking of using the poynting equation to get the fugacity since Psat=611 Pa< 10 bar which means saturation fugacity coefficient=1. But I'm unsure of how to calculate PCF without volume of liquid A, since the number of moles of liquid A was not given and only the density was given.

2023-10-21 08:47:50 UTC

Gibbs free energy of a real liquid consist of its residual and the ideal gas gibbs energy. Why liquid state has ideal gas gibbs free energy? (isnt one liquid phase one gas phase?)

Compressibility

2023-10-22 13:11:37 UTC

What is the difference between 2-parameter and 3-parameter compressibility factor method to calculate fugacity coefficient?

2023-10-23 08:42:51 UTC

hi prof why is the line RKAc1 the vaporisation curve of species 1?

Intuition for fugacity

2023-10-23 13:33:08 UTC

Hi prof, I was abit confused when you mentioned "reference state for fugacity is defined as 0", can I check if my understanding is correct?

Here's how I understand it:
For chemical potential in ideal gas condition, its reference state is defined as negative infinity thus its hard to apply mathematically, which results in the introduction of fugacity, where at ideal gas condition, fugacity is defined as the equivalent of system pressure, thus the change in residual gibbs free energy at ideal gas condition is 0, and hence thats the "reference state is defined as 0" you mentioned during the lecture?

2023-10-24 13:20:52 UTC

hi prof what is the mathematical step for deriving this formula of Pdew i cant seem to get it thanks

2023-10-25 09:28:39 UTC

Hello prof, how do i tackle this problem, im stuck

2023-10-26 12:10:10 UTC

Hi, Prof! I understand that the fugacities of every phase is equal to each other at phase equilibrium, but how are they equal to the system pressure?

2023-10-26 14:53:16 UTC

Hi, Prof Karthiga! For part b), using the virial coefficient method (as Pr = 0.4607 << 1.5), my calculator showed that the fugacity coefficient was 0.7543.

Am I allowed to use the P - Psat < 10 bar condition here to approximate the fugacity coefficient to be 1 as P - Psat is almost lesser than 10 bar?

2023-10-27 09:20:57 UTC

hi prof what is the 2 and 3-parameter compressibility factor method

2023-10-28 09:31:54 UTC

Hi prof, i cant get the answer for qn 4a. where did i go wrong?

2023-10-28 14:13:51 UTC

if Pr>1.5 will it still valid to use virial equation? or can we only use generalised correlation

if Pr<1.5 we can use any method right?

2023-10-31 13:45:39 UTC

prof how do we obtain partial excess gibbs energy from this question

2023-11-01 17:03:56 UTC

prof how do u read the K-factors chart

Prof, what does fugacity coeff. > 1 and < 1 signify?

2023-11-03 13:09:13 UTC

We know that the activity coefficient of a mixture when > 1 means that like species attract each other strongly and < 1 means that unlike species attract each other more.

What can we infer when the fugacity coeff. is > 1 and < 1? Thank you

2023-11-03 16:15:46 UTC

Instead of using the General Equation as shown above. Is it correct to calculate the individual Fi coeffi respectively using virial equation? I did that and i got a slight difference in answer possibly due to some rounding error.

Hi Prof, in poynting equation, the saturated fugacity coefficient will become 1 under what condition? I read the notes that it will become 1 when saturated pressure << but don't quite understand what it means. Thank you.

2023-11-04 01:43:34 UTC

2023-11-04 01:49:41 UTC

For this problem solving question, since the sat pressure <10bar, why we didn't just assume the sat. fugacity coeffient to be 1? The sat. pressure in this qns is 5.267 bar.

2023-11-04 10:18:34 UTC

Hi prof, how do we do part a for the partial excess Gibbs energy part. Do we convert x2 to 1-x1 and differentiate Ge wrt x1 and then do the same for x2?

Prof, is the fugacity coefficient of a liquid/solid zero as there are relatively stronger intermolecular attractions in comparison to the non-existent interactions between those of an ideal gas

2023-11-04 13:27:23 UTC

2023-11-08 10:24:33 UTC

prof where do we find the table for standard delta G for reactions

Help with Z

2024-11-05 05:23:40 UTC

Prof, I know how to get Z for this question via superheated steam tables, but how do I solve with this equation: d(ln𝜙i) = (Z-1)dP/P, especially when P will have to be zero for the lower end of integration?

Lecture recording

2025-10-02 07:05:45 UTC

Are the problem solving lectures recording

Why do we need fugacity when we already have Pressure?

2025-10-02 08:49:02 UTC

What physical situations make them differ significantly?

When the vapor pressure equals system pressure (P = Psat), why does φ at saturation equal both the vapor and liquid phase fugacity coefficients?

2025-10-02 08:50:16 UTC

Usage of Lee Kesler tables to calculate fugacity coefficient at saturation

2025-10-02 15:57:38 UTC

Hi Prof, to my understanding, the Lee Kesler tables include properties for liquids and vapours. The idea of using Tr and Pr to normalise properties using the Theorem of Corresponding States means that the Lee-Kesler Tables should have a saturation curve where the difference in vapour and liquid properties is very obvious (easy to understand and see for the Z table). We were specifically warned by Prof Sergey against using values near the saturation curve (shown explicitly in lecture slides 11:accentric factors and corresponding states) since there is a large gap in properties such as Z, Hr, Sr and phi at these values, and the tables indicate the values for liquid states in italics as well.

With that out of the way, would the method used in Q1 of the problem solving lecture to obtain the fugacity coeff. at saturation through the tables be erroneous? I tried the linear interpolation myself and the values read off the table are very far apart since they are read at the saturation line. It just happens the linear interpolation gives a nice value.

If this method does not hold, can we still use the fugacity coefficients at P = 275bar in a meaningful way, by doing the calculation similar to how we did for a vapour? It seems the Lee Kesler tables are valid for liquids based on the integration of the Z tables for liquids, but the textbook does not elaborate on how the tables are generated, and only does a worked example on a supercritical fluid.

I have attached an attempt by me at using the table to calculate the fugacity coefficient of the liquid using the table.

2025-10-13 07:47:07 UTC

W7 Q3

2025-11-06 08:35:00 UTC

Hi prof, may i see the working for part b)? Im unable to get the fugacity of liquid phase. My PCF is 1.