INTRODUCTION TO
`
`Chemical Engineering Thermodynamics
`
`J. M. SMITH
`Professor of Chemical Engineering
`Northwestern University
`
`H. C. VAN NESS
`
`Associate Professor of Chemical Engineering
`Rensselaer Polytechnic Institute
`
`Second Edition
`
`McGRAW—HILL‘BOOK COMPANY, INC.
`New York Toronto London
`1959
`
`IPR2015-00171
`IPR2015-00171
`Exhibit 1067
`Exhibit 1067
`
`000001
`
`

`
`important concepts
`
`ical—reaction equi-
`the second, because
`'
`However,
`
`of the first edition.
`e new developments
`out-of—date and less
`e covered in a two-
`
`oifering by omitting
`
`- third or fourth year
`
`_ the book who have
`E. W. Comings and
`y offered many Valu-
`ally, much is owed to
`N. Lacey, and H. C.
`the subject.
`
`J. M. Smith
`H. C’. Van Ness
`
`Preface
`
`List 0] Symbols
`
`. Introduction
`
`. The First Law and Other Basic Concepts
`
`. The Ideal Gas
`. Pressure—Volume-Temperature Relations of Fluids
`
`. Heat Effects
`
`. The Second Law of Thermodynamics
`
`. Thermodynamic Properties of Fluids
`
`. Thermodynamics of Flow Processes
`
`. Production of Work from Heat
`
`. Refrigeration
`
`. Thermodynamic Analysis of Processes
`
`. Phase Equilibria
`
`. Chemical-reaction Equilibria
`
`Appendix: Steam Tables
`Name Ihdex
`
`Subject Index
`
`000002
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`000002
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`

`
` SEC. 13-3]
`
`
`408
`CHEMICAL ENGINEERING TI-IERMODYNAMICS
`[CHAR 13
`Equation (13-10) i
`
`
`The ratio of the fugacity at any state to that at the standard state is
`gaseous reaction.‘
`termed the activity a.
`In terms of activities, Eq. (13-6) may be written
`When solids, liqt
`the standard state
`
`AF° = —RT ln “E”?
`a‘}iaB
`
`(13-7)
`
`
`
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`
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`may be impossible.
`expression [Eq. (13
`@‘pr solids and liq
`at 1 atm pressure a
`discussion of stand
`13-3. Efiect of
`
`specifying a definit
`that of the reacti
`values at the stan
`
`this temperature.
`standard heat of re
`The general equ
`ponent system wit
`
`At constant prei
`
`OI‘
`
`1 An alternative dei
`mental concepts thai
`based on the concept
`equilibrium (van’t H
`The vessel is at con
`
`cylinder assemblies f
`shown in Fig. 13-1.
`assumed to be semip-
`in the cylinder to be
`Suppose that gase
`there are a moles of .
`perature of the equil
`1. First, isotherm:
`rately to their fugac
`the vessel. The cha
`
`In this case fi = 1,:
`
`2. The a moles of
`
`
`
`
`
`
`
`
`
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`
`The equilibrium constant K of a chemical reaction is defined by the
`activities of the reactants and products in the equilibrium state as follows:
`
`am
`
`(13 8)
`
`(13-7) results in the basic
`Combination of this definition and Eq.
`relationship between the standard free-energy change and the equilibrium
`constant,
`
`(13-9)
`->K AF° = —RT...1n K
`Qf AF° is negative, K will be greater than unity; if AF° is positive, K
`Will be less than unity. Hence, the equilibrium yield of a reaction is
`high when the standard free—energy
`change has a large negative value and
`low when the free—energy change has a
`
`3_ C, and D
`
`large positive value.
`Standard States. Before Eqs. (13-6)
`to (13-9) can be applied to the cal-
`culation of the equilibrium conversion,
`the standard state of each reactant
`
`Equilibrium mixture of A,
`
`FIG‘ 13'1'.Appa”’t“s in Which? 333°‘
`ous
`reaction occurs at equilibrium
`(Vanni Hoff equilibrium boX)_
`
`and product must be specified. The
`choice is arbitrary, but certain con-
`ventions have proved to be more con-
`venient than others.
`If the standard
`state for a substance is chosen as the
`.
`.
`pure component, specification of the
`temperature and pressure (or fugacity)
`is sufficient to define the state completely.
`If the standard state chosen
`for the substance is a solution, the composition must also be specified.
