Why are composition measures important in chemical engineering processes? Ans: Because composition measures contain information about the chemical species present in the system. A system contains 1000 Kg of sand and 4000 Kg of water. What is the mass fraction of water in the system? Ans: 4000/5000 = 0.8 The molar concentration of a compound in a stream is defined as the number of moles of the compound in the stream divided by the total volume of stream. A mixture has 0.4 moles of water, 1 mole of silica, and 0.6 moles of sodium chloride. What is the molecular mass of this mixture? (H2O: 18.05 g/mol, SiO2: 60.08 g/mol, NaCl: 58.44 g/mol) Ans: The molecular mass of the mixture is the sum of mol fractions times their molecular masses, i.e., (0.4/2)⋅18.05 + (1/2)⋅60.08 + (0.6/2)⋅58.44 = 50.85. What are the natural variables of the Gibbs function? Ans Pressure and temperature (dG = – SdT + VdP) When do you need to consider composition measures in modeling a chemical engineering process? Ans: When the system undergoes chemical reactions; When the system contains more than a single pure substance Suppose a system contains 2 mols of each compound A and B. If the molar Gibbs energy of compound A is 6 J/mol and the molar Gibbs energy of compound B is 2 J/mol, what is the Gibbs free energy of the system? Ans: 2⋅6 + 2⋅2 = 16 The mole fraction of a compound in a system is equal to the number of moles of the compound in the system divided by the total number of moles of all compounds in the system. A stream of 0.5 molar hydrochloric acid at 1 Liter/minute is continuously mixed with a stream of water at 60 Liters/hour. What is the concentration of hydrochloric acid in the product stream? Ans: The concentration in the product stream is the input stream concentration divided by the respective flows, i.e., 0.5/(1 + 1) = 0.25 The molar mass of a mixture of compounds is equal to the molar masses of the compounds weighted by the mole fraction of each compound in the mixture. It is useful to relate the total molar flows of streams with the compositions and flows of each compound in the stream. How does the Gibbs energy of a system with constant composition change with temperature and pressure? Ans: It increases with increasing pressure and decreases with increasing temperature How is it possible for the Gibbs function to change in a system at constant pressure and temperature? Ans: The Gibbs function may change due to chemical reactions transforming one compound into another; The Gibbs function may change due to the transition of a compound from one phase to another; The Gibbs function may change due to material flows in and out of the system. A system has 1 mol of diatomic nitrogen and 1 mol of xenon. What is the mol fraction of xenon in the system? Ans: 0.5 (xi = mi/m) A system has 1 mol of sodium chloride dissolved in 1 Liter of aqueous solution. What is the molar concentration of sodium chloride in the system? Ans: 1 mol/L (Ci = Xi/V) Why is it possible to write the Gibbs free energy of a system as a function of mole fractions instead of number of moles? Ans: Because it is possible to define the system as having a constant total number of moles The mole fraction of a compound in a system is equal to the number of moles of the compound in the system divided by the total number of moles of all compounds in the system. Suppose you determine that a system obeys G = n P1/2 / T3/2. How could you compute the enthalpy of the system? Ans: H = (5/2) n P1/2 / T3/2 What happens to all thermodynamic variables when a system reaches equilibrium? Ans: All thermodynamic properties become constant once equilibrium is reached. Which of the following is the correct formula used to calculate the molar mass of a mixture. Ans: M = ΣiViMi How is it possible for a closed system to contain more than one phase? Ans: Suffice it for the two phases to be nonreacting A system contains two phases and two unreactive compounds. How many variables (T, P, x1, x2) need to be specified to fully characterize the thermodynamic condition of this system? Ans: Two Why is the Gibbs function of a closed two-phase system equal to the sum of the Gibbs functions for each individual phase? Ans: Because the Gibbs free energy is an extensive property. Why do we need solution models to compute the chemical potential of chemical species in solutions? Ans: Because the chemical potential cannot be directly measured How is the change in Gibbs free energy with pressure for a system of constant composition? Ans: By measuring the volume of the system What is a closed system? Ans: A closed system is a system that does not exchange matter with the surroundings. What happens to pressure in a liquid-vapor system when it reaches mechanical equilibrium? Ans: The