Chapter 7: Properties of Solutions (Solution)
Problem 7.1: Prove the following:
(a)\;\;\mu_i={\left(\frac{\partial H^t}{\partial n_i}\right)}_{P,S,n_j} (b)\;\;\mu_i={\left(\frac{\partial A^t}{\partial n_i}\right)}_{T,V,n_j} (c)\;\;\mu_i=-T{\left(\frac{\partial S^t}{\partial n_i}\right)}_{U,V,n_j}Problem 7.2: Discuss the method for the calculation of entropy of solutions.
Problem 7.3: Discuss the variable pressure and variable temperature modifications of Gibbs–Duhem equations.
Problem 7.4: Derive an expression for partial molar volumes \overline{V_1} and \overline{V_2} using the following relation for the molar volume of the binary liquid mixture of components 1 and 2.
V=x_1V_1+x_2V_2+x_1x_2\left[B+C\left(x_2-x_1\right)\right]where x1 and x2 are the mole fractions and V1 and V2 are the molar volumes in the pure state.
Problem 7.5: Describe schematically an experimental technique for the determination of volume change and enthalpy change on mixing.
Problem 7.6: The activity coefficients in a binary mixture based on the Lewis–Randall rule standard state are given by
\ln\left(\gamma_1\right)=Ax_2^2;\;\;\;\;\;\ln\left(\gamma_2\right)=Ax_1^2Derive expressions for activity coefficients based on Henry’s law in terms of composition.
Problem 7.7: Show that Henry’s law constant varies with pressure as
{\left(\frac{\partial\ln K_2}{\partial P}\right)}_T=\frac{V_2^\infty}{RT}where \overline{V_2^\infty} is the partial molar volume of the solute at infinite dilution.
Problem 7.8: The enthalpy of a binary liquid mixture containing components 1 and 2 at 298 K and 1.0 bar is given by
H=400x_1+600x_2+x_1x_2\left(40x_1+4x_2\right)where H is in J/mol. Determine(a) Pure component enthalpies (b) Partial molar enthalpies.
Problem 7.9: The volume of a mixture of two organic liquids 1 and 2 is given by
V=110.0-17x_1-2.5x_1^2where V is the volume in m3/mol at 1.0 bar and 300 K. Find the expressions for \overline{V_1} , \overline{V_2} and ∆V.
Problem 7.10: If the partial molar volumes of species 1 in a binary liquid solution at constant temperature and pressure is given by
\overline{V_1}=V_1+\alpha x_2^2derive the equation for \overline{V_2} . What equation for V is consistent with this?
Problem 7.11: The molar enthalpy of a binary mixture is given by
H=x_1\left(a_1+b_1x_1\right)+x_2\left(a_2+b_2x_2\right)Derive an expression for \overline{H_2} .
Problem 7.12: Using the method of tangent intercepts plot the partial molar volume of HNO3 in aqueous solution at 293 K using the following data where w is the mass percentage of HNO3.
| w | 2.162 | 10.98 | 20.8 | 30.0 | 39.2 | 51.68 | 62.64 | 71.57 | 82.33 | 93.4 | 99.6 |
| ρ×10-3 (kg/m3) | 1.01 | 1.06 | 1.12 | 1.18 | 1.24 | 1.32 | 1.38 | 1.42 | 1.46 | 1.49 | 1.51 |
Problem 7.13: On addition of chloroform to acetone at 298 K, the volume of the mixture varies with composition as follows:
| x | 0 | 0.194 | 0.385 | 0.559 | 0.788 | 0.889 | 1.0 |
| V×106 (m3/kmol) | 73.99 | 75.29 | 76.50 | 77.55 | 79.08 | 79.82 | 80.67 |
where x is the mole fraction of chloroform. Determine the partial molar volumes of the components and plot against x.
Problem 7.14: The partial molar volumes of acetone and chloroform in a mixture in which mole fraction of acetone is 0.5307 are 74.166×10–6 m3/mol and 80.235×10–6 m3/mol respectively. What is the volume of 1 kg of the solution?
Problem 7.15: The volume of a solution formed from MgSO4 and 1.0 kg of water fits the expression
V=1.00121\times10^{-3}+34.69\times10^{-6}\left(m-0.070\right)^2where m is the molality of the solution. Calculate the partial molar volume of the salt and solvent when m = 0.05 mol/kg.
Problem 7.16: Calculate the partial molar volumes of methanol and water in a 40 percent (mol) methanol solution given the following data at 1 bar and 298 K. (x = mole fraction of methanol)
| x | 0 | 0.114 | 0.197 | 0.249 | 0.495 | 0.692 | 0.785 | 0.892 | 1 |
| V×103 (m3/mol) | 0.0181 | 0.0203 | 0.0219 | 0.023 | 0.0283 | 0.0329 | 0.0352 | 0.0379 | 0.0407 |
Problem 7.17: The standard enthalpy of the formation of HCl (in kJ/mol) from the elements at 298 K are given below:
| nw | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 10 | 50 | 100 | ∞ |
| -\triangle H_f^0 | 92.66 | 119 | 141.67 | 149.73 | 156.96 | 158.81 | 161.16 | 162.42 | 166.22 | 166.79 | 205.9 |
Calculate the partial molar enthalpies of HCl and water in a solution containing 10 kmol HCl/m3 of solution.
