## Chap The Mathematics of Thermodynamics

6.1 Mathematical 6.1.1 Exact 6.1.2 The divide-and-hold-constant 6.2 The Maxwell Relations and other Useful Formulas 4 6.3 Thermodynamic Relations for Nonideal Behavior 6 6.3.1 Internal Energy and 6.3.3 Heat 6.4 Nonideal gases with special process restraints 11 6.4.1 Isentropic 6.4.2 Joule Expansion (constant internal energy process) 12 6.4.3 The Joule-Thomson Coefficient (constant enthalpy process) 14

## Joule Expansion constant internal energy process

One of the simplest yet most significant experiments in the long history of thermodynamics was performed in 1843 by James Joule. The experimental setup consisted of two vessels connected by a valve and immersed in a water bath with a thermometer to detect any temperature changes arising from cooling or heating effects in the two vessels. Initially, one vessel contained air at high pressure and the other was evacuated. When the valve connecting them was opened and the gas expanded to fill both...

## Info

(a) Tabulate the activity coefficient of A as a function of composition. (b) Over what range of composition is Henry's law obeyed by A (c) What is the equation for the equilibrium partial pressure of B in the composition range in which component A obeys Henry's law 8.11 Shown below are the free energy Vs composition plots for an A-B binary ststem at four temperatures. The ordinate is the free energy and the abscissa is the mole fraction of component B. (a) On these plots, label the melting...

## Pi J IV J

The path of an isentropic process on p-v coordinates can be inferred from Eq (3.13) The constant in this equation is determined by the properties of the initial state.The isentrope (Fig. 3.4) is of the form p x 1 v7, while the isotherm (Fig. 3.2) varies as p x 1 v. Since 7 > 1 for all gases, the isentrope decreases more rapidly with v than does the isotherm. The work done by isentropically expanding an ideal gas is obtained by substituting Eq (3.15) into Eq (3.1) pv7J PvL -11 --L l tfC&...

## Entropy Change Multicomponent

Fig. 7.1 Two methods of mixing xA moles of pure A with xB moles of pure B to form 1 mole of an ideal gas mixture. The temperature is constant Fig. 7.1 Two methods of mixing xA moles of pure A with xB moles of pure B to form 1 mole of an ideal gas mixture. The temperature is constant The initial states of A and B at pressure p obey the ideal gas law (the mole fractions appear on both sides of these equations to facilitate subsequent explanations). The two pure gases are mixed in two ways. The...

## Simultaneous Gas Phase Reactions

When more than one chemical reaction achieves equilibrium, a mass action law must be written for each. The analytical tool of choice is the reaction progress variable method. Example Dissociation of water vapor. At low temperatures, water vapor (steam) does not dissociate, and chemical reactions can be neglected. At high temperature (> 1500 K), dissociation into H2 and O2 becomes significant at very high temperatures, as might occur in the exhaust of a rocket, the hydroxyl radical, OH,...

## A degenerate eutectic system goldsilicon

The Au-Si eutectic phase diagram is characterized by negligible solubility of A(gold) in B and of B (silicon) in A. The single-phase a and P regions in Fig. 8.10 disappear and points j and l are displaced to their respective temperature axes. Fig. 8.11 shows the gold-silicon phase diagram, which is a prototypical degenerate eutectic system. As shown in the following example, information regarding the nonideality of the liquid phase can be garnered from analysis of the liquidus lines in the...

## Competition for sites CO Vs O on hemoglobin

Binding of oxygen in the blood to hemoglobin is affected by a number of molecules that compete with O2 for the heme sites. In particular, carbon monoxide binding to the heme sites is one to two orders of magnitude greater than oxygen binding. The following analysis of a hypothetical two-site hemoglobin with both O2 and CO dissolved in the blood is given below. 1 If the liquid (blood) is in saturated with O2 in air while the binding equilibria are attained, L, not Lo is constant. However, if the...

## Ion standard state

The chemical potential of an ion in water is given by where a, the activity of the species, is the product of its activity coefficient y and the concentration c , in moles per liter of solution, or M, the molarity1. is the chemical potential of the ion in its standard state, and therein lies the difficulty. The standard state of a component in a nonaqueous solution is the pure substance (see Sect. 10.4). However, this cannot be applied to aqueous ions because a pure-ion state does not exist....