pressure is the same in the liquid and vapor. What are extensive properties? Ans: Extensive properties are properties that depend on the size of the system. What is the aim of solution models? Ans: Solution models aim at describing the interaction between atoms resulting in their chemical potential in a given phase Suppose you determine that a system obeys G = n P1/2 / T3/2. How could you compute the volume of the system? Ans: V = (n/2) P-1/2 / T3/2 What happens to all thermodynamic variables when a system reaches equilibrium? Ans: All thermodynamic properties become constant once equilibrium is reached. Why are partial molar properties also known as response functions? Ans: Because the partial molar quantities are a measure of the response in system properties upon addition of a minute amount of species to the system What is dP for a system in mechanical equilibrium? Ans: dP = 0 Why is the Gibbs function of a closed two-phase system equal to the sum of the Gibbs functions for each individual phase? Ans: Because the Gibbs free energy is an extensive property Why do we need solution models to compute the chemical potential of chemical species in solutions? Ans: Because the chemical potential cannot be directly measured A system with 998 mols of A, initially at equilibrium, receives 2 mols of species A with partial molar enthalpy of 5 Joules/mol. What is the change in molar enthalpy of A in the system? Ans: 0.01 Joules/mol; 2⋅5/1000 What is dT for a system in thermal equilibrium? Ans: dT = 0 A system with 1000 Kilograms of A, initially at equilibrium, receives 1 Kg of species A and shows an entropy change of 50 Joules/Kelvin. What is the partial specific entropy of species A in the system? Ans: It cannot be determined How can you apply the total differential of a thermodynamic function to a system of variable composition if the partial derivatives require the number of mols of each species to be constant? Ans: The total differential can be applied because each partial derivative requires the other variables to be kept constant. A mixture contains 1 mol of A, 2 mols of B, and X mols of C. The partial molar Gibbs energies of A, B, and C are respectively 1 J/mol, 2 J/mol, and 3 J/mol. The mixture has a Gibbs free energy of 20 Joules. How many mols of C are there in the mixture? Ans: 1⋅1+2⋅2+X⋅3=20; X = 5 What is the relationship established by the Gibbs-Duhem equation for a system at constant temperature and pressure? Ans: The mole fractions and partial molar properties of all species in solution cannot vary independently. Why are partial molar properties and partial specific properties readily interchangeable? Ans: Because the number of moles of a species is related to the mass of the species through a simple division by a constant When can you write the total differential of a function as a sum of partial derivatives and differential increments? Ans: When the partial derivatives are taken over independent variables A mixture contains 1 mol of A, 2 mols of B, and 3 mols of C. The partial molar Gibbs energies of A, B, and C are respectively 1 Joules/mol, 2 Joules/mol, and 3 Joules/mol. What is the Gibbs free energy of the mixture? Ans: 14 Joules; 1⋅1 + 2⋅2 + 3⋅3 = 14 Why is it not possible for partial solution of properties of different chemical species to vary independently from each other? Ans: Because the properties of different species in the same solution are related by the Gibbs-Duhem equation Why are partial properties an arbitrary assignment? Ans: Because they represent an arbitrary division of the property, which only truly exists for the whole solution What is the definition of partial specific properties in thermodynamics? Ans: The partial specific property is the change in the total property when an infinitesimal mass of species is added to a finite mass of solution. Given f = x + yz, what is df? Ans: df = dx + ydz + zdy What happens to the molar properties of the solvent when a solution becomes nearly pure? Ans: The partial molar properties approach the value of the molar property of the pure substance. What are the two starting equations used in the derivation of the Gibbs-Duhem equation? Ans: The total differential of solution properties and the summability relations Suppose that two electrically charged atoms, A and B, initially far away from each other are allowed to reach a position of equilibrium at some distance d. How would you distribute the drop in potential energy of the two-atom system to each of the two atoms using the same rationale as in solution thermodynamics? Ans: Half the potential drop would be assigned to each atom. What are Gibbs-Duhem equations used for? Ans: Gibbs-Duhem equations are used to ensure the consistency of thermodynamic models. How can the