# 6: Fugacity

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Distillation Science (a blend of Chemistry and Chemical Engineering)

This is **Part VI, ****Fugacity **of a ten-part series of technical articles on Distillation Science, as is currently practiced on an industrial level. See also **Part I,**** Overview **for introductory comments, the scope of the article series, and nomenclature.

**Part VI, ****Fugacity **deals with the expected departure of apparent pure-component vapor pressure in the vapor-liquid equilibria (VLE) of binary mixtures, as is commonly found in practical application of distillation science. **Part VII, Liquid Activity Coefficients**** **builds on this for mixtures that have greater-than-expected departures from ideal behavior.

This article uses the Vapor Pressure of **Part ****IV **and Equations of State of **Part V**.

The ideal vapor-liquid behavior of volatile fluid mixtures is governed by the combination of Raoult’s Law and Dalton’s Law. Neither of these is a scientific law but rather a representation of ideal behavior, closely approximated when the fluids in question have very similar properties. Raoult’s Law of liquid partial pressures ( PP* _{i}*= X

** VP*

_{i}*) states that the partial pressure of a fluid is the liquid mole fraction times the pure-component vapor pressure. Dalton’s Law (Y*

_{i}*= PP*

_{i}

_{i}*/ Σ{PP*

_{ }*} ) states that the vapor mole fraction of a volatile component is that component’s partial pressure divided by the sum of all partial pressures. If either of these two relationships is extended, with either X*

_{i}*or Y*

_{i }*being zero or unity, a tautology results. The nomenclature used here is: X*

_{i}*and Y*

_{i}*are the liquid and vapor mole fractions of the*

_{i}*ith*component in the mixture; VP

*and PP*

_{i}*are the vapor pressure and partial pressure of the*

_{i}*ith*component.

Combining Raoult & Dalton, for a binary mixture:

\[ Y_{1}= \dfrac{X_{1} \times VP_{1}}{X_{1} \times VP_{1} + X_{2} \times VP_{2}} \label{6-1} \]

\( Y_{2}=\dfrac{X_{2} \times VP_{2}}{X_{1} \times VP_{1} + X_{2} \times VP_{2}} \nonumber\)

However, in actual practice this never exactly works out. The more volatile fluid exerts an effect on the less volatile fluid, seemingly boosting that fluid’s vapor pressure slightly. Conversely, the less volatile fluid slightly reduces the more volatile fluid’s vapor pressure. There are also interactions in the vapor phase due to compressibility (Z_{v}, from **Part V**), which are sometimes significant. These departures from ideal behavior are due to two factors: (1) the fluid’s phase change is at a temperature that is above or below its pure components’ boiling points; (2) interactions occur between molecules with different properties. From a distillation column design perspective, these departures are important, since they make the component separations more difficult.

The technical term for the first factor of departure is fugacity, which has the same units as the fluid’s vapor pressure, including the slight positive or negative secondary effect. The ratio between a liquid’s fugacity and its vapor pressure is the liquid-phase fugacity coefficient; between the vapor’s fugacity and its partial pressure is the vapor-phase fugacity coefficient. For a fluid by itself, fugacity has no meaning.

But in a mixture of two volatile fluids, it is the vapor and liquid fugacities that are in equilibrium with real fluids ( as opposed to a mixture of ideal fluids, such as with Raoult/Dalton).

\[F_{i}^L = \phi_{i}^L \times VP_{i}\label{6-2} \]

where ø^{L}_{i}* _{ }*denotes the liquid fugacity coefficient

\[F_{i}^V = \phi_{i}^V \times PP_{i}\label{6-3} \]

where ø^{V}_{i}* _{ }*denotes the vapor fugacity coefficient

Consider a mixture of "fluid 1" and "fluid 2". At a given temperature, "fluid 1" has a pure-component vapor pressure of VP_{1}. But the presence of "fluid 2" alters that somewhat, so "fluid 1's" fugacity (F_{1}) is different from its pure-component vapor pressure. Conversely the presence of "fluid 1" somewhat alters "fluid 2's" fugacity (F_{2}), away from it's pure-component vapor pressure of VP_{2}. The higher the mole fraction of "fluid 2", the more "fluid 1's " fugacity is altered away from its pure-component vapor pressure. And vice-versa.

