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Introduction to Chemical Engineering Thermodynamics

J. M. Smith, Hendrick C Van Ness, Michael Abbott, Hendrick Van Ness

Chapter 14

TOPICS IN PHASE EQUILIBRIA - all with Video Answers

Educators

Chapter Questions

11:53

Problem 1

The excess Gibbs energy for the system chloroform(1)/ethanol(2) at $328.15 \mathrm{~K}\left(55^{\circ} \mathrm{C}\right)$ is well represented by the Margules equation, written:
$$
G^E / R T=\left(1.42 x_1+0.59 x_2\right) x_1 x_2
$$
The vapor pressures of chloroform and ethanol at $328.15 \mathrm{~K}\left(55^{\circ} \mathrm{C}\right)$ are:
$$
P_1^{\text {sat }}=82.37 \mathrm{kPa} \quad P_2^{\text {sat }}=37.31 \mathrm{kPa}
$$
(a) Assuming the validity of Eq. (10.5), make BUBL P calculations at $328.15 \mathrm{~K}\left(55^{\circ} \mathrm{C}\right)$ for liquid-phase mole fractions of $0.25,0.50$, and 0.75 .
(b) For comparison, repeat the calculations using Eqs. (14.1) and (14.2) with virial coefficients:
$$
B_{11}=-963 \mathrm{~cm}^3 \mathrm{~mol}^{-1} \quad B_{22}=-1,523 \mathrm{~cm}^3 \mathrm{~mol}^{-1} \quad B_{12}=52 \mathrm{~cm}^3 \mathrm{~mol}
$$

Sandra Lundell
Sandra Lundell
Numerade Educator

Problem 2

Find expressions for $\hat{\phi}_1$ and $\hat{\phi}_2$ for a binary gas mixture described by Eq. (3.39). The mixing rule for $\mathrm{B}$ is given by $\mathrm{Eq}$. (11.58). The mixing rule for $\mathrm{C}$ is given by the general equation:
$$
C=\sum_i \sum_j \sum_k y_i y_j y_k C_{i j k}
$$
where Cs with the same subscripts, regardless of order, are equal. For a binary mixture, this becomes:
$$
C=y_1^3 C_{111}+3 y_1^2 y_2 C_{112}+3 y_1 y_2^2 C_{122}+y_2^3 C_{222}
$$

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01:20

Problem 3

A system formed of methane(1) and a light oil(2) at $200 \mathrm{~K}$ and 30 bar consists of a vapor phase containing $95 \mathrm{~mol}-\%$ methane and a liquid phase containing oil and dissolved methane. The fugacity of the methane is given by Henry's law, and at the temperature of interest Henry's constant is $\mathcal{H}_1=200$ bar. Stating any assumptions, estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of pure methane at $200 \mathrm{~K}$ is $-105 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 4

Assume that the last three data points (including the value of $P_1^{\text {sat }}$ ) of Table $12.1, \mathrm{p} .401$, cannot be measured. Nevertheless, a correlation based on the remaining data points is required. Assuming the validity of Eq. (10.5), Eq. (14.28) may be written:
$$
P=x_1\left(\gamma_1 / \gamma_1^{\infty}\right) \mathcal{H}_1+x_2 \gamma_2 P_2^{\text {sat }}
$$
Data reduction may be based on Barker's method, i.e., minimizing the sum of squares of the residuals between the experimental values of $\mathbf{P}$ and the values predicted by this equation (see Ex. 12.1). Assume that the activity coefficients can be adequately represented by the Margules equation.
(a) Show that: $\ln \left(\gamma_1 / \gamma_1^{\infty}\right)=x_2^2\left[A_{12}+2\left(A_{21}-A_{12}\right) x_1\right]-A_{12}$.
(b) Find a value for Henry's constant $\mathcal{H}_1$.
(c) Determine values for parameters $A_{12}$ and $A_{21}$ by Barker's method.
(d) Find values for $\delta y_1$ for the data points.
How could the regression be done so as to minimize the sum of squares of the residuals in $G^E / R T$, thus including the $y_1$ values in the data-reduction process?

