# Chapter 7: Properties of Solutions (Solution)

**Problem 7.1: **Prove the following:

**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.

where x_{1} and x_{2} are the mole fractions and V_{1} and V_{2} 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

Derive 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

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

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

where V is the volume in m^{3}/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

derive 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

Derive an expression for \overline{H_2} .

**Problem 7.12: **Using the method of tangent intercepts plot the partial molar volume of HNO_{3} in aqueous solution at 293 K using the following data where w is the mass percentage of HNO_{3}.

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/m^{3}) | 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×10^{6} (m^{3}/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} m^{3}/mol and 80.235×10^{–6} m^{3}/mol respectively. What is the volume of 1 kg of the solution?

**Problem 7.15: **The volume of a solution formed from MgSO_{4} and 1.0 kg of water fits the expression

where 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×10^{3} (m^{3}/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:

n_{w} | 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/m^{3} of solution.

Calculate the partial molar enthalpies of HCl and water in a solution containing 10 kmol HCl/m^{3} 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/m^{3}) | 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 8.68×10^{4} 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

Do these equations satisfy Gibbs–Duhem equations?

**Problem 7.24: **In a binary mixture, the activity coefficient γ_{1} of component 1, in the entire range of composition, is given by

where 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:

x_{B} | 0 | 0.01 | 0.02 | 0.05 | 0.10 | 0.20 |

γ_{B} | 12.5 | 12.3 | 11.6 | 9.92 | 6.05 | 3.12 |

Find γ_{A} 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.

x_{2} | 0 | 0.00326 | 0.01675 | 0.04856 | 0.0986 | 0.168 | 0.2701 | 0.424 |

γ_{2} | 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 C_{V} 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

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 54×10^{3} kg/h and Cu(NO_{3})_{2}.6H_{2}O at a rate of 64.8×10^{3} 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(NO_{3})_{2} is –302.9 kJ and that of Cu(NO_{3})_{2}.6H_{2}O is –2110.8 kJ. The heat of solution of Cu(NO_{3})_{2}.nH_{2}O at 298 K is –47.84 kJ per mol salt and is independent of n.

**Problem 7.34: **If pure liquid H_{2}SO_{4} 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.H_{2}O (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.H_{2}O (c) = –713.054 kJ

LiCl (5H_{2}O) = –437.232 kJ, H_{2}O (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 H_{2}SO_{4}(3H_{2}O) = –49,000 kJ per kmol H_{2}SO_{4}. Heat of mixing for H_{2}SO_{4}(4H_{2}O) = –54,100 kJ per kmol H_{2}SO_{4}.

**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:

∆H_{S} for LiCl(13.35 H_{2}O) = –33.8 kJ, for LiCl (3.53 H_{2}O) = –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

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

Determine expressions for \overline{H_1^E} and \overline{H_2^E} as functions of x_{1}.

**Problem 7.41: **Given that

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 M^{E}/x_{1}x_{2} as x_{1} → 0 and x_{1} → 1?

**Problem 7.42: **The excess Gibbs free energy is given by

Find expressions for ln γ_{1} and ln γ_{2}.

**Problem 7.43: **Do the following equations satisfy Gibbs–Duhem equations?

Find expressions for G^{E}/RT.

**Problem 7.44:** The excess volume (m^{3}/kmol) of a binary liquid mixture is given by

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 V_{1} = 0.12 m^{3}/kmol and V_{2} = 0.15 m^{3}/kmol.