`Gjor gases it is most convenient to choose the pure component at the
`temperature of the reactioniand at unit fugacity as the standard state.
`‘AB advantage of this choice is that the standard-state pressure approaches
`1 atm for all gases and is exactly 1 atm for an ideal ga@ Free-energy
`values, and hence AF°, are most easily evaluated at this pressure. For
`this standard state each f° is unity, and Eqs.
`(13-6) and (13-7) for a
`gaseous reaction become
`3% AF° = —RTln
`
`= —RT1nK -~
`
`(13-10)
`
`AfB
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`000003
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`

`
`
`412
`CHEMICAL ENGINEERING THERMODYNAMICS
`integrated rigorously. Suppose the data are available in the form of
`power functions
`
` SE0. 13-4]
`
`3. The last
`most conveniel
`mination of K
`in the form 01
`
`energy change
`the standard 1
`used in Eq. (12
`the relation 0
`0£lX.t_11?.,th
`these experim
`great importai
`tion by famot
`-nineteenth ce'
`listed in stant
`
`AH? (see Tab)
`culated by cc
`AH° quantitie
`been calculat
`
`'
`
`where AS° is
`tion of AS° is
`
`cepts of this
`stable state
`
`capacity of a
`One meth(
`utilizes mea
`
`T to nearly :
`it is impossi
`zero. How:
`
`example, to
`Vs. T curve
`to T = 0°K
`
`If the proc-
`material) at
`
`
`
`1 G. N. Le‘
`3 For a res
`
`and Merle R:
`Chap. XXX‘
`
`[CHAR 13
`
`C',,=oz+BT+~/T2
`
`Then, as shown in Chap. 5, the heat of reaction at any temperature T
`is given by the expression
`AH°
`
`+ Au T + “"2” + A—.-73:”
`
`(5-16)
`
`In this equation AH0 is a constant which can readily be evaluated pro-
`vided the standard heat of reaction is known at a single temperature,
`e.g., 25°C. With AH0 determined, AH° from Eq.
`(5-16) can be sub-
`stituted in Eq. (13-18). The integrated result is
`
`g
`
`‘
`
`J
`
`_
`AH0
`Ac:
`A/3
`A7
`(13-20)
`— —fi?+—R-1nT+2—RT—|—€R-T2+C
`Here C is the constant of integration, which may be evaluated from a
`knowledge of the equilibrium constant at one temperature.
`Either Eq. (13-19) or Eq. (13-20) may be used to evaluate the effect
`of temperature on the equilibrium constant, the latter being the more
`accurate but requiring heat-capacity data.
`The variation of the standard free—energy change with temperature
`can be obtained by combining Eqs. (13-20) and (13-9),
`
`(13-21)
`
`-)§"'9(AF° = AHO — Au T ln T — 9; T2 — 561 T3 —_I_T
`where I is a composite constant equal to RC’.
`13-4. Evaluation of Equilibrium Constants. The equilibrium constant
`for a given reaction can be calculated at any temperature by Eq. (13-20),
`provided enough information is known to permit evaluation of the con-
`stants AH0 and C.
`It is presumed that the necessary heat-capacity data
`are available. The general methods in use and the minimum data nec-
`essary for the evaluation of AH9 and C’ may be listed as follows:
`1. K values may be calculated directly by the defining equation for
`K [Eq. (13-8)] from experimental measurements of composition in the
`equilibrium mixture. The methods of doing this in practice are taken
`up in Sec. 13-6. divalues of glggaremlgnown for two temperatures} Eq.
`(13-20) is written for each temperature and the resulting two equations
`are so ved simultaneously for AHO and C.
`I93
`2. If the standard heat of reaction is known for a single temperatu
`e.g., 25°C AHQ may_bgg, m Eq. (5-16). ©nly a_.sin.gle..:¢alue
`determined directly from equi 1 rium measurements, is then needed
`£fQ
`for an evaluation of C by Eq. (13-20).
`
`000004
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`000004
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`

`
`
`
`
`
`in the form of a standard heati reaction AH° and a standard free-
`
`T
`
`413
`CHEMICAL-REACTION EQUILIBRIA
`SE0. 13-4]
`3. The last method is probably the most widely used and is the
`most convenient, because it does not require direct experimental deter-
`mination of K. The method makes use of thermal data only, usually
`
`he constant AHo is determined from
`energy change of reaction AF°.