Calculate the partial molar enthalpies of HCl and water in a solution containing 10 kmol HCl/m3 of solution.
Problem 7.18: The following table gives the molality and density of aqueous solutions of KCl at 298 K. Determine the partial molar volume of KCl at m = 0.3.
| m (mol/kg | 0.0 | 0.1668 | 0.2740 | 0.3885 | 0.6840 | 0.9472 |
| ρ (kg/m3) | 997.07 | 100.49 | 100.98 | 101.271 | 102.797 | 103.927 |
Problem 7.19: Calculate the concentration of nitrogen in water exposed to air at 298 K and 1 bar if Henry’s law constant for nitrogen in water is 8.68×104 bar at this temperature. Express the result in moles of nitrogen per kg water (Hint: Air is 79 percent nitrogen by volume).
Problem 7.20: The partial pressure of methyl chloride in a mixture varies with its mole fraction at 298 K as detailed below:
| x | 0.0005 | 0.0009 | 0.0019 | 0.0024 |
| \overline p (bar) | 0.27 | 0.48 | 0.99 | 1.24 |
Estimate the Henry’s law constant of methyl chloride.
Problem 7.21: Two moles of hydrogen at 298 K and 2.0 bar and 4.0 moles of nitrogen at 298 K and 3.0 bar are mixed together. What is the free energy change on mixing and what would be the value had the pressures been identical initially?
Problem 7.22: Calculate the activity and activity coefficient of acetone based on the Lewis–Randall rule and Henry’s law for the data given in Example 7.12.
Problem 7.23: The activity coefficient data for a binary solution at fixed temperature and pressure are correlated as
\ln\left(\gamma_1\right)=x_2^2\left(0.5+2x_1\right);\;\;\ln\left(\gamma_2\right)=x_1^2\left(1.5+2x_2\right)Do these equations satisfy Gibbs–Duhem equations?
Problem 7.24: In a binary mixture, the activity coefficient γ1 of component 1, in the entire range of composition, is given by
R\ln\left(\gamma_1\right)=Ax_2^2+Bx_2^3where R, A, and B are constants. Derive an expression for the activity coefficient of component 2.
Problem 7.25: For a mixture of acetic acid and toluene containing 0.486 mole fraction toluene, the partial pressures of acetic acid and toluene are found to be 0.118 bar and 0.174 bar respectively at 343 K. The vapor pressures of pure components at this temperature are 0.269 bar and 0.181 bar for toluene and acetic acid respectively. The Henry’s law constant for acetic acid is 0.55 bar. Calculate the activity and activity coefficient for acetic acid in the mixture (a) Based on the Lewis–Randall rule (b) Based on Henry’s law.
Problem 7.26: Calculate the activity and activity coefficients for toluene for the conditions in Exercise 7.24 assuming pure liquid standard state.
Problem 7.27: Partial pressure of ammonia in aqueous solutions at 273 K varies with concentration as:
| x | 0.05 | 0.10 | 0.15 | 0.50 | 1.00 |
| \overline p (bar) | 0.0179 | 0.0358 | 0.062 | 1.334 | 4.293 |
Calculate (a) The activity coefficient of ammonia in 10 mole percent solution using pure liquid standard State (b) The Henry’s law constant if the system obeys Henry’s law.
Problem 7.28: The activity coefficient of n-propyl alcohol in a mixture of water (A) and alcohol (B) at 298 K referred to the pure liquid standard state is given below:
| xB | 0 | 0.01 | 0.02 | 0.05 | 0.10 | 0.20 |
| γB | 12.5 | 12.3 | 11.6 | 9.92 | 6.05 | 3.12 |
Find γA in the solution containing 10 percent (mole) n-propyl alcohol.
Problem 7.29: The activity coefficient of thallium in amalgams at 293 K are given below.
| x2 | 0 | 0.00326 | 0.01675 | 0.04856 | 0.0986 | 0.168 | 0.2701 | 0.424 |
| γ2 | 1.0 | 1.042 | 1.231 | 1.776 | 2.811 | 4.321 | 6.196 | 7.707 |
Determine the activity coefficient of mercury (component 1) at various compositions.
Problem 7.30: A vessel is divided into two parts. One part contains 2 mol nitrogen gas at 353 K and 40 bar and the other contains 3 mol argon gas at 423 K and 15 bar. If the gases are allowed to mix adiabatically by removing the partition determine the change in entropy. Assume that the gases are ideal and CV is equal to 5/2 R for nitrogen and 3/2 R for argon.
Problem 7.31: A stream of nitrogen flowing at the rate of 7000 kg/h and a stream of hydrogen flowing at the rate of 1500 kg/h mix adiabatically in a steady flow process. If the gases are ideal and are at the same temperature and pressure, what is the rate of entropy increase in kJ/h K as a result of the process?