values of the partial molar properties be estimated from knowledge of total solution property and composition in a binary system? Ans: By making a plot of the total solution property versus composition What Maxwell relations could you derive from the total differential of a three variable function df = fx dx + fy dy + fz dz? Ans: (dfy/dz) = (dfz/dy) Why do the molar properties of the solvent approach the molar properties of the pure substance at high solvent concentrations? Ans: Because the action of the solute increases with the relative amount of solute in the solution Since ideal gases always have molecules that do not interact with each other, how is it possible for ideal gas molecules to show differences in thermodynamic properties? Ans: Ideal gases can have different thermodynamic properties because they can have different molecular structures. What are partial properties used for? Ans: Partial properties are used to apportion solution properties to solution components; Partial properties are used to describe the action of each component in a solution; Partial properties are used to compute solution properties from solution compositions. Suppose the molar Gibbs free energy of compound A (GA) in a binary solution of A and B is given by a quadratic fitting of experimental data on the molar fraction of A (xA), such that GA = c (xA + 0.5 xA2) where c is a constant. If GBPure is the molar Gibbs free energy of pure B at the same conditions of T and P as the solution, and xB is the mol fraction of B in the solution, what must be the molar Gibbs free energy of compound B (GB)? Ans: GB = GBPure – (c/2) (1 – xB)2 What can you do with Maxwell relations developed for partial molar quantities? Ans: They can be used to estimate the effects of pressure and temperature on molar partial quantities. Suppose two ideal gases, A and B, are mixed generating a solution with 0.1 mols of A and 0.9 mols of B. What is the molar volume of gas A in this mixture at 300 K and 3 bar? (R = 8.314 ⋅ 10-5 m3⋅Bar/mol⋅K) Ans: The molar volume is the volume of one mole of gas. PV = 1⋅RT; V = RT/P. 8.31 liters/mol Which one of the statements below is not an assumption in the ideal gas model? Ans: The ideal gas cannot be compressed to a volume of zero. Suppose two ideal gases, A and B, are mixed generating a solution where the partial pressure of B is 1.2 bar. What is the partial pressure of A when the mixture sits at 300 K and 3 bar? (R = 8.314 x 10-5 m3⋅bar/mol.K) Ans: PTotal = PA + PB ; PA = (3 – 1.2) bar = 1.8 bar What are Gibbs-Duhem equations used for? Ans: Gibbs-Duhem equations are used to ensure the consistency of thermodynamic models. How can the values of the partial molar properties be estimated from knowledge of total solution property and composition in a binary system? Ans: By making a plot of the total solution property versus composition What is the definition of partial pressure? Ans: The partial pressure is the pressure a gas would exert if it alone occupied the entire volume available What are summability relations used for? Ans: Summability relations are used to compute solution properties from solution compositions. Considering an equimolar binary solution with compounds A and B, what must be the change in partial molar enthalpy of A with the molar fraction of A if the change in molar enthalpy of compound B with molar fraction of B is equal to 5 kJ/mol? Ans: – 5 kJ/mol; – (0.5/0.5)⋅5 = –5 What partial molar property does not follow Gibbs' thorem? Ans: Partial pressures What is the molar volume of an ideal gas at 300 K and 3 bar pressure? (R = 8.314 x 105 m3⋅bar/mol⋅K) Ans: PV = NRT = 1⋅RT; V = RT/P. 8.31 liters/mol Suppose two ideal gases, A and B, are mixed generating a solution with 0.1 mols of A and 0.9 mols of B. What is the partial pressure of gas A when the mixture sits at 300 K and 3 bar? (R = 8.314 x 10-5 m3⋅bar/mol⋅K) Ans: PA = 0.1 ⋅ Ptotal = 0.1 * 3 bar = 0.3 bar Why is the molar enthalpy of an ideal gas independent of pressure? Ans: Because the internal energy of an ideal gas depends only on temperature and PV = NRT Gibbs' theorem states that partial molar properties of gases in mixtures are the same as the properties of the pure gas at the same temperature as the mixture but at a pressure equal to the partial pressure of the gas in the mixture. Suppose two ideal gases, A and B, are mixed, generating a solution with 0.1 mols of A and 0.9 mols of B. What is the molar volume of this mixture at 300 K and 3 bar? (R = 8.314 ⋅ 105 m3⋅bar/mol⋅K) Ans: PV = 1⋅RT; V = RT/P. 8.31 liters/mol What is a valid expression to compute the enthalpy of an ideal gas mass? Ans: H = U + RT What thermodynamic relation is used to compute the partial molar Gibbs energy in ideal gas mixtures? Ans: G = H – TS Why