`q. (5-16$ The Value of AF° is then
`the standard heat of reaction by
`used in Eq. (13-21) to determine I and the constant C is calculated from
`the relation C’ = I/ R.
`Oplgthep third method involves no equilibrium measurements. Since
`these experimental measurements are difficult, the third method is of
`great importance.
`It was developed as a result of considerable investiga-
`tion by famous workers1 in thermodynamics during the last part of the
`nineteenth century. Values of AF? for formation reactions at 25°C are
`listed in standard references along with the standard heats of formation
`AH? §e_(e_"_I‘__2_L_bml_e_1§5_‘H-1). Values of AF° for other reactions at 25°C are cal-
`culated by combining formation reactions exactly in the same way that
`AH° quantities are determined (Chap. 5). The listed values of AF° have
`been calculated from the relation
`AF° = AH° — T AS°
`
`(13-16)
`
`where AS° is the standard entropy change of reaction. The determina-
`tion of AS° is based on the third law of thermodynamics. The basic con-
`cepts of this law are (1) that the entropy of any substance in its most
`stable state at absolute zero temperature is zero and (2) that the heat
`capacity of any substance approaches zero at absolute zero temperature.’
`One method of applying the third law to calculate the absolute entropy
`utilizes measured values of
`the specific heat and latent heats from
`T to nearly absolute zero (all at constant pressure). At the present time
`it is impossible to measure C',, at temperatures all the way to absolute
`zero. However, by carrying the measurements as far as possible (for
`example, to 10 or 12°K) and utilizing concept 2 of the third law, the C1,
`vs. T curve can be extrapolated from the limit of the measurements down
`to T = 0°K. From the second law (Chap. 6),
`
`dQ = T dS
`If the process occurs at constant pressure, dQ = Op dT (for 1 mole of
`material) and
`
`as = 09;”
`
`(13-22)
`
`1 G. N. Lewis, van’t Hoff, and Nernst among others.
`‘For a résumé of the development and significance of the third law see G. N. Lewis
`and Merle Randall, “Thermodynamics and the Free Energy of Chemical Substances,”
`Chap. XXXI, McGraw—Hil1 Book Company, Inc., New York, 1923.
`
`000005
`
`VIODYNAMICS
`
`[CHAR 13
`
`e available in the form of
`
`V2
`
`tion at any temperature T
`
`— + -—
`
`(5-16)
`
`n readily be evaluated pro-
`E at a single temperature,
`um Eq.
`(5-16) can be sub-
`llt is
`
`+ 651% T2 + C
`
`(13-20)
`
`L1 may be evaluated from a
`e temperature.
`used to evaluate the effect
`
`, the latter being the more
`
`y change with temperature
`and (13-9),
`
`7
`
`A FY
`2 ——6Y 13 —_I_T
`
`(13-21)
`
`V L
`
`’-o
`. The equilibrium constant
`temperature by Eq. (13-20),
`ermit evaluation of the con-
`
`uecessary heat—capacity data
`and the minimum data nec-
`be listed as follows:
`
`)y the defining equation for
`hents of composition in the
`g this in practice are taken
`for two temperatures} Eq.
`the resulting two equations
`
`wn for a single temperatureb
`(5-16). ©n1y a_sin.g;la»»-value
`I easurements, is then needed
`
`
`
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`000005
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`
` SEC. 13-5]
`
`the fugacity 0
`expression
`
`where f is the
`pressure of th
`this expressior
`
`where K, is s"
`quantity is :
`composition.
`The fugacit
`data or from T
`values for the
`and given the
`of the da valu
`
`where p repr
`Usually, s‘
`coeflicients E
`that -y value
`assuming th
`ideal gas; sc
`will be a fr
`discussed in
`
`components
`approached
`
`where K. I
`in Eq. (13—'
`Since K
`tion must
`
`‘It is app
`which the st:
`fore the lim
`section.
`2 The wor<
`calculations.
`constant on?
`
`
`420
`CHEMICAL ENGINEERING THERMODYNAMICS
`[CHAR 13
`
`
`
`At 320°C (593°K)
`
`AF° = 6980
`ln K = -5.94
`K = 2.6 X 10‘3
`
`stants from thermal data as illus-
`The results of the calculation of equilibrium con
`nying table with the experimental
`trated in this example are compared in the accompa
`d with the result calculated in Example 13-2.
