For gas and liquid phases in equilibrium, a component in a non-ideal system follows the relation,
For a system in equilibrium, at a given temperature and pressure:
The necessary and sufficient condition for equilibrium between two phases is
During the change of phase of a pure substance:
At the triple point of a pure substance, the number of degrees of freedom is:
The molar excess Gibbs free energy, \(G^E\), for a binary liquid mixture at \(T\) and \(P\) is given by \((G^E/RT) = Ax_1x_2\), where \(A\) is a constant. The corresponding equation for \(\ln \gamma _1\), where \(\gamma _1\) is the activity coefficient
of component 1, is
A change in state involving a decrease in entropy can be spontaneous only if
A liquid mixture contains 30% o-xylene, 60% p-xylene and 10% m-xylene (all percentages in w/w). Which of the following statements would be true in respect of this mixture?
A gas mixture of three components is brought in contact with a dispersion of an organic phase in water. The degrees of freedom of the system are
Maxwell's relation corresponding to the identity, \(dH = T dS + V dP + \sum \mu _i dn_i\) is
At 60oC , vapor pressures of methanol and water are 84.562 kPa and 19.953 kPa respectively. An aqueous solution of methanol at 60oC exerts a pressure of 39.233 kPa; the liquid phase and vapor phase mole fractions of methanol are 0.1686 and 0.5714 respectively. Activity coefficient of methanol is
The number of degrees of freedom for an azeotropic mixture in a two component vapor-liquid equilibria is/are
At a given temperature and pressure, a liquid mixture of benzene and toluene is in equilibrium with its vapor. The available degree(s) of freedom is (are)
If \(T_A\) and \(T_B\) are the boiling points of pure \(A\) and pure \(B\) respectively and \(T_{AB}\) is that of a non-homogeneous immiscible mixture of \(A\) and \(B\), then
An equimolar liquid mixture of species 1 and 2 is in equilibrium with its vapor at 400 K. At this temperature, the vapor pressures of the species are \(P_1^{\text {sat}} = 180\) kPa and \(P_2^{\text {sat}}=120\) kPa. Assuming that Raoult’s law is valid, the value of \(y_1\) is
From the following list, identify the properties which are equal in both vapour and liquid phases at equilibrium.
| P. Density | Q. Temperature |
|---|---|
| R. Chemical potential | S. Enthalpy |
The binary mixture \(A\)-\(B\) forms an azeotrope with a boiling point of 71.8\(^\circ \)C at 1 bar pressure, the azeotropic composition being 55 mol percent \(A\). The pure component vapor pressures at 71.8\(^\circ \)C are: \(A\) = 0.710 bar; \(B\) =
0.743 bar. Estimate the activity coefficient of \(A\) in the liquid, assuming the vapor to be ideal.
An equimolar mixture of benzene and toluene is contained in a piston/cylinder arrangement at a temperature \(T\). What is the maximum pressure (mm Hg) below which this mixture will exist as a vapor phase alone? At the given \(T\), the vapor pressures
of benzene and toluene are 1530 and 640 mm Hg, respectively. Assume that Raoult's law is valid.
A methanol-water vapor liquid equilibrium system is at equilibrium at 60oC and 60 kPa. The mole fraction of methanol in liquid is 0.5 and in the vapor is 0.8. Vapor pressure of methanol and water at 60oC are 85 kPa and 20 kPa respectively.
(i) Assuming vapor phase to be an ideal gas mixture, what is the activity coefficient of water in the liquid phase ?
{#1}
(ii) What is the excess Gibbs free energy (\(G^E\), in J/mol) of the liquid mixture ?
