Info Ied
male fraction salt, xs Fig. 8.16 The NaCl HiO phase diagram On the top of the diagram is a large single-phase region entitled liquid sol'n the common name is brine. On the left-hand ordinate is pure ice and the right hand ordinate x 1, not shown is pure salt. The shaded zone in Fig. 8.16 is the only single-phase solid region in the diagram. It is a compound with the water-to-salt mole ratio denoted by n. It is shown as the region is labeled NaCl H2O x, with x varying from the lower phase...
T
Fig. 1.16 Heat flows between twe system s separated by a diathermal interface and encased in Figure 1.16 shows two systems, called heat reservoirs, labeled 1 and 2, in good thermal contact through a heat-transmitting interface. Taken together, the pair constitutes an isolated system because the encasing boundary is both rigid and adiabatic. No work is done by system 1 on system 2 or vice versa but because T1 T2, heat flows from one system to the other. The direction of the arrows in the diagram...
The Chemical Potential
The thermodynamic terms heat and work can be viewed as the product of a capacity factor or quantity of something and a difference in a potential. Table 7.1 lists several examples of this breakdown of heat work expressions for mechanical, electrical, thermal, and chemical processes. Although rate processes are not within the purview of thermodynamics, they involve the same potentials as those responsible for producing heat or work. The basic rate laws are of the form flux coefficient x potential...
A complex phase diagram ironuranium
Eutectic features often appear in parts of more complex phase diagrams, as shown in the iron-uranium diagram of Fig. 8.14. If the diagram is divided into three parts at 1 3 and 6 7 mole fraction uranium, the result is two simple eutectic diagrams and a more complex diagram. The Fe-rich side resembles Fig. 8.11 with two added features. The first is the number of phases of the pure components. The left hand ordinate of Fig. 8.14 makes provision for three crystallographic modifications of iron The...
Effect of Pressure on GasPhase Chemical Equilibria
Instead of partial pressures, mixture compositions are often more conveniently expressed in terms of the mole fractions of the species present, as in the previous example. The following formulation illustrates how total pressure affects the equilibrium composition. Using Dalton's Rule Eq 7.3 , pi where xi is the mole fraction of species i and p is the total pressure, Eq 9.21 becomes K p reactants products k Products 9 24 Although KP is a function of temperature only, the equilibrium constant in...
Nonideal Liquid and Solid Solutions
Although gas mixtures can for most purposes be treated as ideal, liquid and solid solutions are generally significantly nonideal. The strong intermolecular interactions that are responsible for the existence of pure condensed phases are also the source of their deviations from ideality when mixed in solutions. A binary solution of A and B is ideal if the average of the A-A and B-B intermolecular forces is just equal to the strength of the A-B interaction Ref. 1 contains a thorough explanation...
AS Qh Ql Z
thermal reservoirs r-p t cno ino This portion assumes that the temperature of cooling water for the condenser is 25oC and the high-temperature supply from the boiler steam generator is 320oC Substituting the component entropy gains into Eq 4.14 gives the entropy production per kg of water circulated AStot 0 0.01 0.52 2.53 3.06 kJ kg-K The largest contribution is due to heat transfer over non-zero temperature differences between the cycle components and their associated thermal reservoirs. The...
Problems for Chapter
10.1 The solid-state electrochemical cell NbO Nb Ru electrolyte Ta2 O5 Ta consists on the right of a Ta Ta2O5 couple that produces a fixed electrode potential and on the left a half cell containing a mixture of NbO and Nb dissolved in ruthenium. Ru is inert electrochemically and serves only to dilute the active niobium metal component. The cell operates at 1000 K with various mole fractions of Nb dissolved in ruthenium. For the overall cell reaction NbO Ta Nb 1 Ta2O5 , the standard free energy...