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Problem 5

Assume that the first three data points (including the value of $P_2^{\text {sat }}$ ) of Table 12.1, p. 401, cannot be measured. Nevertheless, a correlation based on the remaining data points is required. Assuming the validity of Eq. (10.5), Eq. (14.28) may be written:
$$
P=x_1 \gamma_1 P_1^{\text {sat }}+x_2\left(\gamma_2 / \gamma_2^{\infty}\right) \mathcal{H}_2
$$
Data reduction may be based on Barker's method, i.e., minimizing the sum of squares of the residuals between the experimental values of $\mathrm{P}$ and the values predicted by this equation (see Ex. 12.1). Assume that the activity coefficients can be adequately represented by the Margules equation.
(a) Show that: $\ln \left(\gamma_2 / \gamma_2^{\infty}\right)=x_1^2\left[A_{21}+2\left(A_{12}-A_{21}\right) x_2\right]-A_{21}$.
(b) Find a value for Henry's constant $\mathcal{H}_2$.
(c) Determine values for parameters $A_{12}$ and $A_{21}$ by Barker's method.
(d) Find values for $\delta y_1$ for the data points.
How could the regression be done so as to minimize the sum of squares of the residuals in $G^E / R T$, thus including the $y_1$ values in the data-reduction process?

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02:53

Problem 6

Work Pb. 14.4 with the data set of Table 12.3 , p. 409.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:46

Problem 7

Work $\mathrm{Pb} .14 .5$ with the data set of Table 12.3 , p. 409.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:11

Problem 8

Use Eq. (14.1) to reduce one of the isothermal data sets identified below, and compare the result with that obtained by application of Eq. (10.5). Recall that reduction means developing a numerical expression for $G^E / R T$ as a function of composition.
(a) Methylethylketone(1)/toluene(2) at $323.15 \mathrm{~K}\left(50^{\circ} \mathrm{C}\right)$ : Table 12.1, p. 401.
(b) Acetone(1)/methanol(2) at $328.15 \mathrm{~K}\left(55^{\circ} \mathrm{C}\right)$ : $\mathrm{Pb} .12 .3$, p. 440.
(c) Methyl tert-butyl ether(1)/dichloromethane(2) at $308.15 \mathrm{~K}\left(35^{\circ} \mathrm{C}\right): \mathrm{Pb} .12 .6, \mathrm{p} .441$.
(d) Acetonitrile(1)/benzene(2) at $318.15 \mathrm{~K}\left(45^{\circ} \mathrm{C}\right): \mathrm{Pb} .12 .9$, p. 443.
Second-virial-coefficient data are as follows:
$$
\begin{array}{l|rrrr}
& \text { Part (a) } & \text { Part (b) } & \text { Part (c) } & \text { Part (d) } \\
\hline B_{11} & -1,840 & -1,440 & -2,060 & -4,500 \\
B_{22} & -1,800 & -1,150 & -860 & -1,300 \\
B_{12} & -1,150 & -1,040 & -790 & -1,000
\end{array}
$$

Robert Zaballa
Robert Zaballa
Numerade Educator
06:18

Problem 9

For one of the substances listed below determine $\mathrm{P}^{\text {sat }} /$ bar from the Redlich/Kwong equation at two temperatures: $T=T_n$ (the normal boiling point), and $T=0.85 T_c$. For the second temperature, compare your result with a value from the literature (e.g., Perry's Chemical Engineers' Handbook). Discuss your results.
(a) Acetylene; (b) Argon; (c) Benzene; (d) n-Butane; (e) Carbon monoxide; (f) $\mathrm{n}$-Decane; (g) Ethylene; ( $h$ )n-Heptane; (i) Methane; $(j)$ Nitrogen.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
00:24

Problem 10

Work $\mathrm{Pb} .14 .9$ for one of the following:
(a) The Soave/Redlich/Kwong equation; (b) The Peng/Robinson equation.