`values given in Example 13-1 an
`________________________________._
`E
`'
`t 1 d t
`X%:::I:l1epl;ea13_21L a’
`Example 13-2
`Example 13-3
`
`
`
`
`
`6.8X10"
`1.3-5X10‘3
`
`
`13.1 X10"
`2.6X10‘3
`
`6.8X10”
`.
`.
`.
`.
`.
`.
`. .
`.
`.
`.
`.
`.
`.
`K at 145°C .
`1.9 X 10"“
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`K at 320°C .
`______________________________________
`imental results and the values calculated indirectly
`The agreement between exper
`cy has not yet been completely accounted
`from thermal data is fair. This discrepan
`t in both the experimental and calculated
`for. However, errors are probably presen
`values.
`It should be noted that the experimental K values have been calculated on
`the assumption that the equilibrium mixture is an ideal gas. Cope‘ has made an
`exhaustive study of this reaction, and he concludes on the basis of all available
`evidence that the most likely values for AHZM, and AF‘;93 are somewhat different
`from the presently accepted ones. His values are
`~ 10,750 cal
`AH§es
`—— 1820 cal
`AF?»
`
`II
`
`The discrepancy between experimental and calculated values of K is narrowed by the
`use of these data.
`13-5. Efiect of Pressure on the Equilibrium Constant. The equi-
`librium constant as defined in terms of activities (often called the true
`equilibrium constant) is not dependent on pressure. As mentioned ear-
`lier, this is apparent from Eq. (13-9), since AF° is based on fixed initial
`and final states and is not influenced by the conditions at any intermediate
`point. On the other hand, the pressure does affect the equilibrium con-
`version for a gas-phase reaction in accordance with the principle that an
`increase in pressure will increase the conversion if there is a. decrease in
`volume accompanying the reaction. This effect of pressure must be
`accounted for in the relationship between the equilibrium constant and
`the equilibrium composition.
`It may be accurately determined provided
`the fugacities of the various components in the equilibrium mixture can
`be related to composition.
`It was found in Chap. 12 that this relation-
`ship wasvery simple for ideal solutions [Eq. (12-24(1)] and that nonideal
`systems could be studied by introducing the activity coefficient 7. Then
`1 C. S. Cope, “Equilibria in the Hydration of Ethylene at Elevated Pressures and
`Temperatures,” Ph.D. dissertation, Yale University, 1956.
`
`
`
`
`
`
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`000006
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`
`
`
`
`SEC. 13-5]
`
`CHEMICAL-REACTION EQUILIBRIA
`
`421
`
`the fugacity of any component in a gaseous mixture is given by the
`expression
`
`.7 = vfy
`
`(13-25)
`
`where f is the fugacity of the pure component at the temperature and
`pressure of the equilibrium mixture and y is its mole fraction. With
`this expression for the fugacity, Eq. (13-10)‘ for K becomes
`
`_ (7f)°(7f)“ 1/‘z/‘la _ (7f)°(vf)‘t
`K ‘ (vf)§(vf)1§ygiy'ia “ <vf>§(~/f>'=.. K”
`
`_
`(13 26’
`
`K ='
`
`where K, is substituted for the ratio of the mole fractions. This latter
`quantity is sometimes called an equilibrium constant2 in terms of
`composition.
`The fugacity of the pure components may be evaluated from Volumetric
`data or from Fig. 12-2 as described in Chap. 12. Since Fig. 12-2 provides
`values for the ratio of fugacity to pressure (called the fugacity coefficient
`and given the symbol <1»), it is convenient to rewrite Eq. (13-26) in terms
`of the 42 values:
`c
`d
`c
`d
`'YC"YD ¢C¢'g K (c+d_a_.1,)
`)
`(
`727% ¢:'.¢>t
`"0
`
`where 11 represents the total pressure.
`Usually, sufficient data are not available for evaluation of the activity
`coefficients as a function of composition of the equilibrium mixture, so
`that 7 values must be assumed equal to unity. This is equivalent to
`assuming that the gas behaves as an ideal solution (not necessarily as an
`ideal gas; see Chap. 12). The error introduced with this simplification
`will be a function of the deviation from ideality of the solution. As
`discussed in Chap. 12, this deviation increases as the dissimilarity of the
`components in the reaction mixture increases and as the critical point is
`approached. With this simplification, Eq. (13-27) becomes
`
`
`
`ERMODYNAMICS
`
`[CHAR 13
`
`istants from thermal data as illus-
`anying table with the experimental
`calculated in Example 13-2.