Reactive gas in contact with a reactive metal
The values of the 02 pressure required for coexistence of M and M02 are usually quite small, because, except for the noble metals, oxides are much more stable than the elemental metals. Reaction 9.3 releases substantial heat, so AHo is large and negative. This term dominates AGo, which is also large and negative. For instance, if AGo -200 kJ mole at 1000 K, Eq 9.33a gives p0 3.6x10-11 atm. From practical considerations, such a low pressure of 02 is difficult to produce and control in a process...
Phase Separation
The single-phase solid and liquid phase regions in Fig. 8.3 show no structure because the A-B solutions were assumed to be ideal. However, if the components exhibit positive deviations from ideality i.e., if the A-B molecular interaction is weaker than the average of the A-A and the B-B interactions , the single-phase solutions separate into two distinct phases, either both liquid or both solid. The system in which phase separation has 3 See Sect. 2.6 for application of the lever rule in...
Binary phase diagrams analytical construction
Binary phase diagrams depict the stable condensed phase or phases formed by a two-component system as a function of temperature and overall composition. The ordinate of a phase diagram is the temperature and the overall composition is the abscissa1. The phase rule Eq 1.21 for a two component system permits F 4 - P degrees of freedom for a two-component system. Since the diagrams deal only with condensed phases, they are minimally affected by total pressure2. Ignoring the total pressure reduces...
Methane Gas
8. . .7 . .71 9.83 . 11.20 12.519 13.767 14.938 16.026 17.036 17.959 18.79 19.551 0.219 '0. 02 299 Til 036 276 lt 3 0 - 17. 861. -16.950 -15.897 14. II -13.396 -8.757 -7.007 -5. 168 -3. 250 -1.261 . 791 897 5. 548 36 9.452 11.68 . 55 .7.175 9.517 51.476 6 0.849 6 . . . 63.983 65.-.57 64.872 68. 231 69.535 70. 86 71.964 73.131 -31. 1 .S -31.230 -35.819 - lt 0.656 - .5.717 -50.987 -56.1.53 -62.107 -67.939 -73.941 -80. lOf -86. .28 -92.901 -99.518 -106.273 -113.162 -120.179 -127.318 -13 57 .
Eutectic Phase Diagram
The binary systems treated in the preceding sections were either ideal melting-solidification or deviated positively from ideality according to regular solution theory phase separation . These simple types are rarely found in real binary systems. First, there may be more than one solid phase, each with a distinct crystal structure, just as there are in pure substances see Sect. 5.6 . Second, the liquid phase and the solid phase s are generally nonideal. The extent of deviation from ideality is...
Criterion of Chemical Equilibrium
As in any system constrained to constant temperature and pressure, the equilibrium of a chemical reaction is attained when the free energy is a minimum. Specifically, this means that dG 0, where the differential of G is with respect to the composition of the mixture. In order to convert this criterion to an equation relating the equilibrium concentrations of the reactants and products, the chemical potentials are the essential intermediaries. At equilibrium, Eq 7.27 provides the equation where...
Standard Free Energy of Formation
Even though the thermochemical database need contain only AGo or, equivalently, AHo and ASo , the number of reactions that would have to be included in such a compilation is intractably large. The key to reducing data requirements to manageable size is to provide the standard free energy changes of forming the individual molecular species from their constituent elements. Particular reactions are constructed from these so-called formation reactions. For molecular compounds containing two or more...
Ql t T J Ql TT
Since temperatures are always positive, and since the initial restriction was T1 gt T2, the above equation shows that Q2, must be positive. That is, the direction of heat flow is from the hot body to the cold body. The above application of the Second Law may seem needlessly formal, but a more challenging analysis of the process depicted in Fig. 1.16 is the following. If the initial state of the isolated system is T1 gt T2, what is the final common equilibrium temperature of the two systems what...