Geno Ellis
Geno Ellis
Numerade Educator
01:11

Problem 11

Departures from Raoult's law are primarily from liquid-phase nonidealities $\left(\gamma_i \neq 1\right)$. But vapor-phasenonidealities $\left(\hat{\phi}_i \neq 1\right)$ also contribute. Consider the special case where the liquid phase is an ideal solution, and the vapor phase a nonideal gas mixture described by Eq. (3.37). Show that departures from Raoult's law at constant temperature are likely to be negative. State clearly any assumptions and approximations.

Manik Pulyani
Manik Pulyani
Numerade Educator
00:57

Problem 12

Determine a numerical value for the acentric factor $\omega$ implied by:
(a) The van der Waals equation; (b) The Redlich/Kwong equation.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:25

Problem 13

Starting with Eq. (14.67), derive the stability criteria of Eqs. (14.68) and (14.69).

Mayank Tripathi
Mayank Tripathi
Numerade Educator
02:06

Problem 14

An absolute upper bound on $G^E$ for stability of an equimolar binary mixture is $G^E=$ $\mathrm{R} \mathrm{T} \ln 2$. Develop this result. What is the corresponding bound for an equimolar mixture containing $\mathrm{N}$ species?

Lijeesh Krishnan
Lijeesh Krishnan
Numerade Educator

Problem 15

A binary liquid system exhibits LLE at $298.15 \mathrm{~K}\left(25^{\circ} \mathrm{C}\right)$. Determine from each of the following sets of miscibility data estimates for parameters $A_{12}$ and $A_{21}$ in the Margules equation at $298.15 \mathrm{~K}\left(25^{\circ} \mathrm{C}\right)$ :
(a) $x_1^\alpha=0.10, x_1^\beta=0.90 ;(b) x_1^\alpha=0.20, x_1^\beta=0.90 ;(c) x_1^\alpha=0.10, x_1^\beta=0.80$.

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Problem 16

Work $\mathrm{Pb}, 14.15$ for the van Laar equation.

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02:50

Problem 17

Considera binary vapor-phasemixturedescribed by Eqs. (3.37) and (11.58). Under what (highly unlikely) conditions would one expect the mixture to split into two immiscible vapor phases?

Vinnu M
Vinnu M
Numerade Educator
00:57

Problem 18

Figures $14.13,14.14$, and 14.15 are based on Eqs. (A) and (F) of Ex. 14.5 with $C_P^E$ assumedpositive and given by $C_P^E / R=3 x_1 x_2$. Graph the corresponding figures for the following cases, in which $C_P^E$ is assumed negative:
(a) $\mathrm{A}=\frac{975}{\mathrm{~T}}-18.4+3 \ln \mathrm{T}$
(b) $A=\frac{540}{\mathrm{~T}}-17.1+3 \ln T$
(c) $A=\frac{1,500}{T}-19.9+3 \ln T$

Cheyenne Whinham
Cheyenne Whinham
Numerade Educator

Problem 19

It has been suggested that a value for $G^E$ of at least $0.5 \mathrm{RT}$ is required for liquidlliquid phase splitting in a binary system. Offer some justification for this statement.

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01:42

Problem 20

Pure liquid species 2 and 3 are for practical purposes immiscible in one another. Liquid species 1 is soluble in both liquid 2 and liquid 3. One mole each of liquids 1,2 , and 3 are shaken together to form an equilibrium mixture of two liquid phases: an a-phase containingspecies 1 and 2 , and a $\beta$-phase containing species 1 and 3 . What are the mole fractions of species 1 in the a and $\beta$ phases, if at the temperature of the experiment, the excess Gibbs energies of the phases are given by:
$$
\frac{\left(G^E\right)^\alpha}{\mathrm{RT}}=0.4 x_1^\alpha x_2^\alpha \quad \text { and } \quad \frac{\left(G^E\right)^\beta}{\mathrm{RT}}=0.8 x_1^\beta x_3^\beta
$$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:42