`
`Example 13-3
`
`Example 13-2
`
`1‘
`
`I’
`
`13.1 x 10-2
`6.8 X 10-2
`
`1.35><10‘3 2.6X10‘3
`
`ld the values calculated indirectly
`lot yet been completely accounted
`h the experimental and calculated
`K values have been calculated on
`n ideal gas. Copel has made an
`des on the basis of all available
`
`tnd AF;’93 are somewhat different
`
`cal
`
`1 t
`
`ed values of K is narrowed by the
`
`brium Constant. The equi-
`tivities (often called the true
`pressure. As mentioned ear-
`AF° is based on fixed initial
`
`onditions at any intermediate
`es affect the equilibrium con-
`ce with the principle that an
`sion if there is a decrease in
`
`effect of pressure must be
`he equilibrium constant and
`urately determined provided
`the equilibrium mixture can
`I Chap. 12 that this relation-
`. (12—24a)] and that nonidcal
`activity coefficient 7. Then
`
`hylene at Elevated Pressures and
`, 1956.
`
`«
`
`K = K.K,p<c+d-H»
`
`(13-28)
`
`where K¢ represents the combination of fugacity coefficients as defined
`in Eq. (13-27).
`Since K is independent of pressure, K, and the equilibrium composi-
`tion must depend on pressure.
`If the reaction mixture behaves as an
`
`1 It is appropriate to recall that Eq. (13-10) is valid only for gaseous systems in
`which the standard states have been taken as the pure gases at unit fugacity. There-
`fore the limitation to gaseous reactions applies to the remaining equations of this
`section.
`2 The word constant is a misnomer as applied to many of the K’s used in equilibrium
`calculations. Nevertheless it is the universal terminology. K,,, for example, can be
`constant only under very special circumstances.
`
`000007
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`000007
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`
`
`
`
` K = K,,p<v+d--=-b>
`.
`(13-29)
`
`422
`
`CHEMICAL ENGINEERING THERMODYNAMICS
`
`[CHAR 13
`
`ideal gas, this effect becomes immediately evident since all the qs values,
`and hence K4,, become unity. Equation (13-27) then reduces to
`
`
`
`
`
`This expression indicates quantitatively the effect of a change in pressure
`on the equilibrium composition. Because K is independent of pressure,
`variation in the pressure term must be balanced by a corresponding
`
`change in Ky, and hence in the composition.
`
`When the ideal-gas law does not apply, the pressure may affect Ky‘
`
`even though the term (3 + d - a — b is zero. This is due to the fact
`
`that K4, [Eq. (13-28)] varies with pressure. However, the effect is usually
`
`small.
`
`In gaseous reactions another equilibrium constant K, is frequently
`used.
`It is defined in terms of the mole fractions and total pressure,
`
`or in terms of the partial pressures, as follows:
`
`
`
`
`It is understood here that partial pressure is defined as the product of
`mole fraction and total pressure (Chap. 4).
`Comparison of Eqs. (13-27) and (13-30) indicates that the true equi-
`librium constant K is related to K,, as follows:
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`-c ‘d
`c
`_ _(Efl’)_(_y_D7’_)‘1 = B6'.7’Z1_> = Kyp(o+d—a—b)
`_
`_
`K —
`(yAp)“(@/B22)”
`flirt
`”
`
`(13.30)
`
`
`
`
`K = K.,K,,K,,
`
`(13-31)
`
`
`
`As stated before, K, can seldom be evaluated and is usually assumed to
`be unity, tantamount to the assumption of an ideal solution. With this
`understanding, Eq. (13-31) assumes the simplified, but approximate, form
`
`(13-31a)
`K = K,K,,
`
`
`
`This expression shows that, although K is independent of pressure,
`K,, is not. The reason is that the extent of the deviations from ideal-gas
`behavior, and hence K.,,, changes with p. On the other hand if the reac-
`tion mixture is an ideal gas, K, is unity and K = K,,.
`The value of, K, can be determined experimentally from measurements
`of total pressure and equilibrium composition, according to Eq. (13-30).