This is how fluid mixtures behave in the real world: Raoult/Dalton is just a introductory concept that would apply in an ideal world.

The technical term for the second factor of departure is the **Liquid Activity Coefficient**, which is covered extensively in Part VII, but initially mentioned here briefly for a more complete understanding.

Fugacity effects are always present whenever two fluids of different volatility are mixed and vaporized or condensed. The Liquid Activity Coefficient effects are always present, but more noticed when the two fluids have significantly different properties, and even more so when the fluids are polar.

This is best illustrated by an example, partially continued from **Part ****IV **and **Part**** V**. At 73.90°C = 347.05°K, TCS has a pure component vapor pressure of 3.50 atmospheres. At that same temperature, STC has a pure component vapor pressure of 1.65 atmospheres. In this example, the liquid mixture is 40% mole fraction TCS, so X_{1}= 0.40 . Therefore STC’s liquid mole fraction, X_{2}= 0.60. *Note: it is conventional notation for the most volatile fluid to have the lowest subscript. *

Per Raoult, the partial pressures PP_{1} and PP_{2} should be \(0.40 \times 3.50=1.40 \nonumber\) and \(0.60 \times 1.65=0.991 \nonumber\) atmospheres, respectively. This liquid mixture at 73.90°C would be expected to boil at \(0.40 + 1.99=2.39 \nonumber\) atmospheres total system pressure; with the TCS mole fraction of the first bubble’s vapor (Y_{1})_{ }being \(1.40/2.39=0.586 \nonumber\)= 0.586 (so Y_{2} is 0.414).

Instead, it is found that first bubble’s vapor has a TCS mole fraction (Y_{1}) of only 0.573 (with Y_{2 }being 0.427) indicating that the TCS was just a little less volatile than expected and the STC was just a little more. While there is just over 2% difference in this case between Raoult/Dalton’s expected and the actual resulting vapor mole fractions, the slight difference is truly present. If the difference in vapor pressures between the two fluids increased, and the overall pressure increased towards critical, the departure from the Raoult & Dalton relationship would then increase.

Two departure-from-ideality factors were purposefully shown in the above example: Fugacity and Liquid Activity Coefficients. The vapor-phase and liquid-phase fugacity coefficients (\(\phi_{i}^V\nonumber\)& \(\phi_{i}^L\nonumber\)) are never exactly unity (as implicit with Raoult &Dalton), but they do often approach unity (especially at low pressures and for molecularly similar fluids). Likewise, the Liquid Activity Coefficients (\(\gamma_{i}\nonumber\)) are never exactly unity for real fluid mixtures (see **Part VII**), but are often set to unity because the fluids are non-interactive. The point to make here is that non-unity fugacity coefficients are just a result of fluids having different volatilities (i.e., they are not fluid-dependent); whereas Liquid Activity Coefficients are specific fluid-system dependent (e.g., TCS-STC has a different set of parameters than DCS-TCS). (*There is no analog to Liquid Activity Coefficients in the vapor phase - vapors just interact due to fugacity effects*.)

In the above example, the overall deviation of 2.3% (0.586 vs 0.573) on TCS mole fraction is almost all due to fugacity (both liquid and vapor), with a lesser amount due to Liquid Activity Coefficients. When designing a distillation for high-reflux purification, the relative importance between these two departure factors often reverses, as well as the overall magnitude.

The general equation (including the \(\gamma_{i}\nonumber\) covered in **Part VII**), is that:

\[Y_{i} \times \pi =PP_{i} =(\phi_{i}^L/ \phi_{i}^V) \times \gamma_{i} \times VP_{i} \times X_{i} \label{6-4} \]

where \(\pi \nonumber\) is the total pressure

Setting aside further discussion of Liquid Activity Coefficients until **Part VII**, the remainder of this article deals with how to determine the fugacity coefficients,\(\phi_{i}^V\nonumber\) & \(\phi_{i}^L\nonumber\). That determination is dependent on the constants of **Part III**, and the calculation of the compressibility factors Z_{v }and Z_{L} from **Part**** V**.