Problem 21

It is demonstrated in Ex. 14.7 that the Wilson equation for $G^E$ is incapable of representing LLE. Show that the simple modification of Wilson's equation given by:
$$
G^E / R T=-C\left[x_1 \ln \left(x_1+x_2 \Lambda_{12}\right)+x_2 \ln \left(x_2+x_1 \Lambda_{21}\right)\right]
$$
can represent LLE. Here, $\mathrm{C}$ is a constant.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:14

Problem 22

Vapor sulfur hexafluoride $\mathrm{SF}_6$ at pressures of about $1600 \mathrm{kPa}$ is used as a dielectric in large primary circuit breakers for electric transmission systems. As liquids, $\mathrm{SF}_6$ and $\mathrm{H}_2 \mathrm{O}$ are essentially immiscible, and it is thereforenecessary to specify a low enough moisture content in the vapor $\mathrm{SF}_6$ so that if condensation occurs in cold weather a liquid-water phase will not form first in the system. For a preliminary determination, assume the vapor phase an ideal gas and prepare the phase diagram [Fig. 14.20(a)] for $\mathrm{H}_2 \mathrm{O}(1) / \mathrm{SF}_6(2)$ at $1600 \mathrm{kPa}$ in the composition range up to 1000 parts per mega parts of water (mole basis). The following approximate equations for vapor pressure are adequate:
$$
\ln P_1^{\text {sat }} / \mathrm{kPa}=19.1478-\frac{5363.70}{T / \mathrm{K}} \quad \ln P_2^{\text {sat }} / \mathrm{kPa}=14.6511-\frac{2048.97}{T / \mathrm{K}}
$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
05:58

Problem 23

In Ex. 14.4 a plausibility argument was developed from the LLE equilibrium equations to demonstrate that positive deviations from ideal-solution behavior are conducive to liquid/liquid phase splitting.
(a) Use one of the binary stability criteria to reach this same conclusion.
(b) Is it possible in principle for a system exhibiting negative deviations from ideality to form two liquid phases?

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:24

Problem 24

Toluene(1) and water(2) are essentially immiscible as liquids. Determine the dewpoint temperatures and the compositions of the first drops of liquid formed when vapor mixtures of these species with mole fractions $z_1=0.2$ and $z_1=0.7$ are cooled at the constant pressure of $101.33 \mathrm{kPa}$. What is the bubble-point temperature and the composition of the last drop of vapor in each case? See Table 10.2, p. 346, for vaporpressure equations.

Penny Riley
Penny Riley
Numerade Educator
06:24

Problem 25

$n$-Heptane(1) and water(2) are essentially immiscible as liquids. A vapor mixture containing $65-\mathrm{mol}-\%$ water at $373.15 \mathrm{~K}\left(100^{\circ} \mathrm{C}\right)$ and $101.325 \mathrm{kPa}$ is cooled slowly at constant pressure until condensation is complete. Construct a plot for the process showing temperature vs. the equilibrium mole fraction of heptane in the residual vapor. See Table 10.2 , p. 346 , for vapor-pressure equations.

James Kiss
James Kiss
Numerade Educator
01:42

Problem 26

Consider a binary system of species 1 and 2 in which the liquid phase exhibits partial miscibility. In the regions of miscibility, the excess Gibbs energy at a particular temperature is expressed by the equation:
$$
G^E / R T=2.25 x_1 x_2
$$
In addition, the vapor pressures of the pure species are:
$$
P_1^{\text {sat }}=75 \mathrm{kPa} \quad \text { and } \quad P_2^{\text {sat }}=110 \mathrm{kPa}
$$
Making the usual assumptions for low-pressure VLE, prepare a $\mathrm{P}-\mathrm{x}-\mathrm{y}$ diagram for this system at the given temperature.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
06:24

Problem 27

The system water(1)/n-pentane(2)/n-heptane(3) exists as a vapor at $101.325 \mathrm{kPa}$ and $373.15 \mathrm{~K}\left(100^{\circ} \mathrm{C}\right)$ with mole fractions $z_1=0.45, z_2=0.30, z_3=0.25$. The system is slowly cooled at constant pressure until it is completely condensed into a water phase and a hydrocarbon phase. Assuming that the two liquid phases are immiscible, that the vapor phase is an ideal gas, and that the hydrocarbons obey Raoult's law, determine:
(a) The dew-point temperature of the mixture and composition of the first condensate.
(b) The temperatureat which the second liquid phase appears and its initial composition.
(c) The bubble-point temperature and the composition of the last bubble of vapor.
See Table 10.2 , p. 346 , for vapor-pressureequations.