{#2}
A binary mixture containing species 1 and 2 forms an azeotrope at 105.4oC and 1.013 bar. The liquid phase mole fraction of component 1 (\(x_1\)) of this azeotrope is 0.62. At 105.4oC, the pure component vapor pressures for species 1 and 2 are 0.878 bar and 0.665 bar, respectively. Assume that the vapor phase is an ideal gas mixture. The van Laar constants, \(A\) and \(B\), are given by the expressions: \[ A = \left [1+\frac {x_2\ln \gamma _2}{x_1\ln \gamma _1}\right ]^2\ln \gamma _1 \quad \quad \quad B = \left [1+\frac {x_1\ln \gamma _1}{x_2\ln \gamma _2}\right ]^2\ln \gamma _2 \] (i) The activity coefficients \((\gamma _1,\gamma _2)\) under these conditions are
{#1}
(ii) The van Laar constants \((A,B)\) are
{#2}
For water at 300oC, it has a vapor pressure 8592.7 kPa and fugacity 6738.9 kPa. Under these conditions, one mole of water in liquid phase has a volume of 25.28 cm3, and that in vapor phase 391.1 cm3. Fugacity of water (in kPa) at 9000 kPa will be
The van Laar activity coefficient model for a binary mixture is given by the form \[ \ln \gamma _1 = \frac {A}{\left (1+\dfrac {A}{B}\dfrac {x_1}{x_2}\right )^2} \qquad \text {and}\qquad \ln \gamma _2 = \frac {B}{\left (1+\dfrac {B}{A}\dfrac {x_2}{x_1}\right )^2} \] Given \(\gamma _1=1.40, \gamma _2=1.25, x_1=0.25, x_2=0.75\), determine the constants \(A\) and \(B\).
For a binary mixture of \(A\) and \(B\) at 400 K and 1 atm, which ONE of the following equilibrium states deviates significantly from ideality?
Given: \[ \ln P_A^{\text {sat}} = 6.2 - \frac {2758}{T} \] where
\(P_A^{\text {sat}}\) = vapor pressure of \(A\), atm
\(T\) = temperature, K
\(p_A\) = partial pressure of \(A\), atm
\(x_A\) = mole fraction of \(A\) in liquid, and
\(y_A\) = mole fraction of \(A\) in vapor
For a binary mixture at constant temperature and pressure, which ONE of the following relations between activity coefficient (\(\gamma _i\)) and mole fraction (\(x_i\)) is thermodynamically consistent?
Consider a binary mixture of methyl ethyl ketone (component 1) and toluene (component 2). At 323 K the activity coefficients \(\gamma _1\) and \(\gamma _2\) are given by \[ \ln \gamma _1 = x_2^2(\psi _1-\psi _2+4\psi _2x_1), \quad \ln \gamma _2=x_1^2(\psi _1+\psi _2-4\psi _2x_2) \] where \(x_1\) and \(x_2\) are the mole fractions in the liquid mixture, and \(\psi _1\) and \(\psi _2\) are parameters independent of composition. At the same temperature, the infinite dilution activity coefficients, \(\gamma _1^\infty \) and \(\gamma _2^\infty \) are given by \(\ln \gamma _1^\infty =0.4\) and \(\ln \gamma _2^\infty =0.2\). The vapor pressures of methyl ethyl ketone and toluene at 323 K are 36.9 and 12.3 kPa respectively. Assuming that the vapor phase is ideal, the equilibrium pressure (in kPa) of a liquid mixture containing 90 mol% toluene is
A binary system consists of n-pentanol (1) and n-hexane (2) at 30\(^\circ \)C. Determine the composition of the vapor of the component-1 (in %) which is in equilibrium with a liquid containing 20 mole % n-pentanol (1) at 30\(^\circ \)C. The following
data are given:
Vapor pressure of n-pentanol (1) at 30\(^\circ \)C, \(P_1^{\text {sat}} = 4.306 \times 10^{-3}\) bar
Vapor pressure of n-hexane (2) at 30\(^\circ \)C, \(P_2^{\text {sat}} = 249.5 \times 10^{-3}\) bar
The activity coefficients
of (1) and (2) at \(x_1\) = 0.2 at 30\(^\circ \)C: \(\gamma _1=2.831, \gamma _2=1.1716\)
Assume that at low pressures, gases are in ideal state, but the liquid phase is not. Use the following equation. \[ y_i\phi _i^v P = x_i \gamma _i P_i^{\text
{sat}} \phi _i^{\text {sat}} \] where \(P\) = total pressure and \(\phi _i^v\) and \(\phi _i^{\text {sat}}\) are the fugacity coefficients of vapor and saturated vapor respectively.
