Systems at Constant Temperature and Pressure

The most familiar transformations occur under conditions of constant temperamre and pressure, so it will be particularly useful to have a criterion of spontaneity and equilibrium that applies to these conditions. 

We can start with Equation (7.3), in which the only restriction is that no work is performed except mechanical work against an external pressure P. If the change is carried out at constant pressure, the pressure P of the system must equal P, so Equation (7.3) can be written as 

If the pressure and temperature are constant, dP and dT are zero, so we can add SdT and VdP to the left side of Equation (7.14) without changing its value  

The function in parentheses in Equation (7.15) is a state function and is called the Gibbs function, or Gibbs free energy, symbolized by G [2] Relationships for G are 

If the temperature and pressure are constant, and if the only constraint on the system is the pressure of the environment. Equation (7.18) provides the criteria of equilibrium and spontaneity. The equality in Equation (7.18) applies to a reversible 

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A thermodynamic function for systems at constant temperature and pressure that indicates whether or not a reaction is favorable (AG < 0), unfavorable (AG > 0), or at equilibrium (AG = 0). 

Definition.—The heat absorbed in producing a change of physical state or chemical composition of a system, at constant temperature and pressure, is called the latent heat of the given transition, and is measured by the number of calories absorbed during the transition of unit mass of the substance from the initial to the final state. 

The second law also describes the equilibrium state of a system as one of maximum entropy and minimum free energy. For a system at constant temperature and pressure the equilibrium condition requires that the change in free energy is zero ... 

Thus, the activity coefficient can be calculated if AGE is known this is the difference between the work of the reversible transition of the real system at constant temperature and pressure from the standard to the actual state and the work in the same process in the ideal system ... 

For a system at constant temperature and pressure, the equilibrium condition is... 

We will consider a reversible change in the system at constant temperature and pressure of the liquid phase pl. Furthermore, we will assume that the equilibrium concentration of all components in the liquid, except for the single component, i, of the solid phase, is fixed. The equilibrium condition yields... 

A chemical reaction can be designated as oscillatory, if repeated maxima and minima in the concentration of the intermediates can occur with respect to time (temporal oscillation) or space (spatial oscillation). A chemical system at constant temperature and pressure will approach equilibrium monotonically without overshooting and coming back. In such a chemical system the concentrations of intermediate must either pass through a single maximum or minimum rapidly to reach some steady state value during the course of reaction and oscillations about a final equilibrium state will not be observed. However, if mechanism is sufficiently complex and system is far from equilibrium, repeated maxima and minima in concentrations of intermediate can occur and chemical oscillations may become possible. 

Note that in Eq. (60) it is the multicomponent Dtj which appear, whereas in Eq. (61) the binary diffusion coefficients Sh, appear.13 For a system at constant temperature and pressure Eq. (61) may also be written... 

Consider a homogeneous solution of w° moles of water, p° moles of a cosolvent P and a moles of a solute X. The Gibbs free energy of the system at constant temperature and pressure is given formally by... 

Systems at constant temperature and pressure, which are common laboratory conditions, have a tendency toward lower enthalpy and higher entropy. A chemical reaction is driven toward the formation of products by a negative value of AH (heat given off) or a positive value of AS (more disorder) or both. When AH is negative and AS is positive, the reaction is clearly favored. When AH is positive and AS is negative, the reaction is clearly disfavored. 

The equilibrium conditions for phase equilibria can be derived in the simplest way using the Gibbs energy G. According to the second law of thermodynamics, the total Gibbs energy of a closed system at constant temperature and pressure is minimum at equilibrium. If this condition is combined with the condition that the total number of moles of component i is constant in a closed system... 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

Thus, the enthalpy is a function of the entropy and the Helmholtz energy is a function of the volume, and each function may be used in place of the other variable. However, the Gibbs energy is a constant for any closed system at constant temperature and pressure, and therefore its value is invariant with the transfer of matter within the closed system. 

Therefore, - AG is a measure of the net work. In other words, the decrease in free energy (- AG) is a measure of maximum net work that can be obtained from a system at constant temperature and pressure. G can also be defined as, the fraction of total energy which is isothermally available for converting into useful work. This decrease in free energy is of great importance in chemistry, especially in physical chemistry. 

The system of equations (2.27) is seen to be rather complicated. Its solution, if obtainable at all in quadratures, must probably be even more complicated. However, in experiments certain conditions which enable the initial equations to be simplified are usually fulfilled. Consider limiting cases of particular interest from both theoretical and practical viewpoints.134,136,139,140 The process of growth of the ApBq and ArBs layers will be analysed in its development with time from the start of the interaction of initial substances A and B up to the establishment of equilibrium at which, according to the Gibbs phase rule (see Refs 126-128), no more than two phases should remain in any two-component system at constant temperature and pressure. 

These considerations lead to the following criterion of stability for a singlephase binary system. At constant temperature and pressure, AG and its first and second derivatives must be continuous functions of x, and the second derivative must everywhere satisfy the inequality... 

A system at constant temperature and pressure is at disequilibrium until all of its Gibbs free energy, G, is used up. In the equilibrium condition the Gibbs free energy equals zero. 

This method provides the exact solutions for ideal systems at constant temperature and pressure. It is successful in describing diffusion flow in (i) nearly ideal mixtures, (ii) equimolar counter diffusion where the total flux is zero (Nt = 0), (iii) diffusion of one component through a mixture of n — 1 inert components, and (iv) pseudo-binary case and the diffusion of two very similar components in a third. 

Using the forces and flows identified in Eq. (7.1), and the Gibbs-Duhem equation for an n-component system at constant temperature and pressure, we obtain... 

Example 8.10 Time variation of affinity We wish to derive the time variation of affinity in an open system at constant temperature and pressure. The affinity for reaction / is... 

In other words, as expected, at a spontaneous evolution of the system at fixed p and T, its Gibbs potential decreases, dG < 0. Thus, the rate of entropy pro duction and energy dissipation in an open system at constant temperature and pressure is proportional to the rate of decreasing its Gibbs potential due to occurrence of irreversible spontaneous processes inside the system. 

Figure 3. A complete ystematization of modem thermodynamics. Note (AG )t.p, (AG2)t.p and (AG)t.p are Gibbs free energy changes of reaction 1, reaction 2 and the whole system at constant temperature and pressure, respectively. X and J represent thermodynamic force and thermodynamic flux for irreversible process, respectively.

If a small change is made in the system at constant temperature and pressure, such that the number of moles of the constituent 1 is increased by dni, of 2 by dn2, or, in general, of the constituent i by dn,, the total change dX in the value of the property X is given by... 

The integral h can be transformed by using the Gibbs-Duhem Eq. (7) for a binary system at constant temperature and pressure ... 

The meaning of these equations can best be understood with reference to a geometrical representation of the mean molar Gibbs free energy g. For a system at constant temperature and pressure, we construct the surface g X2, )... 

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