James Kiss
James Kiss
Numerade Educator
02:40

Problem 28

Work the preceding problem for mole fractions $z_1=0.32, z_2=0.45, z_3=0.23$.

Debasish Das
Debasish Das
Numerade Educator

Problem 29

The Case I behavior for SLE (Sec. 14.6) has an analog for VLE. Develop the analogy.

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Problem 30

An assertion with respect to Case II behavior for SLE (Sec. 14.6) was that the condition $z_i \gamma_i^3=1$ corresponds to complete immiscibility for all species in the solid state. Prove this.

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01:36

Problem 31

Use results of Sec. 14.6 to develop the following (approximate) rules of thumb:
(a) The solubility of a solid in a liquid solvent increases with increasing $\mathrm{T}$.
(b) The solubility of a solid in a liquid solvent is independent of the identity of the solvent species.
(c) Of two solids with roughly the same heat of fusion, that solid with the lower melting point is the more soluble in a given liquid solvent at a given $\mathrm{T}$.
(d) Of two solids with similar melting points, that solid with the smaller heat of fusion is the more soluble in a given liquid solvent at a given $\mathrm{T}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
05:40

Problem 32

Estimate the solubility of naphthalene(1) in carbondioxide(2) at a temperature of 353.15 $\mathrm{K}\left(80^{\circ} \mathrm{C}\right)$ at pressures up to $300 \mathrm{bar}$. Use the procedure described in $\mathrm{Sec} .14 .7$, with $l_{12}=$ 0.088 . Compare the results with those shown by Fig. 14.22. Discuss any differences. $P_1^{\text {sat }}=0.0102$ bar at $353.15 \mathrm{~K}\left(80^{\circ} \mathrm{C}\right)$.

Ly Tran
Ly Tran
Numerade Educator
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Problem 33

Estimate the solubility of naphthalene(1) in nitrogen(2) at a temperature of $308.15 \mathrm{~K}$ $\left(35^{\circ} \mathrm{C}\right)$ at pressures up to $300 \mathrm{bar}$. Use the procedure described in Sec. 14.7 , with $l_{12}=0$. Compare the results with those shown by Fig. 14.22 for the naphthalene/ $\mathrm{CO}_2$ system at $308.15 \mathrm{~K}\left(35^{\circ} \mathrm{C}\right)$ with $l_{12}=0$. Discuss any differences.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:20

Problem 34

The qualitative features of SVE at high pressures shown by Fig. 14.22 are determined by the equation of state for the gas. To what extent can these features be represented by the two-term virial equation in pressure, Eq. (3.37)?

A. Elizabeth Hildreth
A. Elizabeth Hildreth
Numerade Educator
01:05

Problem 35

The UNILAN equation for pure-species adsorption is:
$$
n=\frac{m}{2 s} \ln \left(\frac{c+P e^s}{c+P e^{-s}}\right)
$$
where $m, s$, and c are positive empirical constants.
(a) Show that the UNILAN equation reduces to the Langmuirisotherm for $s=0$.
(b) Show that Henry's constant $\mathrm{k}$ for the UNILAN equation is:
$$
k(\text { UNILAN })=\frac{m}{c s} \sinh s
$$
(c) Examine the detailed behavior of the UNILAN equation at zero pressure ( $\mathrm{P} \rightarrow 0$, $n \rightarrow 0$ ).

