Thermodynamic Relations Between

In this section, we explore the relationship between the activity coefficients of the different species in a mixture. Using the Gibbs-Duhem equation, we will show that the activity coefficients are not independent. Their interrelationship will motivate development of a new type of thermodynamic property—excess Gibbs energy. Finally, we will illustrate an application of these principles by coming up with a way to test the quality of experimental data and see whether they are thermodynamically consistent. 

Activity Coefficient Relationships Using the Cihhs—Duhem Equation 

In Section 6.3, we used the Gibbs—Duhem equation to provide a relationship between the partial molar properties of different species in a mixture. We can use this equation to relate the activity coefficients of different species in a mixture as well. We begin by writing Equation (6.19) in terms of partial molar Gibbs energy, that is, chemical potential  

We now want to rewrite this equation in terms of the activity coefficient. From the 

Expanding the fugacity of species i in the mixture using Equation (7.32)  

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The experiments result in an explicit measure of the change in the shock-wave compressibility which occurs at 2.5 GPa. For the small compressions involved (2% at 2.5 GPa), the shock-wave compression is adiabatic to a very close approximation. Thus, the isothermal compressibility Akj- can be computed from the thermodynamic relation between adiabatic and isothermal compressibilities. Furthermore, from the pressure and temperature of the transition, the coefficient dO/dP can be computed. The evaluation of both Akj-and dO/dP allow the change in thermal expansion and specific heat to be computed from Eq. (5.8) and (5.9), and a complete description of the properties of the transition is then obtained. 

The ability of N to exist in its compounds in at least 10 different oxidation states from —3 to +5 poses certain thermodynamic and mechanistic problems that invite systematic treatment. Thus, in several compounds N exists in more than one oxidation state, e.g. [N- "H4] + [N "02] , [N-" H4] + [N 03] , [N-"2H5] + [N 03]-, [N- "H4] + [N-3 3]-, etc. Furthermore, we have seen (p. 423) that, under appropriate conditions, NH3 can be oxidized by O2 to yield N2, NO or NO2, whereas oxidation by OCl yields N2H4 (p. 427). Likewise, using appropriate reagents, N2H4 can be oxidized either to N2 or to HN3 (in which the average oxidation number of N is — ). The thermodynamic relations between these various hydrido and 0x0 species containing N can be elegantly codified by means of their... 

Since optical measurements of monolayers at the water-oil interface are rather difficult to carry out, a configuration was suggested where a monolayer at the water-air interface was in contact with an oil lens which was partly wetting the monolayer [23]. The thermodynamic relation between this monolayer and that residing at the water-oil interface was discussed. This configuration was utilized in the X-ray diffraction experiments [24] where the structural changes of dipalmitoyl phosphatidylcholine (DPPC) and DPPE were followed. 

In the case of hydrogen, for example, at a temperature of 2500 K, the equilibrium constant for dissociation has the value, calculated from the thermodynamic relation between the Gibbs energy of formation and the equilibrium constant of 6.356 x 10 4 and hence at a total pressure of 10 2 atmos, the degree of dissociation is 0.126 at 2500 K, which drops to 8.32 x 10 3 at 2000 K. 

According to these data the heat of activation for the decomposition of nitric oxide, to which reaction the factor k refers, is A = 82 10 kcal/mole.10 It should be especially noted that there is no systematic divergence between the data on the formation and on the decomposition of nitric oxide this fact justifies the assumption that the rate of decomposition is directly proportional to the square of the nitric oxide concentration.11 The investigation covered the temperature range from 2000°K to 2900°K in which the rate varies by a factor of 300. As appears from Fig. 13, except for the scattering due to the inevitable errors of the experiments and computations, the points actually do fall on a straight line in the coordinates lg kr, 1/Tm, i.e., the Arrhenius temperature dependence of the reaction rate holds. The thermodynamic relation between the rates of the direct and reverse reactions permits determining the heat of activation A for the formation of nitric... 

Formulation of the mathematical model here adopts the usual assumptions of equimolar overflow, constant relative volatility, total condenser, and partial reboiler. Binary variables denote the existence of trays in the column, and their sum is the number of trays N. Continuous variables represent the liquid flow rates Li and compositions xj, vapor flow rates Vi and compositions yi, the reflux Ri and vapor boilup VBi, and the column diameter Di. The equations governing the model include material and component balances around each tray, thermodynamic relations between vapor and liquid phase compositions, and the column diameter calculation based on vapor flow rate. Additional logical constraints ensure that reflux and vapor boilup enter only on one tray and that the trays are arranged sequentially (so trays cannot be skipped). Also included are the product specifications. Under the assumptions made in this example, neither the temperature nor the pressure is an explicit variable, although they could easily be included if energy balances are required. A minimum and maximum number of trays can also be imposed on the problem. 

The conditions concerning the temperature and pressure are rather obvious. Those concerning the chemical potential are not so obvious, but they are extremely important. It is these conditions that lead to all the thermodynamic relations between different phases at equilibrium. 

So as there is a thermodynamic relation between entropy and internal energy, the unknown quantities q, H in equations (8.9) and (8.10) can be connected with each other and also can be determined production of entropy S through other quantities. The density of total energy E in equation (8.9) can be represented as a sum of the kinetic energy and the thermodynamic total energy of the resting volume... 

Table 4-1 lists the thermodynamic relations between pressure, volume, and temperature of an ideal gas and its internal energy U and entropy 5. We see that the definition of an ideal gas leads to the conclusion that the pressure exerted by... 