is there always a positive increase in entropy when mixing two ideal gases? Ans: Because the number of possible system configurations increases considerably when more than one type of atom is available; Because each of the two gases occupies the entire volume available as if it were alone. Why is it evident from equation 10.6 that the chemical potential is a good criterion for equilibrium? Ans: Because a difference in chemical potentials would imply a process with negative change in Gibbs function What is the fugacity of a gas in its pure state at the reference pressure? Ans: The fugacity of the pure gas is equal to one bar Why are there two components in the partial molar entropy of an ideal gas in equation 10.23? Ans: Because the molar entropy of an ideal gas in a mixture is not equal to the molar entropy of the pure gas Why is mixing enthalpy equal to zero in mixtures of ideal gases? Ans: Because the molecules of ideal gases do not interact with one another How can you obtain the fugacity coefficient of a gas in a mixture of real gases? Ans: By integrating the compressibility factor and pressure starting at the limit of zero pressure What are some of the major pitfalls of using equality in chemical potential as equilibrium criterion? Ans: The chemical potential goes to minus infinity when system pressures approach zero; The chemical potential does not have one absolute reference state; The chemical potential goes to minus infinity when mol fractions approach zero When is the fugacity of a gas equal to its pressure? Ans: When the gas behaves ideally What would be the change in Gibbs function for a molecule going from liquid to vapor if the fugacity of the vapor was higher than the fugacity of the liquid? Ans: The vaporization reaction would have a positive change in Gibbs function What happens to the compressibility factor as a gas approaches the limit of zero pressure? Ans: Zi tends to unity because all gases tend to ideal behavior at very low pressures What is the residual Gibbs energy? Ans: The residual Gibbs energy is the difference between the actual Gibbs energy and the Gibbs energy of an ideal system Suppose a pure substance is in equilibrium with its own vapor at a certain condition of temperature and pressure. What must be the ratio of fugacity coefficients in the liquid and vapor? Ans: The ratio of fugacity coefficients must be equal to one Consider a mixture of two ideal gases, A and B, in a container at some equilibrium pressure and temperature. What would you say are the chances of observing all molecules of A crowd on the left side of the container while all molecules of B crowd on the right side of the container? Ans: Negligibly small (This is the reverse process of mixing. It doesn't happen because it requires a negative change in entropy.) What is the meaning of Zi in equation 10.35? Ans: Zi is the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure How can you compute the first ratio in the factorization of subcooled liquid fugacity? Ans: By computing the fugacity coefficient of pure vapor in equilibrium with liquid What is the fugacity coefficient? Ans: The fugacity coefficient is the ratio of fugacity to actual gas pressure Why is the third ratio in the factorization of subcooled liquid fugacity usually small? Ans: Because pressure effects on the Gibbs free energy of condensed phases are small when compared with thermal effects Why is the ratio of fugacity coefficients in liquid and vapor in a VLE independent of saturation pressure? Ans: Because the saturation pressure is used as denominator in both fugacity coefficients Why is the second ratio in the factorization of subcooled liquid fugacity equal to one? Ans: Because the ratio corresponds to the equilibrium pressures of vapor and liquid 1.2 What can be said about a residual property? Ans: A residual property is equal to the difference between the actual property in a system and the property of an ideal gas at same (T,P,x) What is the residual Gibbs free energy of an ideal mixture? Ans: The residual Gibbs free energy of an ideal mixture is always equal to zero What are mixing rules? Ans: Mixing rules are expressions relating pure-species data with solution models The knowledge of the _____ at conditions of interest is sufficient information to compute residual Gibbs energy and therefore fugacity coefficient. Ans: residual enthalpy of a substance; residual entropy of a substance; compressibility as a function of pressure Identify the equation proposed by Prausnitz et al. to calculate Vcij for species in gas mixtures. Ans: How do you define fugacity in solutions of liquids or gases? Ans: In solutions of liquids or gases, fugacity is defined by analogy with the partial pressures of ideal gases What is measured