Narayan Hari
Narayan Hari
Numerade Educator
01:05

Problem 36

In Ex. 14.10, Henry's constant for adsorption $k$, identified as the intercept on a plot of $n / \mathrm{P}$ vs. $n$, was found from a polynomial curve-fit of $n / \mathrm{P}$ vs. n. An alternative procedure is based on a plot of $\ln (P / n)$ vs. n. Suppose that the adsorbateequation of state is a power series inn: $z=1+B n+C n^2+\ldots$. Show how from a plot (or a polynomial curve-fit) of $\ln (P / n)$ vs. $n$ one can extract values of $\mathrm{k}$ and B.

Narayan Hari
Narayan Hari
Numerade Educator
00:58

Problem 37

It was assumed in the development of Eq. (14.105) that the gas phase is ideal, with $\mathrm{Z}=1$. Suppose for a real gas phase that $\mathrm{Z}=\mathrm{Z}(T, \mathrm{P})$. Determine the analogous expression to Eq. (14.105) appropriate for a real (nonideal) gas phase.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:29

Problem 38

Use results reported in Ex. 14.10 to prepare plots of $\Pi$ vs. $n$ and $z$ vs. $n$ for ethylene adsorbed on a carbon molecular sieve. Discuss the plots.

Chai Santi
Chai Santi
Numerade Educator
01:05

Problem 39

Suppose that the adsorbate equation of state is given by $z=(1-b n)^{-1}$, where $\mathrm{b}$ is a constant. Find the implied adsorption isotherm, and show under what conditions it reduces to the Langmuir isotherm.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 40

Suppose that the adsorbateequation of state is given by $z=1+\beta n$, where $\beta$ is a function of T only. Find the implied adsorption isotherm, and show under what conditions it reduces to the Langmuir isotherm.

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01:06

Problem 41

Derive the result given in the third step of the procedure for predicting adsorption equilibria by ideal-adsorbed-solution theory at the end of Sec. 14.8.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:46

Problem 42

Consider a ternary system comprising solute species 1 and a mixed solvent (species 2 and 3). Assume that:
$$
\frac{G^E}{R T}=A_{12} x_1 x_2+A_{13} x_1 x_3+A_{23} x_2 x_3
$$
Show that Henry's constant $\mathcal{H}_1$ for species 1 in the mixed solvent is related to Henry's constants $\mathcal{H}_{1,2}$ and $\mathcal{H}_{1,3}$ for species 1 in the pure solvents by:
$$
\ln \mathcal{H}_1=x_2^{\prime} \ln \mathcal{H}_{1,2}+x_3^{\prime} \ln \mathcal{H}_{1,3}-A_{23} x_2^{\prime} x_3^{\prime}
$$
Here $x_2^{\prime}$ and $x_3^{\prime}$ are solute-free mole fractions:
$$
x_2^{\prime}=\frac{x_2}{x_2+x_3} \quad x_3^{\prime}=\frac{x_3}{x_2+x_3}
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:07

Problem 43

It is possible in principle for a binary liquid system to show more than one region of LLE for a particular temperature. For example, the solubility diagram might have two side-by-side "islands" of partial miscibility separated by a homogeneous phase. What would the AG vs. $x_1$ diagram at constant T look like for this case? Suggestion: See Fig. 14.11 for a mixture showing normal LLE behavior.

Nicole Smina
Nicole Smina
Numerade Educator
01:20

Problem 44

With $\bar{V}_2=V_2$, Eq. (14.132) for the osmotic pressure may be represented as a power series in $x_1$ :
$$
\frac{\Pi V_2}{x_1 R T}=1+\mathcal{B} x_1+\mathcal{C} x_1^2+\cdot \cdot
$$
Reminiscent of Eqs. (3.11) and (3.12), this series is called an osmotic virial expansion. Show that the second osmotic virial coefficient $\mathcal{B}$ is:
$$
\mathcal{B}=\frac{1}{2}\left[1-\left(\frac{d^2 \ln \gamma_2}{d x_1^2}\right)_{x_1=0}\right]
$$
What is $\mathcal{B}$ for an ideal solution? What is $\mathcal{B}$ if $G^E=A x_1 x_2$ ?

Emily Harris
Emily Harris
Numerade Educator