Questions and Answers of Introduction To Chemical Engineering Thermodynamics

Consider a binary liquid mixture for which the excess Gibbs energy is given by GE/RT = Ax1x2. What is the minimum value of A for which liquid/liquid equilibrium is possible?To the xy diagram provided
For a 35-wt-% aqueous solution of H2SO4 at 300 K, what is the heat of mixing ΔH in kJ·kg−1?
A large quantity of very dilute aqueous NaOH solution is neutralized by addition of the stoichiometric amount of a 10-mol-% aqueous HCl solution. Estimate the heat effect per mole of NaOH neutralized
A large quantity of very dilute aqueous HCl solution is neutralized by the addition of the stoichiometric amount of a 10-mol-% aqueous NaOH solution. Estimate the heat effect per mole of HCl
Ten thousand (10,000) kg·h−1 of an 80-wt-% H2SO4 solution in water at 300 K is continuously diluted with chilled water at 280 K to yield a stream containing 50-wt-% H2SO4 at 330 K.(a) What is the
Determine the heat of mixing ΔH of sulfuric acid in water and the partial specific enthalpies of H2SO4 and H2O for a solution containing 65-wt-% H2SO4 at 300 K.
It is proposed to cool a stream of 75-wt-% sulfuric acid solution at 330 K by diluting it with chilled water at 280 K. Determine the amount of water that must be added to 1 kg of 75-wt-% acid before
Develop Eq. (11.12) for ΔSid by appropriate application of Eqs. (5.36) and (5.37) to a mixing process.Eq. (11.12)Eq. (5.36)Eq. (5.37) Δsid-RΣ x; Inx (11.12) i
The following liquids, all at atmospheric pressure and 300 K, are mixed: 25 kg of pure water, 40 kg of pure sulfuric acid, and 75 kg of 25-wt-% sulfuric acid.(a) How much heat is liberated if mixing
A 90-wt-% aqueous H2SO4 solution at 25°C is added over a period of 6 hours to a tank containing 4000 kg of pure water also at 25°C. The final concentration of acid in the tank is 50-wt-%. The
Consider a closed vessel of fixed volume containing equal masses of water, ethanol, and toluene at 70°C. Three phases (two liquid and one vapor) are present.(a) How many variables, in addition to
The pressure above a mixture of chloroform and tetrahydrofuran at 50°C is measured to be 62 kPa. What are the possible compositions of the liquid and vapor phases?To the Pxy diagram for
The pressure above a mixture of chloroform and tetrahydrofuran at 50°C is measured to be 52 kPa. What are the possible compositions of the liquid and vapor phases?To the Pxy diagram for
Consider a binary (two-species) system in vapor/liquid equilibrium. Enumerate all of the combinations of intensive variables that could be fixed to fully specify the intensive state of the system.
What is the composition of the azeotrope for the chloroform(1)/tetrahydrofuran (2) system? Would this be called a high boiling or low-boiling azeotrope?To the Pxy diagram for
Consider a closed vessel initially containing 1 mol of pure ethyl acetate at 74°C and 100 kPa. Imagine that pure ethanol is slowly added at constant temperature and pressure until the vessel
Consider a chloroform (1)/tetrahydrofuran(2) mixture with x1 = 0.80, initially at 50°C and 70 kPa. Describe the evolution of phases and phase compositions as the pressure is gradually reduced to 50
Consider an chloroform(1)/tetrahydrofuran(2) mixture with x1 = 0.90, initially at 50°C and 50 kPa. Describe the evolution of phases and phase compositions as the pressure is gradually increased to
Consider a closed vessel initially containing 1 mol of tetrahydrofuran at 50°C and 52 kPa. Imagine that pure chloroform is slowly added at constant temperature and pressure until the vessel contains
A mixture of chloroform and tetrahydrofuran is heated in a closed system at 120 kPa to a temperature of 75°C, and two phases are observed to be present. What are the possible compositions of the
A chloroform and tetrahydrofuran mixture is heated in a closed system at 120 kPa to a temperature of 70°C, and two phases are observed to be present. What are the possible compositions of the liquid
Consider a chloroform(1)/tetrahydrofuran(2) mixture with x1 = 0.80, initially at 70°C and 120 kPa. Describe the evolution of phases and phase compositions as the temperature is gradually increased
Consider a chloroform(1)/tetrahydrofuran(2) mixture with x1 = 0.20, initially at 70°C and 120 kPa. Describe the evolution of phases and phase compositions as the temperature is gradually increased
Consider a chloroform(1)/tetrahydrofuran(2) mixture with x1 = 0.10, initially at 80°C and 120 kPa. Describe the evolution of phases and phase compositions as the temperature is gradually reduced to
Consider a chloroform(1)/tetrahydrofuran(2) mixture with x1 = 0.90, initially at 76°C and 120 kPa. Describe the evolution of phases and phase compositions as the temperature is gradually reduced to
A certain gas is described by the equation of state:Here, b is a constant and θ is a function of T only. For this gas, determine expressions for the isothermal compressibility κ and the thermal
A Carnot engine operates between an infinite hot reservoir and a finite cold reservoir of total heat capacity CtC.(a) Determine an expression for the work obtained as a function of CtC, TH (=
A single gas stream enters a process at conditions T1, P1, and leaves at pressure P2. The process is adiabatic. Prove that the outlet temperature T2 for the actual (irreversible) adiabatic process is
Assuming the validity of Eq. (6.89), derive Edmister’s formula for estimation of the acentric factor:where θ ≡ Tn / Tc , Tn is the normal boiling point, and Pc is in (atm). 3 log P. - 1 1-0
A vessel contains 1 kg of H2O as liquid and vapor in equilibrium at 1000 kPa. If the vapor occupies 70% of the volume of the vessel, determine H and S for the 1 kg of H2O.