Problem 7.32: The molar volume of a binary liquid mixture is given by
90\times10^{-3}x_1+50\times10^{-3}x_2+x_1x_2\left(6\times10^{-3}x_1+9\times10^{-3}x_2\right)Obtain expressions for \overline{V_1} & \overline{V_2} and show that they satisfy Gibbs–Duhem equations.
Problem 7.33: Water at a rate of 54×103 kg/h and Cu(NO3)2.6H2O at a rate of 64.8×103 kg/h are mixed together in a tank. The solution is then passed through a heat exchanger to bring the temperature to 298 K, same as the temperature of the components before mixing. Determine the rate of heat transfer in the exchanger. The following data are available. The heat of formation at 298K of Cu(NO3)2 is –302.9 kJ and that of Cu(NO3)2.6H2O is –2110.8 kJ. The heat of solution of Cu(NO3)2.nH2O at 298 K is –47.84 kJ per mol salt and is independent of n.
Problem 7.34: If pure liquid H2SO4 is added to pure water both at 300 K to form a 20 percent (weight) solution, what is the final temperature of the solution? The heat of the solution of sulphuric acid in water is H2SO4 (21.8 H2O) = –70×103 kJ/kmol of sulphuric acid. Standard heat of formation of water = –286 kJ/mol. The mean heat capacity of sulphuric acid may be taken from the Chemical Engineer’s Handbook.
Problem 7.35: LiCl.H2O (c) is dissolved isothermally in enough water to form a solution containing 5 mol of water per mole of LiCl. What is the heat effect? The following enthalpies of formation are given:
LiCl (c) = –409.05 kJ, LiCl.H2O (c) = –713.054 kJ
LiCl (5H2O) = –437.232 kJ, H2O (l) = –286.03 kJ
Problem 7.36: Calculate the heat effects when 1.0 kmol of water is added to a solution containing 1.0 kmol sulphuric acid and 3.0 kmol of water. The process is isothermal and occurs at 298 K. Data: Heat of mixing for H2SO4(3H2O) = –49,000 kJ per kmol H2SO4. Heat of mixing for H2SO4(4H2O) = –54,100 kJ per kmol H2SO4.
Problem 7.37: A single effect evaporator is used to concentrate a 15 % (weight) solution of LiCl in water to 40 %. The feed enters the evaporator at 298 K at the rate of 2 kg/s. The normal boiling point of a 40 % LiCl solution is 405 K and its specific heat is 2.72 kJ/kg K. For what heat transfer rate in kJ/h, should the evaporator be designed? The heat of solution of LiCl in water per mole of LiCl at 298 K are:
∆HS for LiCl(13.35 H2O) = –33.8 kJ, for LiCl (3.53 H2O) = –23.26 kJ. Enthalpy of superheated steam at 405 K, 1 bar = 2740.3 kJ/kg. Enthalpy of water at 298 K = 104.8 kJ/kg. The molecular weight of LiCl = 42.39.
Problem 7.38: The excess Gibbs free energy of solutions of methylcyclohexane (MCH) and tetrahydrofuran (THF) at 303 K are correlated as
G^E=RTx\left(1-x\right)\left[0.4587-0.1077\left(2x-1\right)+0.0191{(2x-1)}^2\right]where x is the mole fraction of methyl cyclohexane. Calculate the Gibbs free energy change on mixing when 1 mol MCH and 3 mol THF are mixed.
Problem 7.39: Derive the relation between the excess Gibbs free energy of a solution based on the Lewis– Randall rule and that based on the asymmetric treatment (Lewis–Randall rule for solvent and Henry’s law for solute) of solution ideality.
Problem 7.40: The excess enthalpy of a solution is given by
H^E=x_1x_2\left(40x_1+20x_2\right)\;\frac J{mol}Determine expressions for \overline{H_1^E} and \overline{H_2^E} as functions of x1.
Problem 7.41: Given that
M^E=x_1x_2\left[A+B\left(x_1-x_2\right)+C\left(x_1-x_2\right)^2\right]derive expressions for \overline{M_1^E} and \overline{M_2^E} . What are the limiting values for \overline{M_1^E} and \overline{M_2^E} and ME/x1x2 as x1 → 0 and x1 → 1?
Problem 7.42: The excess Gibbs free energy is given by
\frac{G^E}{RT}=-3x_1x_2\left(0.4x_1+0.5x_2\right)Find expressions for ln γ1 and ln γ2.
Problem 7.43: Do the following equations satisfy Gibbs–Duhem equations?
\ln\left(\gamma_1\right)=Ax_2^2+Bx_2^2\left(3x_1-x_2\right);\;\;\ln\left(\gamma_2\right)=Ax_1^2+Bx_1^2\left(x_1-3x_2\right)Find expressions for GE/RT.
Problem 7.44: The excess volume (m3/kmol) of a binary liquid mixture is given by
V^E=0.01x_1x_2\left(3x_1+5x_2\right)at 298 K and 1 bar. Determine \overline{V_1} and \overline{V_2} for an equimolar mixture of components 1 and 2 given that V1 = 0.12 m3/kmol and V2 = 0.15 m3/kmol.