Solving for the Equilibrium Composition
The law of mass action for a particular reaction Eq 9.24 is but a single equation with more than one variable. As an example, Eq 9.21a contains three unknown mole fractions. There are two principal methods for incorporating the conservation equations into the analysis the element conservation method and the reaction progress variable method. Both of these methods require the following input information The temperature and total pressure AHo and ASo of the reaction This information fixes KP by...
s k log W
Implicit in Boltzmann's equation is the Third law of thermodynamics, which states that the entropy of crystalline solids is zero at 0 Kelvin. This due to the perfectly-ordered arrangement of the atoms and the cessation of their vibration. The W in Boltzmann's equation Fig. 1.6 is the number of ways that a large number of indistinguishable particles can be arranged. If all atoms in the solid are in their equilibrium positions and their lowest vibrational state, only one arrangement is possible,...
Stability diagrams
Equations 9.33a and 9.33b are plotted in Fig. 9.6. These plots are called stability diagrams because the lines separate regions in which only one of the two phases is present. The line represents the p - T combinations where both the metal and its oxide coexist. The oxide-metal stability diagram is similar to the p-T phase diagram of a single substance such as water, where lines separate existence regions of solid, liquid, and vapor phases see Figs. 5.1 and 5.3 . The zones above and below the...
Info Bcc
For T lt 553 K v x 10s U8 - 2.0 x 103T 3.42 xl T2 h 130 4.99 T - 293 - 2.95 x lO- T1 - 2931 3.1 x W3- 2933 s 0.28 4.9907293 - 5.9 x HH T - 293 4.7 x lO fF - 293J Note The last row in each table gives saturation values. Example The coefficient of compressibility at 240 C 513 K between 5 and 10 MPa is determined from the numerical entries in the 5 MPa and 10 MPa subtables Because v x 103 is given to only three significant figures, the above value of 0 is not very accurate. Using values from the...
Isentropic process
Isentropic expansion of an ideal gas was treated in Sect. 3.5. Here, the same process is analyzed without the restriction of ideality. Equation 6.23b is divided by dv while holding s constant, which produces the relation To illustrate the effect of gas nonideality on property changes during an isentropic expansion, the right hand side of Eq 6.30 is evaluated for a Van der Waals gas obeying the equation of state in the form given by Eq 2.5 Substituting the above EOS into Eq 6.28 yields In the...
Heat Capacities
Subtracting Eq 6.23b from Eq 6.24b gives Dividing by dv and holding p constant gives Inverting the partial derivative on the left hand side yields For an ideal gas, the product of T and the two partial derivatives is equal to the gas constant. For nonideal gases, on the other hand, the two heat capacities can differ significantly from R see problem 6.8 . For condensed phases, the first partial derivative in Eq 6.25 is replaced by av and the second by Eq 6.8 , yielding For solids or liquids with...
Thermodynamic Relations for Nonideal Behavior
In Chapters 2 and 3, numerous property relations were presented for ideal gas and idealized solids. The latter are characterized by constant coefficients of thermal expansion and compressibility and obey the equation of state given by Eq 2.18 . For these substances, the specific heats and hence the internal energy and enthalpy are functions of temperature but are independent of pressure or specific volume the entropy of the ideal gas varies with T and v or p according to Eqs 3.9 and 3.10 . The...
Info Bqw
Fig. 1.1 Kelvin and the gas thermometer Work in a mechanical sense was a well-known concept very early in history. Perhaps the most fundamental form of work is that done against or by gravity in raising or lowering weights. Other devices that produce or accept work include springs, piston cylinders, moving an object against a resisting force, turbines, electrons moving in a electric potential gradient and muscle contraction. Heat, however, is none of these and cannot be classified as work....
Zv V
V j njV or v j XjV 7.2a Absent pV work and heat exchange with the surroundings, the First law requires that the internal energy of the mixture be equal to the sum of the values of the pure components, or where u is the molar internal energy of component i and u is the internal energy per mole of mixture or solution. Because mixing occurs at constant pressure and there is no change in system volume, a similar equation applies to the enthalpy Since the specific heats CV and CP are temperature...