At 318 K and a total pressure of 24.4 kPa, the composition of the system ethanol(1) and toluene(2) at equilibrium is \(x_1\) = 0.3 and \(y_1\) = 0.634. The saturation pressure at the given temperature for pure components are \(P_1^{\text {sat}}\) = 23.06
kPa and \(P_2^{\text {sat}}\) = 10.05 kPa respectively.
Calculate the value of \(G^E/RT\) for the liquid phase.
The activity coefficients of benzene (\(A\)) - cyclohexane (\(B\)) mixtures at 40\(^\circ \)C are given by \(RT \ln \gamma _A = b x_B^2\) and \(RT \ln \gamma _B = b x_A^2\). At 40\(^\circ \)C, \(A\) and \(B\) form an azeotrope containing 49.4 mol% \(A\)
at a total pressure of 202.6 mm Hg. If the vapor pressures of pure \(A\) and \(B\) are 182.6 and 183.5 mm Hg respectively, calculate the total pressure (mm Hg) of the vapor at 40\(^\circ \)C in equilibrium with a liquid mixture containing 12.6 mol%
\(A\).
Determine the mole fraction of methane, \(x_1\), dissolved in a light oil at 200 K and 20 bar. Henry’s law is valid for the liquid phase, and the gas phase may be assumed to be an ideal solution. At these conditions:
Henry’s law constant for methane
in oil = 200 bar
fugacity coefficient of pure methane gas = 0.90
mole fraction of methane in the gas phase, \(y_1\) = 0.95
A mixture of \(A\) and \(B\) conforms closely to Raoult’s law. The pure component vapor pressures \(P_A^{\text {sat}}\) and \(P_B^{\text {sat}}\) in kPa are given by (\(T\) in \(^\circ \)C) \[ \begin {align*} \ln P_A^{\text {sat}} &= 14.27 - \frac
{2945}{T + 224} \\
\ln P_B^{\text {sat}} &= 14.20 - \frac {2973}{T + 209} \end {align*} \] If the bubble point of a certain mixture of \(A\) and \(B\) is 76\(^\circ \)C at a total pressure of 80 kPa, find the mole fraction of \(A\) in the
first vapor that forms.
Consider a binary liquid mixture at equilibrium with its vapour at 25oC. Antoine equation for this system is given as \( \displaystyle \log _{10}P_i^{\text {sat}} = A - \frac {B}{T+C}\) where \(T\) is in oC and \(P\) in Torr. The Antoine constants (\(A\), \(B\), and \(C\)) for the system are given in the following table.
| Component | \(A\) | \(B\) | \(C\) |
|---|---|---|---|
| 1 | 7.0 | 1210 | 230 |
| 2 | 6.5 | 1206 | 223 |
The vapour phase is assumed to be ideal and the activity coefficients (\(\gamma _i\)) for the non-ideal liquid phase are given by
\[ \begin {align*} \ln (\gamma _1) &= x_2^2[2-0.6 x_1] \\ \ln (\gamma _2) &= x_1^2[1.7+0.6x_2] \end {align*} \]
If the mole fraction of component 1 in liquid phase (\(x_1\)) is 0.11, then the mole fraction of component 1 in vapour phase (\(y_1\)) is ____________
A binary mixture of components (1) and (2) forms an azeotrope at 130oC and \(x_1=0.3\). The liquid phase non-ideality is described by \(\ln \gamma _1=Ax_2^2\) and \(\ln \gamma _2=Ax_1^2\), where \(\gamma _1, \gamma _2\) are the activity coefficients, and \(x_1, x_2\) are the liquid phase mole fractions. For both components, the fugacity coefficients are 0.9 at the azeotropic composition. Saturated vapor pressures at 130oC are \(P_1^{\text {sat}}=70\) bar and \(P_2^{\text {sat}}=30\) bar. The total pressure in bars for the above azeotropic system (up to two decimal places) is ____________
A sparingly soluble gas (solute) is in equilibrium with a solvent at 10 bar. The mole fraction of the solvent in the gas phase is 0.01. At the operating temperature and pressure, the fugacity coefficient of the solute in the gas phase and the Henry’s law constant are 0.92 and 1000 bar, respectively. Assume that the liquid phase obeys Henry’s law. The mole percentage of the solute in the liquid phase, rounded to 2 decimal places, is ____________
Generate \(P\)-\(x\)-\(y\) data for a binary system at a temperature at which \(P_1^{\text{sat}}\) = 84.562 kPa, and \(P_2^{\text{sat}}\) = 19.953 kPa. Assume system to follow Raoult’s law. (Determine \(P\) and \(y_1\) values for \(x_1\) = 0.2, 0.4, 0.6, and 0.8.)