`Alternatively, it can be evaluated by Eq. (13-31a), provided the assump-
`tion of ideal solutions is justified. Figure 13-2 shows K1. as a function of
`temperature for a number of reactions. These values are valid over a
`pressure range in which the reaction mixture behaves essentially as an
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
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`000008
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`

`
` SE0. 13-6]
`
`
`424
`CHEMICAL ENGINEERING THERMODYNAMICS
`[CHAR 13
`ideal gas, 13.9., where K, is independent of pressure and is equal to the
`
`
`true equilibrium constant.
`The effect of pressure on the conversion for liquid- and solid—phase
`
`
`reactions is usually negligible because of the small change in volume with
`
`
`pressure in such cases.
`
`
`...—‘1..... .1. mm ’\4}+A
`13-6. Calculation of the Equilibrium Conversion. Once the equilib-
`ri}_1_r_r_r_constant is known, the calculation of the composition at equilibrium
`
`
`is a simple process, providedflthat only one phase exists and this phase
`
`
`is an ideal solution. The principle of the conservation of mass (material
`balance) and the definition of the equilibrium constant in terms oi7FoTrI-“’
`
`
`position [as, for example, by M(13¢2‘8) for a gaseous reaction] are all
`
`
`that is necessary.
`If more than one phase is present, the problem is
`
`
`more complicated, but it may be handled by introducing the criteria of
`
`
`phase equilibria developed in Chap. 12. This is required because at
`
`
`equilibrium there can be no tendency for a change to occur, either with
`
`
`respect to a transfer of material from one phase to another or with respect
`
`
`to conversion of one chemical species to another. The procedures are
`
`
`developed and illustrated for gas, liquid, and heterogeneous reactions in
`
`the remainder of this section.
`
`‘
`
`GAS-PHASE REACTIONS
`
`
`
`Let the moles of
`
`(a)
`
`Mol
`
`M0
`
`M01
`
`Substituting in '.
`
`(b) The total p
`number of moles C
`K, does not chang
`
`(c) In this case
`in the number of 1
`result and the frat
`
`(d)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Example 13-4. The water-gas-shift reaction
`
`00(9) + H2O(g) ~> 002(9) + H2(g)
`
`is carried out under the different sets of conditions described below. Calculate the
`fraction of the steam decomposed in each case. Assume that the mixture behaves as
`an ideal gas.
`(a) The reactants consist of 1 mole of water vapor and 1 mole of CO. The tem-
`perature is 1530°F and the total pressure 1 atm.
`(b) Same as (a) except that the total pressure is 10 atm.
`(c) Same as ((1) except that 2 moles of N2 is included in the reactants.
`(d) The reactants are 2 moles of H20 and 1 mole of CO. Other conditions are the
`same as in (a).
`‘
`(e) The reactants are 2 moles of CO and 1 mole of H20.
`(f) The initial mixture consists of 1 mole of H20, 1 mole of C0, and 1 mole of CO2.
`Other conditions are the same as in (a).
`'
`(g) Same as (a) except that the temperature is raised to 2500°F.
`Solution. From Fig. 13-2, at 1530°F, and Eq. (13-30),
`
`
`
`
`
`
`
`
`
`
`
`
`
`(e) Here the i
`
`(f) In this ca:
`
`(g) From Fig.
`
`The conversic
`the reaction is 6
`Example 13-
`vapor-phase hyn
`13-1, and assuu
`
`
`
`000009
`
`The method of relating initial composition and the conversion to the
`equilibrium constant involves the application of Eqs. (13-28) to (13-31a).
`The following examples illustratepthe method of solution:
`
`000009
`
`

`
`
`
`
`
`SEC. 13-6]
`
`CHEMICAL-REACTION EQUILIBRIA
`
`433
`
`i.e., solving the three
`From this point on the problem is simply one in algebra,
`equations (A), (B), and (C) for 0:, 5, and 6. Since KI and K11 are both relatively
`large, there will be little unreacted alcohol at equilibrium. This simplifies the solu-
`tion, for, approximately,
`
`at + 53 = 1
`
`If 01 is assumed = 0.883, then
`
`B = 1 - (1 = 0.117
`
`
`
`
`
`
`
`
`
`
`Using these values in Eqs. (I), (II), and (III) to find 5 values gives essentially the
`same result, 6 = 0.08.
`At equilibrium:
`
`M0168 C4H5 .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`
`
`Moles C2H5OH .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`. 0.000
`
`
`Moles C2114 .
