Thermodynamic equilibrium, criterion pressure

For the description of /-independent phases with k components, we therefore need f(k - 1) independent data on composition. To these we still have to add data on temperature and pressure, so we have altogether/( - l) + 2 intensive data, if the temperature and pressure are equal in the whole system. If the system considered is in equilibrium, the intensive criterion of the thermodynamic equilibrium must be fulfilled, fhus the chemical potentials of all the k components in all the/phases have to be equal. This criterion thus defines fhe number of binding conditions between the intensive variables. This number is kif- 1), because the number of binding conditions is one less than the number of phases. Then the difference between both the quantities defines the number of intensive variables, which are independent in a system with A components and/phases being in equilibrium -the variance, or the number of degrees of freedom v... 

The surface area expansion process in Figure 3.5 must obey the basic thermodynamic reversibility rules so that the movement from equilibrium to both directions should be so slow that the system can be continually relaxed. For most low-viscosity liquids, their surfaces relax very rapidly, and this reversibility criterion is usually met. However, if the viscosity of the liquid is too high, the equilibrium cannot take place and the thermodynamical equilibrium equations cannot be used in these conditions. For solids, it is impossible to expand a solid surface reversibly under normal experimental conditions because it will break or crack rather than flow under pressure. However, this fact should not confuse us surface tension of solids exists but we cannot apply a reversible area expansion method to solids because it cannot happen. Thus, solid surface tension determination can only be made by indirect methods such as liquid drop contact angle determination, or by applying various assumptions to some mechanical tests (see Chapters 8 and 9). 

Certain equilibrium states of thermodynamic systems are stable to small fluctuations others are not. For example, the equilibrium state of a simple gas is stable to all fluctuations, as are most of the equilibrium states we will be concerned with. It is possible, however, to carefully prepare a subcooled liquid, that is, a liquid below its normal solidiflcation temperature, that satisfies the equilibrium criteria. This is an tin-.stable equilibrium. state because the slightest disturbance, such as tapping on the. side of the containing ve.s.sel, will cause the liquid to freeze. One sometimes encounters mixtures that, by the chemical reaction equilibrium criterion (see Chapter 13). should react however, the chemical reaction rate is so small as to be immeasurable at the temperature of interest. Such a mixture can achieve a state of thermal equilibrium that is stable with respect to small fluctuations of temperature and pressure. If, however, there is a sufficiently large, but temporary, increase in temperature. so that die rate of the chemical reaction is appreciable for some period of time) and then the system... 

To illustrate the use of this equilibrium criterion, consider the very simple, initially nonuniform system shown in Fig. 7.1-1. There a single-phase, single-component fluid in an adiabatic, constant-volume container has been divided into two subsystems by an imaginary-boundary. Each of these subsystems is assumed to contain the same chemical species of uniform thermodynamic properties. However, these subsystems are open to the flow of heat and mass across the imaginary internal boundary, and their temperature and-pressure need not be the same. For the composite system consisting of the two subsystems, the total mass (though, in fact, we will use number of moles), internal energy, volume, and entropy, all of which are extensive variables, are the sums of these respective quantities for the two subsystems, that is. 

Solubility is defined by the thermodynamic equilibrium of a solute between two phases, which in the context of this chapter are a solid phase and a liquid solution phase.The criterion for equilibrium between coexisting phases is that the temperature, pressure and molar free energies or chemical potentials of each individual species in each phase are equal.For a co-crystal, however, the sum of the molar free energies or chemical potentials of each co-crystal component plays a key role in determining phase equilibria. The molar Gibbs energy of the co-crystal A B in equilibrium with a solution phase is given by ... 

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

Thermodynamics is used to predict whether reactants have a spontaneous tendency to change into products. This tendency is associated with a decrease in the free energy or Gibbs energy of the system (G) to a minimum. As a consequence, the thermodynamic criterion for spontaneous change at constant temperature and pressure is AG < 0. Under standard conditions (concentrations = 1 M, and P = 1 atm), the standard Gibbs energy variation (AG°) is related with the equilibrium constant (A) by equation 11 ... 

The thermodynamic criterion for the equilibria CaCO, ) = Ca0(,) + C02 (,) is AG ° = -RT n Kp, where AG° is the change in Gibbs free energy of the reactants and products in their standard state, R is the gas constant, and Kp is the equilibrium constant. For this equilibria, A p = pco, for pressure in units of atmospheres. Values for AG are tabulated in the form AG° = a+ bT combining these expressions yields an exponential relationship between the partial pressure of CO2 and temperature for the above equilibria. Complete derivations and discussion of these equations may be found in physical chemistry textbooks such as references [13] and [14]. 

Select the criterion to be used for thermodynamic consistency. Deviations from thermodynamic consistency arise as a result of experimental errors. Impurities in the samples used for vapor-liquid equilibrium measurements are often the source of error. A complete set of vapor-liquid equilibrium data includes temperature T. pressure P. liquid composition x, and vapor composition y,. Usual practice is to convert these data into activity coefficients by the following equation, which is a rearranged form of the equation that rigorously defines K values (i.e., defines the ratio y, /x, under Related Calculations in Example 3.1) ... 

If a liquid mixture at temperature T and pressure P is in equilibrium with a vapor mixture at the same temperature and pressure, therefore at an equilibrium condition, the thermodynamic criterion will be... 

Or, again, take the case of pure benzene on the one hand and a saturated solution of benzene in water on the other, both systems being at the same temperature A saturated solution of benzene is necessarily in equilibrium with pure liquid benzene itself because of the fact of saturation The conclusion to he drawn from the thermodynamic criterion considered is, that under these conditions, (8A)TV = o, and therefore, if we imagine one mole of benzene transferred from the pure benzene to the saturated solution, the work must be zero That is, there must be the same vapour pressure over the pure benzene as there is over its saturated solution in water, the vapour in each case being benzene vapour In the case of a hydrated salt on the one hand and the saturated solution of the salt on the other, the conditions are more complex We shall consider this point in Chap X in connection with the application of the Phase Rule to two component systems... 

There are five linear hydrodynamic equations containing the seven fluctuations (pi, u x,ii y, uu,pi,si, 7i). The local equilibrium thermodynamic equations of state can be used to eliminate two of the four scalar field quantities (pi, si, Ti, pi). In this chapter we chose the temperature and number density as independent variables, although we could equally well have chosen the pressure and entropy. One useful criterion for choosing a particular set is that the equilibrium fluctuations of the two variables be statistically independent. The two sets (pi = dp, T = ST) and (pi = Sp, si = Ss) both involve two variables that are statistically independent, that is, statistical independence of the two variables simplifies our analysis considerably. It is particularly convenient to chose the set (Pi,Ti) over the set (pi, si) because the dielectric constant derivatives (de/dp)T and (de/dT) are more readily obtained from experiment (other than light scattering) than are (ds/dS)p and (ds/dp)s-... 

It is a derived thermodynamic property, unlike the measured thermodynamic properties, temperature and pressure, that provide the criteria for thermal and mechanical equilibrium, respectively Although the chemical potential is an abstract concept, it is useful since it provides a simple criterion for chemical equilibria of each species i. 

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