For either phase, the equation for fugacity coefficient, as derived from thermodynamics in integral form is:

\[\operatorname{Ln}(\varnothing)=\int_0^P \left[\left(\dfrac{V}{R T}\right)- \dfrac{1}{P}\right] d P=\int_0^P \left[ Z - \dfrac{1}{P}\right] dP\]

Fortunately, the Peng-Robinson Equation of State (P-R EOS) is good enough for most distillation design to evaluate fugacity coefficients based on some simplifications (see **Part V**). Admittedly the P-R EOS does not exactly fit chlorosilanes (and similar fluids), and a better EOS would be preferable, if it exists. However P-R is adequate to the task, especially with the “work-arounds” given below.

To evaluate \(\phi_{i}^V \nonumber\) there is an almost linear relationship between Z_{v} and P until close to P_{r}=0.4. So that allows Equation 6-5 to be reduced to \[Ln(\phi)=( Z_{V}^{T,P} -Z_{V}^{sat}) \label{6-6} \]

where \(Z_{V}^{T,P} \nonumber\) and \(Z_{V}^{sat} \nonumber\) are respectively the component’s vapor-phase compressibility at the temperature and pressure of the boiling mixture, and at the pure-component saturation condition (T & VP). For P_{r}>0.4, Equation 6-5 tends to give low values of \(\phi^{\nu} \nonumber\).

Since many industrial processes operate at higher pressures (closer to critical than P_{r}=0.4) another "work-around" is needed. Between 0.3< P_{r}<0.90, a series expansion cubic polynomial fits well for Z_{v} as a function of P_{r}, which then expresses the above thermodynamic integral form (Z_{v}-1)/P_{r }to the function:

\[ Ln(\phi^{\nu}) = a_{3}(P_{r2}^{3}-P_{r1}^{3})+ a_{2}(P_{r2}^{2}-P_{r1}^{2})+a_{1}(P_{r2}-P_{r1})+a_{0}[(LnP_{r2})-Ln(P_{r1})] \label{6-7} \]

where P_{r}_{2} and P_{r}_{1}_{ }are the reduced pressures of the mixture boiling pressure and the component’s saturation pressure. The constants for Equation (*\ref{6-7}*), a_{3} through a_{0}, are -0.175248; 0.393866; -0.887714; and -0.0141409, respectively.

For \(\phi_{i}^L \nonumber\), the evaluation of is simpler, since Z_{L} is fairly insensitive to pressure, simplifying the relationship for the *ith *component to be:

\[Ln(\phi^{L})=[Z_{L} \times (\pi -VP)/VP] \label{6-8} \] where \(\pi \nonumber\) is the system total pressure, VP is the pure component’s vapor pressure and \(Z_{L} \nonumber\) is the compressibility at the component’s saturation temperature/pressure.

To the extent that the system pressure is greater than the *ith *component’s vapor pressure (i.e., that liquid component is forcibly sub-cooled), the value of \(\phi_{i}^{L} \nonumber\) will be greater than unity. To the extent that the system pressure is less than the *ith *component’s vapor pressure (i.e., that liquid component is forcibly super-heated), the value of \(\phi_{i}^{L} \nonumber\) will be less than unity.

The pattern for \(\phi_{i}^{V} \nonumber\) is usually the reverse as a result of compressibility changes, so the value of \(\phi_{i}^{L}/ \phi_{i}^{V} \nonumber\) in Equation (*\ref{6-4}*) above shows how the distinct difference in Raoult/Dalton departure results. In the example above (40% molar TCS in STC @ 73.9°C saturation), the values are given in Table 6-1:

TCS | STC | |
---|---|---|

\(\phi^{L} \) | 0.9976 | 1.0039 |

\(\phi^{V} \) | 1.0488 | 0.9562 |

\(\phi^{L}/ \phi^{V} \) | 0.9512 | 1.0498 |

The careful reader will note an apparent discrepancy between the overall 2.3% ideality departures reported in the example above and the fugacity ratio of Table 6-1. Just the fugacity effects should make the TCS mole fraction departure about 4.8% lower than Raoult/Dalton expectation, but the Liquid Activity Coefficient effects restores about half of the departure. Thus it can be seen that fugacity effects and Liquid Activity Coefficient effects can be counteractive.

The “take-away” from this article is:

- Raoult & Dalton describe ideal behavior of vaporizing liquid mixtures, but do not completely describe the behavior of real fluids
- All departure factors must be considered: liquid fugacity, vapor fugacity and Liquid Activity Coefficients