`.
`.
`.
`.
`.
`.
`.
`,
`.
`.
`.
`.
`. 0.803
`
`
`Moles CH;,CHO .
`.
`.
`.
`.
`.
`.
`.
`.
`. 0.037
`
`
`Moles H20 .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`. 0.963
`.
`Moles H2 .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`. 0.117
`
`.
`.
`Total .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`. 2 .000
`
`The conversion to butadiene is 8 per cent
`
`
`
`f = fa:
`
`(13-32)
`
`where f is the fugacity of the pure component at the reaction pressure
`and temperature and 3: is its mole fraction at equilibrium.
`Since pres-
`sure usually has a very small effect on the properties of liquids, f can be
`replaced by f°, the standard—state fugacity, without introducing appre-
`ciable error. Hence
`
` LIQUID—PI-IASE REACTIONS
`
`The problem of calculating the equilibrium composition of liquid—pha.se
`
`reactions is chiefly one of relating the activity to the composition for
`
`substitution in Eq. (13-8). The standard states for the components of
`liquid-phase reactions may be chosen as the pure liquids at 1 atm pres-
`
`sure and the reaction temperature. This is analogous to the choice of
`
`standard states for gas—phase reactions.
`If the equilibrium mixture is an ideal solution, the fugacity of each
`
`component is given by
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`>DYNAMICS
`
`[CHAP, 13
`
`100 atm
`
`of ethyl alcohol to butadiene
`tactions involved are
`
`427°C are
`
`Then suppose that
`action I
`action 11
`
`will be
`
`8 equilibrium-constant equa-
`9 an ideal gas,
`
`— e)(a + e)
`Emma
`
`+5;
`
`(4)
`
`<3)
`
`(0)
`
`1 the over-all reaction. The
`ducts which are assumed to
`
`‘would be present were the
`ions: 2C2H50H_; 04H“ +
`
`and the activity is given by
`
`f = °a:
`_
`a = 1., = x
`
`(13-33)
`
`(13-34)
`
`
`
`
`Thus for ideal solutions, Eq. (13-8) reduces to
`
`
`
`
`K — "3”‘i’ = K,
`— a:jx’,’_.,
`
`(13-35)
`
`
`
`000010
`
`000010
`
`

`
`
`
`434
`
`CHEMICAL ENGINEERING THERMODYNAMICS
`
`[CHAR 13
`
`SEC. 13-6]
`
`
`
`If the solution is not ideal, the activity coefficient may be introduced,
`as in Eq. (13-25), to take into account deviations from ideality. The
`fugacity of each component in the equilibrium mixture is then
`
`_
`and the activity is
`
`f = was
`a =fio = 7a:
`
`(13-36)
`(13-37)
`
`The equilibrium constant now becomes
`
`K = 'Y2,‘7dD
`72173’; $21903’;
`
`= "/€,"Y£1lJ K
`'Y1i'Yi3
`
`x
`
`This is the general relationship for K when the standard states are taken
`as the pure components at 1 atm.
`The practical difficulty with Eq. (13-38) is that data are rarely avail-
`able for evaluating the activity coefiicients, and one usually has no alter-
`native but to assume the solution ideal, so that all the 1/’s are unity.
`In
`other words, when the standard states are taken as the pure components,
`the approximate equation (13-35) must generally be used. This does not
`always result in so large an error as might be expected, because the ratio
`of the 7’s in Eq. (13-38) may be nearly unity even when the individual 1/s
`are not.
`
`HYP°l“s“bfi‘§gLr}'m°‘a‘
`/
`
`
`
`line drawn tange
`is valid in the ca:
`
`it is possible to c:
`Henry’s law to a
`widely used as a
`The standard-
`
`Hence, for any or
`to hold
`
`and
`
`The obvious adx
`
`simple relations]
`Henry’s law is '
`centration of 1 1:
`is a real state 0
`data are availai
`for otherwise tl
`
`(13-9).
`For solutes in
`
`used, and this
`thermodynamics
`liquid solutions
`equilibrium com
`concentration 0'
`then made to ti.
`
`When liquid 2
`Eq. (12-1), a re
`with the equati
`able choice in '
`consider a react
`tion of C.