Show that isobars and isochores have positive slopes in the single-phase regions of a TS diagram. Suppose that CP = a + bT, where a and b are positive constants. Show that the curvature of an isobar
Show that the Clapeyron equation for liquid/vapor equilibrium may be written in the reduced form: d In Psat lv ΔΗ lv AH where AH = RT. dT, TAZ
A Carnot engine with H2O as the working fluid operates on the cycle shown in Fig. 8.2. The H2O circulation rate is 1 kg⋅s−1. For TH = 475 K and TC = 300 K, determine:(a) The pressures at states
For comparison of Diesel- and Otto-engine cycles:(a) Show that the thermal efficiency of the air-standard Diesel cycle can be expressed aswhere r is the compression ratio and rc is the cutoff ratio,
An air-standard Diesel cycle absorbs 1500 J⋅mol−1 of heat (step DA of Fig. 8.10, which simulates combustion). The pressure and temperature at the beginning of the compression step are 1 bar and
An air-standard gas-turbine cycle is modified by the installation of a regenerative heat exchanger to transfer energy from the air leaving the turbine to the air leaving the compressor. In an optimum
Air enters a gas-turbine engine (see Fig. 8.11) at 305 K and 1.05 bar and is compressed to 7.5 bar. The fuel is methane at 300 K and 7.5 bar; compressor and turbine efficiencies are each 80%. For one
Air-standard power cycles are conventionally displayed on PV diagrams. An alternative is the PT diagram. Sketch air-standard cycles on PT diagrams for the following:(a) Carnot cycle(b) Otto cycle(c)
Devise a general scheme for analyzing four-step air-standard power cycles. Model each step of the cycle as a polytropic process described byPVδ = constantwhich implies thatTP(1 − δ) ⁄δ =
An easy way to rationalize definitions of cycle performance is to think of them as:Thus, for an engine, thermal efficiency is η = |W|/|QH|; for a refrigerator, the coefficient of performance is
In comparing the performance of a real cycle with that of a Carnot cycle, one has in principle a choice of temperatures to use for the Carnot calculation. Consider a vapor-compression refrigeration
Rework the preceding problem for methane entering at 200 bar and precooled to 240 K by external refrigeration.
The condenser of a home refrigerator is commonly underneath the appliance; thus, the condensing refrigerant exchanges heat with household air, which has an average temperature of about 21°C. It is
A common misconception is that the coefficient of performance of a refrigerator must be less than unity. In fact, this is rarely the case. To see why, consider a real refrigerator for which ω = 0.6
What is the partial molar temperature? What is the partial molar pressure? Express results in relation to the T and P of the mixture.