## Tjw

Fig 4.12 The Rankine cycle process diagram superimposed on the EOS of water Pump The efficiency of the pump in Fig. 4.11 is calculated using Eq (4.13a). For compressed liquid water, v 0.001 m3 kg. The pump's inlet pressure is 15 kPa and the outlet pressure is 2 MPa. Using these values in Eq (4.13a) yields (w3-4)rev 0.001(2000 - 15) 2 kJ kg The actual work required for pump operation is 4 kJ kg, so the pump efficiency is 50 . The turbine and pump efficiency calculations indicate the presence of...

## Interphase Equilibrium

The two-headed arrows in Fig. 8.1 indicate chemical equilibrium of components A and B between phases I and II. According to the discussion in Sect.1.11.14, the criterion of chemical equilibrium at fixed T and p is the minimization of the total free energy of the contents of the cylinder-piston in the figure. Since the total free energy is the sum of those of the two phases, this criterion is The free energy of a phase is related to the chemical potentials of its components by Eqs (7.27) Using...

## Heatup behavior

The solid-to-liquid transformation in the eutectic system is more complex than the melting process in the ideal-solution phase diagram discussed in connection with Fig. 8.3. In Fig. 8.10, suppose the system starts at temperature T1 with a mole fraction B a bit greater than the terminal solubility at o. The initial state is in the a+P two-phase region. Upon increasing the temperature holding the overall composition constant, the P phase disappears when the terminal solubility curve is reached at...

## Fig An aqueous electrochemical cell with an active metal halfcell and a hydrogen halfcell

The left hand half-cell containing the metal M and its ion Mz+ in solution is in equilibrium with electrons in the circuit to the left of the potentiometer. The half-cell reaction describing this equilibrium is The right hand compartment comprises a hydrogen half-cell. Gaseous H2 saturates the solution with molecular hydrogen, which, by virtue of rapid reaction on the inert metal electrode, maintains equilibrium with hydrogen ions in solution and electrons in the right side of the external...

## Problems for Chap

4.1 A cyclical engine receives 325 kJ of heat from a reservoir at 1000 K, delivers 200 kJ of work, and rejects 125 kJ of heat to a reservoir at 400 K. Does this engine violate either the First law or the Second law 4.2 Consider a Carnot cycle of 25 efficiency using water as the working fluid. Heat transfer to the engine takes place at 300oC, during which the water changes from saturated liquid to saturated vapor. Similarly, heat rejection occurs isothermally as the (a) Sketch this Carnot cycle...

## Rt

The partial pressure a vapor in equilibrium with the liquid is psat. Making this substitution for p in the above equation and integrating from a reference pressure psat,ref at temperature T where gg gg,ref yields psat,ref is the saturation pressure in the absence of any external agent such as an inert gas to cause the total pressure to differ from the saturation pressure Pressure in excess of the saturation pressure alters the free energy of this phase by When integrated from psat,ref to some...

## Ideal gas mixtures

Ideality in a multicomponent system means that the components interact with each other on a molecular level with the same intensity as the component molecules interact amongst themselves. In an ideal gas mixture, there are no intermolecular interactions in either the pure components or in the mixture. The mixing process starts from prescribed quantities of the pure components at specified temperature and pressure. The mixture is at the same T and p. If the mixture or solution is ideal, this...

## One Component Phase Diagrams

The most familiar graphical representation of the phase relationships of a pure substances is the p-T diagram, as illustrated for water and carbon dioxide in Fig. 5.4. The most familiar graphical representation of the phase relationships of a pure substances is the p-T diagram, as illustrated for water and carbon dioxide in Fig. 5.4. Fig. 5.4 p-T phase diagrams for water and carbon dioxide Fig. 5.4 p-T phase diagrams for water and carbon dioxide p-T diagrams are comprised of three lines that...

## Internal Energy and Enthalpy

Consider the internal energy to be a function of temperature and specific volume, or u T,v . The total differential is The coefficient of dT is by definition the heat capacity at constant volume, CV. The coefficient of dv is obtained from the fundamental differential du Tds - pdv by dividing by dv and holding T constant Where the partial derivative involving the entropy has been replaced by the Maxwell relation of Eq 6.19 . The final result for du is This form is applicable to nonideal gases...