Equilibrium
The equal sign in Eq 9.5 signifies that the equilibrium state has been achieved. By convention, the molecular species on the left-hand side of the reaction are called reactants and those on the right hand side are termed products. At equilibrium, there is no fundamental distinction between reactants and products Eq 9.5 could just as well have been written with C and D on the left and A and B on the right. As long as the element ratios are the same, the equilibrium composition does not depend on...
Chemical Potentials in Gas Mixtures
The analysis in Sect. 7.2.2 of the entropy change associated with mixing of ideal gases at fixed T and p was based on the absence of an entropy change if the pure gases are at the partial pressures that they will have in the mixture. Since the gases are ideal, neither is there an enthalpy change in the mixing process. With both the enthalpy and entropy of each species unaltered, the Gibbs free energy must also remain constant during this mode of mixing. Since the partial molar Gibbs free energy...
Info Flg
state of the system Fig.1.18 Equilibrium conditions for certain restraints In the left-hand sketch, the state of the system might be temperature nonuniformity inside the impervious boundary. Or, it could be an unequal distribution of the material within the boundary, as illustrated in the example of Fig. 1.17. Off-equilibrium states in the right-hand diagram most often represent a chemically reactive mixture that has not attained chemical equilibrium. In a multiphase system, additional...
The zirconiumhydrogen phase diagram
The Sieverts' law behavior illustrated in Fig. 9.9 does not increase the concentration of A indefinitely as pA increases. There is a limit that the metal can accept without precipitating a new phase. This limit is called the terminal solubility. At this limit additional gas in the solid ends up in forming a M-A compound called a hydride if A H, a nitride if A N, and an oxide of A O. This process is shown by the zirconium-hydrogen phase diagram in Fig. 9.9. The Zr-H system exhibits three...
T Oio
5.9 The melting point of iodine is 113oC. The vapor pressures of the solid and liquid states are given by lnpsat A - B T DlnT, where Bs 8240, Bl 7381, Ds -2.51, Dl 5.18, As 34.16, and Al 47.83 a Calculate the triple point temperature and pressure of iodine. Why does this result differ from the reported melting temperature b Consider iodine vapor initially at 0.04 atm and 150oC. For the following processes, determine which condensed phase first appears and at what p and T. Sketch the p-T...
Method of Lagrange multipliers
In order to understand the new computational method, a mathematical detour into the theory of Lagrange multipliers is necessary. Consider a function F n1, n2, . The values of n1, n2, at which F is a minimum are to be determined. The system is subject to the following constraints V n1, n2, 0 and W n1, n2, 0 9.62 Where F, V and W are specified functions of the mole numbers of all species, ni. The differential of F is dF fn1dn1 fn2dn2 0 9.63 The differential is a minimum so dF is set equal to...
Rt Nxl
For r 3, this equation yields the pair of solutions xBI 0.07, xBII 0.93. The exact correspondence of these phase compositions with those obtained by the analytical method is expected because both methods are based on the same model. Problem 8.3 offers an additional exercise in analyzing phase separation in a binary regular solution. Real systems exhibiting phase separation The symmetry of the two-phase boundary in Fig. 8.4 arises from the use of regular solution theory to account for...
Info Aez
a calculate the molar enthalpy relative to the initial pure components for both cases. The specific heats of sulfuric acid and water are 147.7 and 75.4 J mole-K, respectively. Assume ideality only for the purpose of calculating the specific heat of the mixture. The density of sulfuric acid is 1.84 g cm3. b plot the data according to the method treated in problem 7.26 and determine the partial molar quantities. At low acid concentrations, the partial molar volumes of sulfuric acid and water in...