by the ratio of fugacity to partial pressure in a solution of real gases? Ans: The deviation of the real system from the behavior of ideal gases What conditions should be satisfied in order to apply the virial expansion? Ans: The virial expansion is applicable at low to moderate temperatures and pressures What is omega in equation (10.66)? Ans: Omega is a measure of the shape of molecules; Omega is difference in reduced vapor pressure from that of simple fluids at a particular reduced temperature; Omega is a measure of the size of molecules Where can you find values of pure-species virial coefficients? Ans: Values of pure-species virial coefficients are found from experimental data What is the condition of equilibrium in a multi-component system with many phases? Ans: The fugacities of each component in each phase must be equal What is the fugacity coefficient of a species in an ideal gas mixture? Ans: The fugacity coefficient of a species in an ideal gas mixture is equal to one What does B12 represent in the virial expansion of a binary gaseous solution? Ans: B12 represents the interactions of molecules of type 1 with other molecules of type 2 What is the reference state used to compute the partial molar Gibbs free energy in condensed solutions? Ans: The pure substances at the same conditions of temperature and pressure as the solution Why are the fugacity coefficients of components in ideal mixtures equal to the fugacity coefficients of pure species at the same T and P? Ans: Because the fugacities of ideal solutions follow the Lewis-Randall rule A mathematical formalism of excess properties is defined, which is analogous to that of residual properties, but with ideal-solution behavior rather than ideal- gas state behavior as a basis. As a solution approaches purity, the excess properties of the solvent vanish to zero. Why can we expect the ideal solution model developed for gases to be a good starting point in modeling condensed phases? Ans: Because the entropy associated to mixing should also be present in condensed phases How can you compute the fugacity of a solution component using the Lewis-Randall rule? Ans: By multiplying the fugacity of the pure component at the same T and P as the solution by the mol fraction of the component in the solution What is a solution excess property? Ans: An excess property is the difference between the actual value of an extensive solution property and the value of the property in an ideal solution Match the excess properties on the left with the way they are determined on the right: What can be said about excess properties? Ans: Excess properties are strong functions of temperature; Excess properties are weak functions of pressure 1.3 In the thermodynamic study of mixing processes, a standard process takes the pure substances at the temperature and pressure of the solution as the starting point. The best way to visualize the standard mixing process is to imagine an enclosure fitted with a piston, a partition, and a heat reservoir. Mixing is initiated by removing the partition. The piston is allowed to move so that the pressure remains constant, while heat is removed or given to the system so as to keep the temperature also constant. In a standard mixing process, the heat transfer is equal to the total enthalpy change of the system _____. Ans: because the pressure is kept constant during the process. An engineer is working on the production of refractory insulation for steam reformers. The target mixture has mole fractions of silica, alumina, and magnesia of 0.2, 0.3, and 0.5, respectively. The thermal capacities of silica, alumina, and magnesia are 20, 30, and 50 J/mol⋅K, respectively, at the temperature and pressure of paste mixing. If the thermal capacity of the mixture is 50 J/mol⋅K, what is the thermal capacity change of mixing? Ans: 12 J/mol⋅K What is an excess property? Ans: An excess property is the difference between the actual value of the property and the value it would have in an ideal solution. A scientist attempts to obtain a uniform mixture of ethyl alcohol and water at 500 K and 5 bar. What should be the reference state if the standard mixing process is chosen to obtain the desired results? Ans: Pure ethyl alcohol and water at 500 K and 5 bar The equilibrium state of a standard mixing process depends on _____. Ans: the final solution temperature; the initial quantity of compounds; the final solution pressure Ideal mixtures have zero excess volume and enthalpy. What is the heat of mixing in a standard mixing process? Ans: The heat of mixing is the change in enthalpy during a standard mixing process. An engineer is working on the production of saline solutions for nasal spraying. The solution must have a mole fraction of 0.2 