## Excess properties

In this theory, the molecules mix randomly as they do in ideal solutions, so that That is, there is no tendency for either like or unlike molecules to cluster. When sex 0, the excess Gibbs free energy reduces to the excess enthalpy. The analytical formulation of hex in terms of composition is restricted by the limiting behavior of the solution enthalpy h as the solution approaches pure A and pure B. In these limits, h hA and h hB, respectively. Examination of Eq 7.21 shows that to satisfy these...

## 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 Gas Phase 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

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...

## 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...

## 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...

## 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...

## 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...

## 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....

## 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,...

## 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...

## Water With Quality Of X Is Contained In A Rigid Vessel

2.9 a A container contains half liquid water and half vapor by volume at 1 atm pressure. Estimate the quality of this mixture. b A steam water mixture of 50 quality is contained in a rigid vessel. If heat is added to this system, does the quality increase or decrease c A 15-liter rigid vessel contains 10 kg of water liquid vapor . If the vessel is heated, will the liquid level rise or fall At what temperature does one of the two phases disappear 2.10 For the p-v-T values in table A-l at 100 C...

## Graphical Representations Of The Equation Of State Of Water

Water is the prime example of a condensable substance, meaning that in many practical situations it can exist as a vapor or a liquid and less frequently as a solid . Water is a goldmine of detailed thermodynamic data and applications of thermodynamic theory. Van Ness 1983 and Potter and Somerton 1993 provide very readable expositions of this topic with many illustrative problems. The equations of state of water, both volumetric and thermal, exhibit features of the-gas equations of state...

## Gas dissolution

The second example of the application of Henry's law involves the dissolution of the so-called permanent gases in condensed phases. Examples are helium solution in glass and oxygen dissolution in blood. To describe these equilibria, the chemical potentials of the gas species A in the two phases are equated. In the gas phase, the chemical potential of A is given by Eq 8.4 . In applying Eq 8.3 to A in the condensed phase, however, the assignment of the reference free energy gA s or l poses a...

## The Steam Tables

The graphical representations of Figures 2.8 and 2.9 are valuable for understanding the general features of the p-v-T properties of water but are of little use for quantitative analysis. For this purpose, extensive tables of the thermodynamic properties of water have been compiled. Such tabular information is available for many condensable substances in addition to water. For the latter, the property listings are called the steam tables, although they contain data for liquid water and ice as...

## Partial molar properties

For simplicity, the theory of nonideal solutions is presented only for the enthalpy, but the same formulas apply to all properties. The enthalpy of a nonideal solution is of the same form as that for an ideal solution Eq 7.2c . The sole change is replacement of the molar enthalpy of the pure components hi by a quantity called the partial molar enthalpy, denoted by The partial molar property for species i depends on temperature and on the composition of the solution but the effect of total...

## The element conservation method

Element conservation is expressed as ratios of the number of moles of one element to the number of moles of another element. For a reaction involving N molecular species, N-2 of these mole ratios are required. This method is best explained by applying it to the methane combustion reaction of Eq 9.1 . For this reaction, the mass-action law is To solve for the 4 mole fractions in this equation requires three other equations. The first is the summation of the mole fractions xCH4 xO2 xCO2 xH2O 1 B...

## Metalnonmetal the FeO phase diagram

In addition to the metal-metal phase diagrams discussed to this point, the other large class of phase diagrams are the metal-nonmetal systems. In particular, the most important nonmetal is oxygen. Of all the metal oxygen systems, the Fe O phase diagram, shown in Fig. 8.15, has received the closest study. On the diagram, the concen- Fig. 8.15 The Fe O phase diagram with oxygen pressure isobars. Fig. 8.15 The Fe O phase diagram with oxygen pressure isobars. tration units are the O Fe atom ratio...

## Ibu

Fig. 8.10 Phase diagram derived from Fig. 8.9 after Ref. 1 Temperature T6 is the melting point of pure A, and is the first temperature at which a solid phase appears from the liquid. At this point, the free energy-composition plot shows that gL ga. For the entire composition range, the liquid is the stable phase because its free energy is lower than that of either of the two solid phases. Transferring this information from the T6 plot in Fig. 8.9 to the T6 isotherm in the phase diagram of Fig....

