Systems at Constant Temperature and Volume

When the value of DQ obtained from the first law is substituted into liquation (6.108), the result is 

If the only restraint on the system is the pressure of the environment, then the only work is mechanical work against the external pressure P. Therefore, DW is equal to —P dV and Equation (7.1) becomes 

As the volume is constant, P dV equals zero and can be omitted. Because the temperature is constant, (SdT) can be added to the left side of Equation (7.3) without changing its value. Thus, 

The terms in parentheses in Equation (7.4) are equal to the differential of the function TS, and Equation (7.4) can be written as 

If the temperature and volume are constant, and if the only constraint on the system is the pressure of the environment. Equation (6.106) and Equations (7.1) through (7.6) provide the criteria of equilibrium and spontaneity. The equality in Equation (7.1) apphes to a reversible change, and as no exchange of work occurs with the environment, the reversible change must be in a system at equilibrium. Similarly, 

---

Thus for a system at constant temperature and volume, the equilibrium condition is... 

It is also possible to develop the equilibrium and stability conditions for systems subject to other constraints. For a closed system at constant temperature and volume, the energy and entropy balances are... 

In considering the equilibrium state of a system at constant temperature and volume, we construct a closed system which consists of the system (subsystem 1) under consideration and a thermal reservoir (subsystem 2) with the temperamre T. When the two systems are brought into thermal contact, energy is exchanged between subsystem 1 and subsystem 2. Because the... 

So far we have nsed moles or concentrations in stoichiometric calcnlations. However, it is equally valid to use pressures for a gas-phase system at constant temperature and volume because in this case pressure is directly proportional to the number of moles ... 

The second law of thermodynamics provides the general criterion for spontaneous processes No process can decrease the entropy of the universe. For a closed simple system at constant pressure and temperature the Gibbs energy G cannot increase, and for a closed simple system at constant temperature and volume the Helmholtz energy A cannot increase. 

Consider a system at constant temperature and volume, with a state vector q and an energy function U q), such that at equilibrium, the probability of finding the system in state q is described by the Boltzmann distribution... 

The suffix x indicates that besides T, all the variables xu a, . . . during the change of which external work is done, are maintained constant (adynamic condition). Thus, if the only external force is a normal and uniform pressure p, then x — v, the volume of the system, and (11) is the condition of equilibrium at constant temperature and volume. 

The calculation is based on the rule of thermodynamics, which states that a system will be in equilibrium when the Gibbs free energy is at a minimum. Cl The objective then is the minimization of the total free energy of the system and the calculation of equilibria at constant temperature and volume or at constant pressure. It is a complicated and lengthy calculation but, fortunately, several computer programs are now available that considerably simplify the task. PI... 

Once the energy of the solute/solvent system has been determined, using some variant of a classical or a QM/MM force field, it can be used to find the free energy of solvation, which is the property of real interest. This is given, at constant temperature and pressure or at constant temperature and volume, respectively, by Eqs. (20) and (21) ... 

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. 

Consider a film of liquid on a wire balance as shown in Fig. 1. A force is applied to a frictionless, movable wire, stretching the film, which adheres to the wire. The process occurs at constant temperature and volume of the liquid. The reversible work done as the movable wire moves a distance dx (at constant T and V) is equal to the increase in the Helmholtz free energy of the system ... 

The physical meaning of the Helmholtz free energy is similar to that of the Gibbs energy, both being criteria to define - equilibrium. The equilibrium criterion in a closed system, which is only capable of doing P-V work and held at constant temperature and volume, is the minimum of Helmholtz energy. See also Helmholtz. 

The second law requires that, for a spontaneous change at constant temperature and volume, dF < 0. Alternatively, under conditions of constant temperature and pressure gives dG < 0. That is, the total free energy of the system decreases spontaneously at constant T and P until it reaches a minimum at equilibrium, dG = 0. The minimization of the Gtotai is one of two criteria defining an equilibrium state. 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26) ... 

By definition, the integral heat of adsorption is defined as the amount of heat evolved by the system when " or n are adsorbed at constant temperature and volume. Thus, since no volume work is done, the integral heat is obtained in accordance with the first law of thermodynamics as the final minus the initial internal energy of the system ... 

The element of work involved in enlarging the surface area at constant temperature and volume of the system is given by... 

The enthalpy of a reaction, 6B°, is the heat transfer between a system and its surroundings for a process at constant pressure, but not at constant temperature and volume (VO. For example, consider the formation of liquid water from gaseous hydrogen and oxygen at 25°C, which, with respective volumes and AHf values given beneath it is written... 

The treatment, in this chapter has been limited to nonflow systems at constant temperature and constant volume so that the rate can be represented by the rate of change of concentration of a reactant or product. With these restrictions there will be little change in the total pressure for a gaseous reaction unless the total number of moles changes. However, if there is an increase in number of moles as the reaction proceeds, there will be an increase in pressure. This increase is uniquely related to the extent of reaction. Hence measuring the total pressure as a function of time is a suitable method for studying the kinetics of such a system and has been widely used. 

According to the Gibbs phase rule, at constant temperature and volume the binary two-phase model system has only one degree of freedom, meaning that only one variable in eq. (II.4) is independent. It is possible then to replace partial derivatives by full ones. The treatment is simpler for a surface that is equimolecular with respect to solvent, for which T, = 0 the individual subscripts are no longer needed and can be omitted (i.e., p = p2 and T = T2), and eq. (II.4) becomes... 

Since T5 en > 0. this equation establishes that more work is needed to drive the system from state I to state 2 if the process is carried out irreversibly than if it were carried out reversibly. Conversely, if we are interested in the amount of work the system can do on its surroundings at constant temperature and volume in going from state 1 to state 2 (so that VV., is negative), we find, by the same argument, that more work is obtained if the process is carried out reversibly than if it is carried out irreversibly. 

Thus, for phase equilibrium to exist in a closed, nonreacting multicomponent system at constant energy and volume, the pressure must be the same in both phases (so that mechanical equilibrium exists), the temperature must be the same in both phases (so that thermal equilibrium exists), and the partial molar Gibbs energy of each species must be the same in each phase fso that equilibrium with respect to species diffusion exists). ... 

In practice this cannot be done in any simple manner. On the other hand, we can keep the temperature of a system constant by holding it in a temperature-controlled bath. We are therefore more interested in knowing the conditions for equilibrium when a system is maintained either (a) at constant temperature and volume, or (b) at constant temperature and pressure. ... 

---