Show that:(a) The “partial molar mass” of a species in solution is equal to its molar mass.(b) A partial specific property of a species in solution is obtained by division of the partial molar
For the system methane(1)/ethane(2)/propane(3) as a gas, estimate(a) Through application of Eq. (10.64).(b) Assuming that the mixture is an ideal solution.(a) Eq. (10.64)where the dummy indices i and
Given below are values of GE /J·mol−1, HE /J·mol−1, and CEP / J · mol −1· K−1 for some equimolar binary liquid mixtures at 298.15 K. Estimate values of GE, HE, and SE at 328.15 K for one
If the molar density of a binary mixture is given by the empirical expression:find the corresponding expressions for V̅1 and V̅2 p= ag + ajx1 +a2xf
For a ternary solution at constant T and P, the composition dependence of molar property M is given by:M = x1 M1 + x2 M2 + x3 M3 + x1 x2 x3Cwhere M1, M2, and M3 are the values of M for pure species
For a particular binary liquid solution at constant T and P, the molar enthalpies of mixtures are represented by the equation
A pure-component pressure pi for species i in a gas mixture may be defined as the pressure that species i would exert if it alone occupied the mixture volume. Thus,where yi is the mole fraction of
If for a binary solution one starts with an expression for M (or MR or ME) as a function of x1 and applies Eqs. (10.15) and (10.16) to find M̅1 and M̅2 (or M̅R1and M̅R2 or M̅E1 and M̅E2) and
Analogous to the conventional partial property M̅i , one can define a constant-T, V partial property M̅i :Show that M̅i and M̅i are related by the equation:Demonstrate that
Justify the following equations: ( - VR aln ƏT P.x RT RT2 T.x GR Ex, In ộ Exd In ộ = 0 (const T, P) RT
For the system ethylene(1)/propylene(2) as a gas, estimate(a) Through application of Eqs. (10.63).(b) Assuming that the mixture is an ideal solution.(a) Eqs. (10.63) fi. f2, ô1, and 2 at t = 150°C,
Make use of Eqs. (3.36), (3.61), (3.62), (6.54), (6.55), (6.56), (6.70), (6.71), (10.62), and (10.69)–(10.74), to estimate V, HR, SR, and GR for one of the following binary vapor mixtures:(a)
For a multicomponent mixture containing any number of species, prove thatwhere the summation is over all species. Show that for a binary mixture this result reduces to Eqs. (10.15) and (10.16).Eq.
Rationalize the following expression, valid at sufficiently low pressures, for estimating the fugacity coefficient: ln ϕ ≈ Z − 1.
The following empirical two-parameter expression has been proposed for correlation of excess properties of symmetrical liquid mixtures:Here, quantities A and B are parameters that depend at most on
An engineer claims that the volume expansity of an ideal solution is given byIs this claim valid? If so, show why. If not, find a correct expression for βid. pid = Exißi %3D i
Laboratory A reports the following results for equimolar values of GE for liquid mixtures of benzene(1) with 1-hexanol(2):GE = 805 J·mol−1 at T = 298 K GE = 785 J·mol−1 at T = 323
The following expressions have been proposed for the partial molar properties of a particular binary mixture:M̅1 = M1 + Ax2 M̅2 = M2 + Ax1Here, parameter A is a constant. Can these
Two (2) kmol·hr−1 of liquid n-octane (species 1) are continuously mixed with 4 kmol·hr−1 of liquid iso-octane (species 2). The mixing process occurs at constant T and P; mechanical power
Fifty (50) mol·s−1 of enriched air (50 mol-% N2, 50 mol-% O2) are produced by continuously combining air (79 mol-% N2, 21 mol-% O2) with a stream of pure oxygen.All streams are at the constant
A simple expression for ME of a symmetrical binary system is ME = Ax1x2. However, countless other empirical expressions can be proposed which exhibit symmetry. How suitable would the two following
Commonly, if ME for a binary system has a single sign, then the partial properties M̅E1 and M̅E2 have the same sign as ME over the entire composition range. There are occasions, however,
At 25°C and atmospheric pressure the volume change of mixing of binary liquid mixtures of species 1 and 2 is given by the equationΔV = x1 x2 (45x1  + 25x2)where ΔV is in cm3·mol−1. At these
Following are data for GE and HE (both in J·mol−1) for equimolar mixtures of the same organic liquids. Use all of the data to estimate values of GE, HE, and TSE for the equimolar mixture at
The volume change of mixing (cm3·mol−1) for the system ethanol(1)/methyl butyl ether(2) at 25°C is given by the equationΔV = x1 x2 [ − 1.026 + 0.0220 (x1− x2)]Given that V1 = 58.63 and
Starting with Eq. (6.9), show that isobars in the vapor region of a Mollier (HS) diagram must have positive slope and positive curvature.Eq. (6.9) dH = T dS + V dP (6.9)
The PVT behavior of a certain gas is described by the equation of state:P (V−b) = RTwhere b is a constant. If in addition CV is constant, show that:(a) U is a function of T only.(b) γ = const.(c)
A pure fluid is described by the canonical equation of state: G = Γ ( T ) + RT ln P, where Γ(T ) is a substance-specific function of temperature. Determine for such a fluid expressions for V, S, H,
A pure fluid, described by the canonical equation of state: G = F(T) + KP, where F(T) is a substance-specific function of temperature and K is a substance-specific constant. Determine for such a
Determine expressions for GR, HR, and SR implied by the three-term virial equation in volume, Eq. (3.38).Eq. (3.38) PV 1+ RT B C Z= (3.38) %3! V V2
Determine expressions for GR, HR, and SR implied by the van der Waals equation of state, Eq. (3.39).Eq. (3.39) RT a (3.39) %3D V - b
Determine expressions for GR, HR, and SR implied by the Dieterici equation:Here, parameters a and b are functions of composition only. RT а P = V - b exp - VRT
The state of 1(lbm) of steam is changed from saturated vapor at 20 (psia) to superheated vapor at 50 (psia) and 1000(°F). What are the enthalpy and entropy changes of the steam? What would the
Propane gas at 100°C is compressed isothermally from an initial pressure of 1 bar to a final pressure of 10 bar. Estimate ΔH and ΔS.