## Composition

Composition is not a variable in one-component systems. A two-component system possesses a single composition variable. In general, a C-component system is characterized by C-1 independent compositions. Several measures of composition or concentrations are available. The most commonly used is the mole fraction, which for component i is defined by xj where n n 7.1 n is the total moles in the system and ni is the number of moles of component i. By definition, the sum of the mole fractions is...

## Mixtures Vs solutions

Before starting, clarification of the terms mixture and solution is in order. These terms are nearly, but not quite, synonymous. A solution unequivocally refers to a homogeneous system of two or more components. This term is applied to liquids and solids, but not to gases. Salt dissolved in water is an aqueous solution of NaCl a gold-silver alloy is a solid solution of these two elements. However, air is a mixture of oxygen and nitrogen plus minor species , not a solution of O2 in N2. A...

## The Clapyron Equation

The phase rule Eq 1.21 allows one degree of freedom for a two-phase, one-component system. As the temperature is changed, the pressure must also adjust in order to maintain both phases in equilibrium. The criterion of Eq 5.2 determines the p - T relationship for co-existence of two-phases. If Eq 5.2 is satisfied at a particular combination of T and p, and the temperature is incremented by dT, the pressure must change by dp to maintain both phases in equilibrium. As a result of these changes,...

## The Lever Rule

At temperature-composition points within a single-phase region, the abscissa of the phase diagram is the actual composition. In two-phase regions, on the other hand, the abscissa gives the overall composition of the two co-existing phases. Figure 8.16 shows a two-phase region shaded area bounded on the left by phase I and on the right by phase II. This diagram represents any of the two-phase regions with single-phase neighbors in the phase diagrams depicted in this chapter. At point P in the...

## Common tangent rule

The foundation of the graphical method is called the common tangent rule. This is the proof that a graphical construction applied to free energy curves provides the link to the phase diagram. The common tangent rule states that the compositions of the two coexisting equilibrium phases lie at the points of common tangency of the free energy curves. The two phases may be both solids, both liquids, or one solid and one liquid. For generality, the two phases are labeled I and II. As usual, the...

## Dissociative dissolution and Sieverts law

A very important exception to purely physical nondissociative dissolution occurs when diatomic gases such as O2, N2 and H2, enter metals. Many metals form M-O, M-N, and M-H bonds that are sufficiently strong to break the bond of the diatomic molecule and absorb the gas in atomic form.1 Palladium of cold fusion fame is an example of this process. Denoting the diatomic gas molecule by A2, the controlling equilibrium is The equilibrium condition for this reaction in terms of chemical potentials is...

## Psychrometry Gas Vapor Mixtures

The gas - vapor mixture of greatest practical importance is the air - water system1. The thermodynamic behavior of this mixture affects the weather and engineered devices such as air conditioners. In these applications, air can be treated as an inert ideal gas. Water vapor can be considered to behave ideally because its concentration in air is low. In this two-component gas mixture, air is treated as a single species. The gas and the vapor obey Dalton's rule Sect. 7.2 pw xw p pa xa p 5.11 where...

## Mixtures and Solutions defined measures of composition

Up to this stage, only systems containing one component have been analyzed. These simple substances can exist at equilibrium in up to three phases. This limit is imposed by the phase rule Eq 1.21 , F C 2 - P where C is the number of components, P the number of phases and F the number of degrees of freedom, or adjustable thermodynamic variables. The objective of the present chapter is to understand the thermodynamics of single-phase P 1 , two-component C 2 systems. The characteristics of...

## Exact differentials

Let x and y be two specified properties. Denoting one of the dependent properties by z, the relation between the three is written as z x,y . The total differential of z is The partial derivatives depend on x and y as expressed by the functions M and N in the second equality of Eq 6.1 Total differentials have the mathematical characteristic of being exact or inexact. The distinction is determined by comparison of the mixed second derivatives of z The partial derivatives depend on x and y as...

## Transformation of Graphite to Diamond

Another notable solid-solid equilibrium is the graphite-to-diamond transition in the element carbon. Graphite is fairly common in the earth's crust but the rarity of diamond is the origin of its value. Under normal terrestrial conditions 300 K, 1 atm the two forms of carbon are not in equilibrium and so, thermodynamically speaking, only one form should exist. The stable form is the one with the lowest Gibbs free energy. At 300 K, the enthalpy difference between diamond and graphite is Ahd-g...