sodium chloride. The molar enthalpies of water and sodium chloride are, respectively, 30 J/mol and 50 J/mol. If the molar enthalpy of the solution is 40 J/mol, what is the enthalpy change of mixing? Ans: 6 J/mol Property changes of mixing depends on ____ Ans: Temperature; Composition; Pressure what is the standard state used as a reference in mixing processes? Ans: The pure chemical species at the temperature and pressure of the solution The excess properties VE and HE are equal to their corresponding property changes of mixing. Ans: T Based on the figure, what can be said about the mixing of 1 mole of ethanol with 1 mole of water at 50oC? Ans: The mixing process is exothermic. What thermodynamic relation connects the three curves in each of the plots? Ans: G = H – TS The enthalpy of a solution can be computed from the enthalpies of each solution component plus the heat of mixing. Based on the figure, what can be said about the mixing of 1 mole of ethanol with 1 mole of water at 90oC? Ans: The mixing process is endothermic. What is the origin of heats of mixing? Ans: Heats of mixing come from intermolecular interactions and do not change the chemical composition of molecules. For most systems, the Gibbs energy change of mixing is always positive. Ans: F What could be used as the horizontal axis in a binary H-x diagram? Ans: The mole fraction of one component; The partial pressure of one component; The mass fraction of one component What condition must apply in order for the H-x diagram to be used in computing the final temperature of mixing two binary solutions? Ans: The mixing process must be done in a very well-insulated container, so that it is safe to assume the process is adiabatic. In order to remain at the same temperature, a container must lose 50 J when 0.1 mole of sulfuric acid are added to 0.9 mole of water in the container. If the molar enthalpies of water and sulfuric acid are 20 J/mole and 40 J/mole, respectively, at the temperature and pressure of the solution, what is the molar enthalpy of the solution? Ans: –28 J What is the origin of heats of reaction? Ans: Heats of reaction come from intramolecular changes, resulting in changes to the chemical composition of compounds. Why can you arbitrarily assign a value of zero to pure compounds when drawing a HEx diagram? Ans: Because the HE-x diagram represents only the excess heat of mixing for a single temperature and not the total enthalpy of the solution Why is the product of the concentrations always included in the regressions of excess heat of mixing? Ans: The product of the concentrations is added to ensure that excess properties converge to zero for the pure compounds. An H-x diagram is a useful way to represent enthalpy data for binary solutions. How much heat should be removed from a beaker with sulfuric acid when a small amount of water is added to it in order to keep the temperature of the solution constant? Ans: 2280 kJ/Kg; see the right-side axis in the figure. The enthalpy change is equal to zero in an adiabatic process at constant pressure _____. Ans: because an adiabatic process cannot exchange heat with the surroundings How can you compute the heat generated or absorbed during a solution process from the heat of solution? Ans: By multiplying the heat of solution and the mole fraction of the solvent When 1 mol of LiCl is mixed with 12 mol of H2O, the process is represented by _____. Ans: LiCl (s) + 12H2O (l) → LiCl (12H2O) An HE-x diagram represents the relationship between the molar excess enthalpy of a solution and the composition of the solution. The excess property takes the form of a symmetric parabola with a maximum or minimum at _____. Ans: x1 = 0.5 Why is it convenient to use heats of formation of water complexes to convey information about heats of solution? Ans: Because solutions represented as hydrated complexes can be used easily in chemical equations How much heat should be removed from a beaker with water when a small amount of sulfuric acid is added to it in order to keep the temperature of the solution constant? Ans: 735 kJ/Kg; see the left axis in the figure. The heat of solution is the heat generated or absorbed when solids or gases are dissolved in liquids. What would be a chemical formula to represent a solution with 2 moles of sulfuric acid and 10 moles of water? Ans: H2SO4(5H2O) The heats of solution can be computed from the heats of formation of hydrated complexes. Why is there only one curve in an excess enthalpy-composition diagram? Ans: Because the molar enthalpies of pure components are taken as zero What may happen if water is added to acid instead of acid being added to water? Ans: Boiling and sputtering due to the large release of heat associated with the excess enthalpy of water