`The
`
`phase with simu
`phase equilibrit
`evaluated from
`
`ponent, z'.e., un
`other hand, the
`
`
`
`For components which are known to be present in high concentration,
`Eq. (13-33) is usually nearly correct, because the Lewis and Randall rule
`always becomes valid for a component
`as its concentration approaches as = 1.
`[See Eq. (12-241)). The proof is analo-
`gous to Example 12-9.]
`For components which are at low
`concentration in aqueous solution, a
`different procedure has been Widely
`adopted, because in this
`case Eq.
`(13-33) is frequently not even approxi-
`mately correct. The method is based
`on the use of fictitious or hypothetical
`standard states. The standard state of
`
`m, molality
`FIG. 13-4. Standard state for dilute
`aqueous solutions.
`
`the solute is taken as the hypothetical
`state which would exist if the solute
`
`the way to a molality of 1.
`obeyed Henry’s law all
`molality and fugacity, Henry's law is
`-
`
`In terms of
`
`f = km
`
`(13-39)
`
`and it is always valid for a component whose concentration approaches
`zero. This hypothetical state is illustrated in Fig. 13-4. The dashed
`
`000011
`
`000011
`
`

`
`
`
`NAMICS
`
`[CHAR 13
`
`SEO. 13-6]
`
`CHEMICAL-REACTION EQUILIBRIA
`
`
`
`435
`
`line drawn tangent to the curve at the origin represents Henry’s law and
`is valid in the case shown to a molality much less than unity. However,
`it is possible to calculate the properties the solute would have if it obeyed
`Henry’s law to a concentration of 1 molal, and this hypothetical state is
`Widely used as a standard state.
`The standard-state fugacity is
`
`f° = lcm° = k(1) = 19
`Hence, for any component at a concentration low enough for Henry’s law
`to hold
`ll
`
`and
`
`‘hi
`
`a
`
`km = f°m
`
`I1
`
`J’
`
`= 111,
`
`(13-40)
`
`The obvious advantage of this standard state is that it provides a very
`simple relationship between activity and concentration for cases where
`Henry’s law is valid.
`Its range does not commonly extend to a con-
`centration of 1 molal.
`In the rare case where it does, the standard state
`is a real state of the solute. This procedure is useful only where AF°
`data are available in terms of the hypothetical 1-molal standard state,
`for otherwise the equilibrium constant cannot be evaluated from Eq.
`(13-9).
`For solutes in nonaqueous solvents, similar procedures are occasionally
`used, and this subject is treated in advanced textbooks on chemical
`thermodynamics. The use of special standard states for components of
`liquid solutions sometimes allows a more accurate determination of the
`equilibrium composition, but often at least one component will exist at a
`concentration outside the range of use of such methods. Recourse is
`then made to the approximate equations for ideal solutions.
`
`nt may be introduced,
`s from ideality. The
`:ture is then
`
`(13-36)
`
`(13-37)
`
`(13-38)
`
`hdard states are taken
`
`1 data are rarely avail-
`le usually has no alter-
`the 'y’s are unity.
`In
`5 the pure components,
`e used. This does not
`
`eted, because the ratio
`when the individual -y’s
`
`in high concentration,
`Lewis and Randall rule
`
`valid for a component
`ion approaches as = 1.
`. The proof is analo-
`12-9.]
`its which are at
`
`low
`
`1 aqueous solution, a
`ure has been widely
`be
`in this case Eq.
`ntly not even approxi-
`The method is based
`
`titious or hypothetical
`The standard state of
`
`en as the hypothetical
`ld exist if the solute
`
`y of 1.
`
`In terms of
`
`(13-39)
`
`centration approaches
`ig. 13-4. The dashed
`
`
`
`HETEROGENEOUS REACTIONS
`
`When liquid and gaseous phases are both present in a reaction mixture,
`Eq. (12-1), a requirement of phase equilibrium, must be satisfied along
`with the equation of ehemica1—reaction equilibrium. There is consider-
`able choice in the method of treatment of such cases. For example,
`consider a reaction between gas A and water B to form an aqueous solu-
`tion of C. The (reaction may be assumed to occur entirely in the gas
`phase with simultaneous transfer of material between phases to maintain
`phase equilibrium.
`In this method the equilibrium constant would be
`evaluated from AF° based on the standard state for gases for each com-
`ponent, i.e., unit fugacity at the temperature of the reaction. On the
`other hand, the reaction may be assumed to occur in the liquid phase, in
`
`000012
`
`000012