Prove thatFor an ideal gas with constant heat capacities, use this result to derive Eq. (3.23c).Eq. (3.23c) Cy ( aT Cp ( aT ds = dP + dV T av P V
A tank of 4 m3 capacity contains 1500 kg of liquid water at 250°C in equilibrium with its vapor, which fills the rest of the tank. A quantity of 1000 kg of water at 50°C is pumped into the tank.
A well-insulated tank of 50 m3 volume initially contains 16,000 kg of water distributed between liquid and vapor phases at 25°C. Saturated steam at 1500 kPa is admitted to the tank until the
Starting with Eq. (6.9), show that isotherms in the vapor region of a Mollier (HS) diagram have slopes and curvatures given by:Here, β is volume expansivity. If the vapor is described by the
The temperature dependence of the second virial coefficient B is shown for nitrogen in Fig. 3.8. Qualitatively, the shape of B(T) is the same for all gases; quantitatively, the temperature for which
For the reversible isothermal compression of a liquid for which β and κ may be assumed independent of pressure, show that:Do not assume that V is constant at an average value, but use Eq. (3.6) for
In general for an arbitrary thermodynamic property of a pure substance, M = M(T,P); whenceFor what two distinct conditions is the following equation true? ƏM dT + ƏT we, dM = dP dP T
The enthalpy of a pure ideal gas depends on temperature only. Hence, Hig is often said to be “independent of pressure,” and one writes (∂Hig / ∂P)T = 0 . Determine expressions for (∂Hig /
As noted in Ex. 6.6, Δ H lv is not independent of T; in fact, it becomes zero at the critical point. Nor may saturated vapors in general be considered ideal gases. Why is it then that Eq. (6.89)
The derivative (∂U / ∂V)T is sometimes called the internal pressure and the product T(∂P / ∂T)V the thermal pressure. Find equations for their evaluation for:(a) An ideal gas;(b) A van der
(a) A pure substance is described by an expression for G(T, P). Show how to determine Z, U, and CV, in relation to G, T, and P and/or derivatives of G with respect to T and P.(b) A pure substance is
Rationalize the following approximate expressions for solid/liquid saturation pressures:(a) Psatsl = A + BT ;(b) Psatsl = A + BlnT
Real-gas behavior for turbomachinery is sometimes empirically accommodated through the expression Ẇ = 〈Z〉 Ẇig , where Ẇig is the ideal-gas mechanical power and 〈Z〉 is some suitably
As suggested by Fig. 3.1, the slope of the sublimation curve at the triple point is generally greater than that of the vaporization curve at the same state. Rationalize this observation. Note
An ideal gas with constant heat capacities enters a converging/diverging nozzle with negligible velocity. If it expands isentropically within the nozzle, show that the throat velocity is given
A gas enters a converging nozzle at pressure P1 with negligible velocity, expands isentropically in the nozzle, and discharges into a chamber at pressure P2. Sketch graphs showing the velocity at the
For an adiabatic gas compressor, the efficiency with respect to isentropic operation η is a measure of internal irreversibilities; so is the dimensionless rate of entropy generation SG /R ≡ ṠG
For isentropic expansion in a converging/diverging nozzle with negligible entrance velocity, sketch graphs of mass flow rate ṁ, velocity u, and area ratio A/A1 versus the pressure ratio P/P1. Here,
For a pressure-explicit equation of state, prove that the Joule/Thomson inversion curve is the locus of states for which:Apply this equation to (a) the van der Waals equation; (b) the Redlich/Kwong
A fan is (in effect) a gas compressor which moves large volumes of air at low pressure across small (1 to 15 kPa) pressure differences. The usual design equation is:where subscript 1 denotes inlet
Two nonconducting tanks of negligible heat capacity and of equal volume initially contain equal quantities of the same ideal gas at the same T and P. Tank A discharges to the atmosphere through a

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