Info Eqv
a Starting from the fundamental differentials for du and dh, derive expressions for du dp T and du dT p in terms of CP and the equation-of-state variables a, P and v. Use Maxwell's equations where needed. b what temperature change at 1 atm produces the same change in internal energy as a pressure increases from 1 atm to 100 atm The specific heat of the solid is 400 J kg-K, T 300 K and the p-v-T properties of the substance are density 8.8 g cm3 a 5x10- K- P 9x10- MPa- . Assume that the density...
Info Idk
q w RT I V2 RTlnf V2 v v v For the specified initial state, the ideal gas law gives v1 0.0149 m3 mole. For the final state, the gas law gives v2 0.0446 m3 mole. Equation 3.5 gives w q 9800 J mole For this case, the temperature and the initial and final pressures are chosen to be the same as those in Example a . The specific volumes and internal energies of these states are obtained from steam table A.3 v1 0.825 m3 kg 0.0148 m3 mole u 3661.8 kJ kg v2 2.475 m3 kg 0.0446 m3 mole u2 3663.1 kJ kg...
Vaporization or sublimation
Application of the Clapyron equation to vapor-liquid equilibria is identical to that for vapor-solid equilibrium, so only the former is presented. When one of the phases is a vapor, its molar volume is so much larger than that of the condensed phase that the latter can be neglected in Avvap. In addition, assuming the vapor to behave ideally is generally adequate. With these two approximations the volume change on vaporization is Avtr Avvap vg - vl S vg S RT psat where psat is the vapor...
Maxwell Relations and other Useful Formulas
The fundamental differentials described in Section 1.10 are of the form of Eq 6.1 . They provide the starting point for obtaining many useful thermodynamic relations. The fundamental differentials are represented as exact differentials of u s,v , h s,p , f T,v , and g T,p du _ I I ds 1 I dv _ Tds - pdv 6.9 dh ds d dp _ Tds vdp 6.10 df J j dT I dv _ -sdT - pdv 6.11 dg _l gj dT ldp dp _ -sdT vdp 6.12 Note that each of the energy-like properties has a pair of natural variables associated with it....
Info Gzd
Fig. 4.5 Process diagrams for the power cycle of Fig 4.4 on the EOS of water 4.1.3 The First law for heat engines In all four prior illustrations of heat engines, the 1st law is Because of the cyclic nature of the system, there is no change in internal energy or any other property of the working fluid in each cycle. 4.2 The Second Law applied to heat-engine cycles The following qualitative constraints on the heat-engine cycles discussed in Sect. 4.1 were discovered in the nineteenth century and...
Equilibrium between two phases
The equilibrium criterion of minimum Gibbs free energy Sect. 1.11 can be applied to any of the phase transitions described in the previous section. At fixed pressure and temperature, let the system contain nI moles of phase I and nII moles of phase II, with molar Gibbs free energies of gI and gII, respectively. The total Gibbs free energy of the two-phase mixture is The requirement of equilibrium is that G remain unchanged at its minimum value for any variations in the state of the system....
Dissolution of Gases in Metals
The treatment of chemical equilibrium in the preceding sections of this chapter gives the impression that all that is required is searching the available database for AGo of the reaction, using this information to calculate the equilibrium constant and then applying the law of mass action. This approach may also require estimation of activity coefficients if solid or liquid solutions are involved, and specification of the total pressure if gas mixtures are part of the reaction. This equilibrium...
Binary phase diagrams by the graphical method
The cases of melting of two-component ideal solutions and of phase separation in a regular solution described in the Sect. 8.4 were easily treated by analytical methods. However, as the nonideal behavior of the liquid and solid solutions become more complicated i.e., do not follow regular solution theory , the analytical methods based on Eq 8.2 as the starting point quickly become sufficiently complex to preclude derivation of simple formulae such Eqs 8.12 and 8.21 . The graphical method does...
Activity and Activity Coefficient
Although the thermodynamic behavior of species in solution is ultimately tied to their chemical potentials, a connection between this property and the concentration of the component is needed. This connection is made via a quantity called the activity of a solution species. The activity is a measure of the thermodynamic strength of a component in a solution compared to that of the pure substance the purer, the stronger. As an example, when alcohol is mixed with water its effectiveness is...