0200-6-td At high temperature and pressure \(\ce{N2}\) obeys the equation of state \(P(V-b) = RT\). Calculate the fugacity of \(\ce{N2}\) at 1000\(^\circ\)C
and 1000 atm, if \(b = 39.1\) cm\(^3\)/mol. Calculate the fugacity coefficient also.
Calculate the fugacity of \(\ce{CO2}\) gas at 100 atm pressure and 75\(^\circ\)C assuming that the gas obeys van der Waals equation of state. The constants for this equation of state are: \[a = 3.606 \times 10^6 \text{ atm.cc$^2$/gmol$^2$} \ \ b = 42.88 \text{ cc/gmol}.\]
For a system consisting of components 1 and 2, the VLE data at 45\(^\circ\)C were reported by an experiment. Test whether the following data are thermodynamically consistent or not.
| \(P\) (torr) | 315.32 | 339.70 | 397.77 | 422.46 | 448.88 | 463.92 | 472.84 | 485.16 | 498.07 | 513.20 |
| \(x_1\) | 0.0556 | 0.0903 | 0.2152 | 0.2929 | 0.3970 | 0.4769 | 0.5300 | 0.6047 | 0.7128 | 0.9636 |
| \(y_1\) | 0.2165 | 0.291 | 0.4495 | 0.5137 | 0.5832 | 0.6309 | 0.6621 | 0.7081 | 0.7718 | 0.9636 |
The saturated vapor pressures of component 1 and 2 are 512.4 and 254.4 torr, respectively.
Identify the extensive and intensive properties from the given list:
(a) chemical potential, (b) entropy, (c) fugacity, (d) enthalpy, (e) activity coefficient.
For the binary system methanol(1) and benzene(2), the recommended values of the Wilson parameters at 68\(^\circ \)C are \(\Lambda _{12}\) = 0.1751 and \(\Lambda _{21}\) = 0.3456. The vapor pressures of pure species at 68\(^\circ \)C are \(P_1^{\text {sat}}\)
= 68.75 kPa and \(P_2^{\text {sat}}\) = 115.89 kPa. Show that the given system can form an azeotrope at 68\(^\circ \)C. Assume that the vapor behaves like an ideal gas.
In a binary mixture the activity coefficient \(\gamma_1\) of component 1, in the entire range of composition is given by \[R \ln \gamma_1 = A x_2^2 + B x_2^3\] where \(R\), \(A\) and \(B\) are constants.
Derive the expression for the activity coefficient of component 2.
The Excess Gibbs free energy for cyclohexanone(1)/phenol(2) is given by \[\frac{G^E}{RT} = -2.1x_1x_2\] where \(x_1\) and \(x_2\) are the mole fractions of components 1 and 2 in the liquid phase. The vapor pressures of components at 417 K are \(P_1^{\text{sat}} = 75.2\) kPa and \(P_2^{\text{sat}} = 31.66\) kPa.
Derive expressions for activity coefficients of each component as a function of composition.
Verify whether the expressions derived in (a) satisfy the Gibbs-Duhem equation.
Determine the equilibrium pressure \(P\) and vapor composition for a liquid composition \(x_1=0.8\) and 417 K. Assume vapor phase to be ideal gas.
Last Modified on: 24-Oct-2022
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