It is noted that electrolytes may have a drastic effect on /7 j (because they may lead to a double layer at the LG border) but hardly any on cos a (because they affect the SL and LG interfacial tensions to only a minor extent, see fig. 11.3.73). There also are solutes for which these trends are the other way around. Below, a thermodynamic relation between cos a, and the integral of /7(h) will be derived, see [5.3.9]. 

The Gibbs-Duhem equation for ternary mixtures is used to analyze the quality of experimental data pertaining to the solubility of drugs and other poorly soluble solids in a binary mixed solvent. In order to test the quality of the data, a thermodynamic consistency test is suggested. This test is based on the thermodynamic relation between the solubilities of a solid in a binary mixed solvent at two different compositions and the activity coefficients of the constituents of the solute-free mixed solvent. The suggested test is applicable to all kinds of systems with the following limitations (1) the solubility of the solid should be low, (2) the above two compositions of the mixed solvent should be close enough to each other. 

Another relevant thermodynamic relation between the state functions is... 

By combining the total energy balance with simple thermodynamic relations between state variables, a transport equation that must be satisfied by the entropy density field ps is obtained. 

Koenig, F.O. (1950). On the thermodynamic relation between surface tension and curvature. J. Chem. Phys., 18, 449-59. 

Chemical equilibrium in homogeneous systems—Dilute solutions—Applicability of the Gas Laws—Thermodynamic relations between osmotic pressure and the lowering of the vapour pressure, the rise of boiling point, the lowering of freez ing point of the solvent, and change in the solubility of the solvent in another liquid—Molecular weight of dissolved substances—Law of mass action—Change of equilibrium constant with temperature and pressure... 

Thermodynamic Relations between the Osmotic Pressure of Dilute Solutions and other Properties of such Solutions... 

Koenig (18) has analyzed the thermodynamic relations between the potentials described above. He points out that certain long-standing assumptions attributed to Kelvin (19), Bridgman (20) and Lorentz (21) are strictly non-thermodynamic and that these assumptions have not been... 

Establish the so-called Four thermodynamic relations between py v, Ty , when any two are taken as independent variables. 

Experimental specific heats, C (p,T), are known to increase apparently without limit on the close approach to the critical point. This nonanalytic behavior influences a far greater portion of the P(p,T) surface than generally is appreciated. The thermodynamic relation between specific heats and the equation of state along isotherms is... 

This is the product rule for the divergence of the product of scalar and a vector. The thermodynamic relation between specific enthalpy and specific internal energy via Legendre transformation i%h = u + p/p (see equation 29-20). Hence, the second term on the right side of (25-25) and the second and sixth terms on the right side of (25-26) can be combined as follows ... 

There is a thermodynamic relation between the adsorption and the surface-tension changes. Creation of concentration differences by accumulation of molecules in a boundary layer would, in itself, represent an increase in free energy, but it is compensated by the fact that the surface free energy is correspondingly lowered, since a is decreased. An equilibrium between the two effects is maintained. The simplest derivation of the well-known Gibbs relation which expresses the balance will now be given. 

Under these conditions it is very rare to obtain macrovoids (typical from instantaneous danix-ing). Therefore, adding solvent to the precipitation bath limits the formation of a macrovoids in a system in which the thermodynamic relations between its components would produce instantaneous demixing, being likely to generate these macrovoids. 

Alternatively, one can simply use the matrix relation between the A and B matrices, and a thermodynamic relation between the isochoric and isobaric chemical potential derivatives to provide the elements of the A matrix. The relevant expressions are... 

In Ref. [3], heat-conduction formulas at the skin surface are derived. For example, a high-temperature spot in the IR image of the face, caused by severe acute respiratory syndrome (SARS), are important to detect early for further immediate isolation of those infect to prevent outbreaks of such an epidemic. The thermodynamic relation between the blood flow rate at the skin level, blood temperature at the body core, and the skin temperature is used to convert IR intensity to temperature and, therefore, to the blood flow rate. In addition, the breathing rate can be estimated during the dialogue of fhe patient or user with the personnel [7]. 

Anomalous slope of x or e in the vicinity of the phase transition is a reason for the anomalous behavior also for quantities d,e,c, s (tensor indexes are omitted for simplicity) measured at constant electric field. It follows from the thermodynamic relations between the electromechanical coefficients shown in Table 4.3. On the contrary, no anomaly in the temperature dependencies for quantities g,h,cP,P measured at constant electric displacement D (or constant polarization P) is observed. 

The subsequent three chapters are devoted to the electric double-layer structure at the interface between immiscible electrolytes examined by the electrocapillary curves method (Prof. Senda and coauthors) and by measurement of the electric double-layer capacity (Dr. Samec and Dr. Mare ek) as well as to the investigation of the Galvani and Volta potentials in the above-mentioned systems (Prof. Koczorowski). These chapters will be of interest to many electrochemists since the results obtained here are comparable with the thoroughly studied metal/electrolyte solution interface. An insignificant potential shift in the compact layer at the interface between immiscible electrolytes in the absence of specific ion adsorption - this is the main conclusion arrived at by the authors of Chaps. 4 and 5. Chapter 6 deals with the scale of potentials in a system of immiscible electrolytes and the thermodynamic relation between the distribution coefficients and the Volta potentials. 

The theory of the cryoscopic method, which is also called the freezing point lowering method, has been given by Rossini [44-tay/ros, 44-tay/ros, 50-ros]. For the equilibrium between a crystalline phase consisting of the major component alone and a liquid phase corrsisting of the major component and minor components, the thermodynamic relation between the temperature of eqitilibriirm and the composition of the liquid phase, for an ideal or sufficiently dilute solution, is [similiarto equation (1.4)] ... 

J. W. Cahn, Thermodynamic Relations between Phase Separation and Crystallization, in press. 

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