Scope
The preceding chapter dealt with the chemical properties of species in single mixture or solution phases. The free energy of a solution or mixture and the chemical potentials of its constituents were quantified. In the present chapter, these properties are applied to determine the phases present and their compositions when a two-component system, or binary system, achieves equilibrium. The two components, denoted by A and B, distribute between two phases labeled I and II. This system, shown in...
Dextran Peg Thermodynamics
Given ZA, ZB, JA and JB, Eqs 11.61a,b , 11.62 and 11.63 are to be solved for X, Y, U and V. Figure 11.19 shows a calculated phase diagram, with coexisting, equilibrium phase concentrations B on the ordinate and A on the abscissa3. The locus of equilibrium points lie along the curve, which is called the binodal. The region below the curve represents a single homogeneous aqueous phase containing polymers A and B in this case dextran and polyethylene glycol, or PEG . Solutions with overall...
Oxygen isobars on a phase diagram
Phase diagrams are a convenient vehicle for displaying the equilibrium oxygen pressures generated by a metal and its oxides. In elements with multiple oxidation states and or crystal structures, the equilibrium may not involve only the metal and an oxide, as in the MO2 M couple discussed in the previous section. In particular, two-phase regions separating two different oxides are represented by reactions of the following type The stoichiometric coefficients w and z are determined by balances on...
Excess Gibbs free energy and the entropy of mixing
As in Eq 7.21 for the enthalpy, the molar Gibbs free energy of a solution g can be written in terms of pure-component contributions gA and gB and an excess value gex . However, an important contribution needs to be added. For a binary solution, the terms contributing to g are g XAgA XBgB geX Agmix 7.33 gex contains the effects of solution nonideality. The last term on the right hand side arises from the entropy of mixing, and is present in ideal as well as nonideal solutions. According to the...
Free energy composition curves
In the free energy formula of Eq 7.36 , nonideality is expressed by the general form gex, the excess free energy. The simplifications used in the prior analyses of ideal melting and phase separation, namely neglecting sex and confining hex to the regular-solution model, are not valid for most binary systems. In order to construct phase diagrams by the common-tangent technique, more elaborate solution models are needed to relate free energy to composition for all likely phases. Figure 8.9 shows...
Ellingham Diagram
Fig. 9.8 Ellingham diagram for the free energy of formation of oxides Fig. 9.8 Ellingham diagram for the free energy of formation of oxides From Fig. 9.8, AG Cu0 -48 kcal mole -203 kJ mole . However, this reaction does not represent an equilibrium situation Cu and CuO can never coexist because the lower oxide Cu2O intervenes. Nonetheless, the standard free energy change for the Cu2O CuO equilibrium can be obtained from AGC o cuo 2AGCu cuo - AGC o 2x -203 - -230 -176 kJ mole Therefore, at 400oC,...
Problems for Chap Xhk
8.1 Benzene and toluene form nearly ideal solutions. At 20 C the vapor pressures of pure benzene and toluene are 74 Torr and 22 Torr, respectively. A 1 1 solution is prepared, and the external pressure is varied such that it boils at 20 C. Assuming isothermal boiling, calculate a The total pressure when the solution first begins to boil. b The composition of the vapor at the onset of boiling. c The total pressure when only a few drops of liquid remain. 8.2 The vapor pressures of dilute...
Solubility Products
Special treatment is accorded to equilibrium constants that determine the maximum concentrations in water of the cation and anion of a solid ionic compound. Alternatively, this equilibrium can be regarded as determining the maximum concentrations of the cation and anion in water without precipitating the solid compound of the two species. Dissolution of an ionic solid in water is more complex than dissolution of a nondissociating species such as sugar . In the latter case, there exists a unique...