## Info

1.10 The fundamental differentials

1.10.1 internal energy

In a closed system, the 1st Law for a differential changes is: du = 8q - 8W(PV) - SWext

The work term has been divided up into an expansion/contraction (pV) term and the last term representing all other forms of work (such as electrical work supplied by a battery or chemical work expended as ATP induces muscle contraction). If the heat and pV work terms are reversible, they can be replaced by Tds and pdV, respectively, and the above equation becomes:

similar differentials can be obtained from the last three energy-like properties.

1.10.2_Enthalpy h = u + pv ^ dh = du + pdv + vdp, then use Eq (1.15):

1.10.3 Helmholz free energy f ■ u - Ts ^ df = du - Tds - sdT , then use Eq (1.15):

1.10.4 Gibbs free energy g ■ h - Ts ^ dg = dh - Tds - sdT ^ use Eq (1.16):

1.10.5 Working forms

If the external work term is removed, these equations become:

The importance of these equations cannot be overestimated. In their present form, they are the starting points for most thermodynamic analyses of one-component substances. With an additional term, Eq (1.18a) constitutes the basis of chemical thermodynamics. From them, the several conditions of equilibrium in a closed system can be derived (see below).

Another significant aspect of these equations relates to the processes to which they apply. Their derivation was premised on the assumption of reversibility. However, the end result contains only state functions (properties), so that they apply to irreversible processes as well.

The importance of these four equations is reflected in their collective name. The most common is fundamental differentials. They are also known as the Gibbs equations. When rearranged, the first two (Eqs (1.15a) and (1.16a)) are termed the Tds equations.

### 1.11 Equilibrium

In the preceding sections, the term equilibrium has been used without definition. For thermodynamics to be of use, the state of the system under examination must be at equilibrium, which means (at a minimum) that it has no tendency to change with time if unprovoked. However, there are many caveats to this definition. Time-independence is not a foolproof criterion; a closed system with Q = W would be at steady-state, but not necessarily at equilibrium if the heat input or output required temperature gradients within the system. To accommodate this possibility, uniformity of all properties through the system (each phase, if more than one) is also mandated. Moreover, time-independence is not a necessary condition; in a reversible process, the system changes with time, yet the rate of change must be slow enough for the system to pass through a large number of equilibrium states. If any of these states is frozen in time, it would be indistinguishable from a truly time-invariant system.

Two aspects of equilibrium are internal (within each phase of the system) and external (between system and surroundings). In a multi-phase system, the requirement of equilibrium between all phases falls somewhere between internal and external equilibrium, and is covered in

1.11.1 Internal equilibrium

Internal equilibrium means that the matter in the system is uniform on a molecular level, or that it has no concentration, pressure, or temperature gradients within it. For example, the gas in a box in which all gas molecules spontaneously occupy one half of the volume with the other half empty is not an equilibrium system. Macroscopic thermodynamics has nothing to say about such a situation. Microscopic or statistical thermodynamics shows that such a state cannot be ruled out, but is highly improbable.

In purely mechanical systems, equilibrium is represented by the state of the system that has the lowest energy. Thus, the equilibrium state of a pendulum is achieved when the weight hangs motionless in a vertical position. In thermodynamic systems, on the other hand, there are many criteria for equilibrium, depending on the constraints placed on the system. Constraining a system means fixing at least two of its properties. Depending on the constraint, equilibrium is expressed as the state in which a particular thermodynamic property is a maximum or a minimum. Examples of thermodynamic equilibrium are given below.

Consider an isolated system. Because its boundary is rigid and cannot pass matter, the system can exchange neither heat nor any form of work. These conditions imply constraints of fixed internal energy and volume, dv = du =0 which reduces Eq (1.15), to:

The subscripts on ds indicate that any change in s must occur at constant internal energy and volume. Swext can be thought of as virtual work, which could be realized if the system could communicate with its surroundings.

An example is a box containing a pendulum with its arm at maximum swing. If connected by a line to a pulley and weight outside the box, external work could be performed by the pendulum in lifting the weight. Without such a line, the potential for work cannot be realized.

If the boundary does not even permit virtual work (Swext = 0), the system is said to be in equilibrium, and Eq (1.19) reduces to:

In words, this brief requirement is:

At equilibrium, the entropy of an isolated system is a maximum

Equation (1.19a) does not say whether the extremum is a maximum or a minimum, but the proof is as follows. If the system is not in equilibrium, it is still capable of doing external work. In so doing, the left-hand side of Eq (1.19) is positive, and so must be the right-hand side. Therefore, for any isolated system out of equilibrium, dsu>v > 0, or the entropy increases until the maximum is attained and Eq (1.19a) applies.

A different equilibrium criterion applies when the process constraints are the more common ones of constant temperature and pressure. In this case, equilibrium is attained with the Gibbs free energy of the system is a minimum. This is proven by setting dT = dp = 0 in Eq (1.18), resulting in:

The equilibrium criterion is obtained from the inability-to-do-work criterion by setting Swext = 0, which results in:

At equilibrium, the Gibbs free energy of a system held at constant T & p is a minimum

That the equilibrium state is a minimum is demonstrated by considering Eq (1.20). A system at constant T and p that is out of equilibrium can still perform real external work (not just virtual work), and in so doing its free energy decreases. Therefore for any isothermal, isobaric system out of equilibrium, dgT,p < 0, or the free energy decreases until equilibrium is achieved7. This equilibrium condition is particularly useful in dealing with chemical reactions (Chap. 9) or in assessing the stability of the various phases (vapor, liquid or solid) of a system (Chap. 5).

Figure 1.18 shows the conditions represented by Eqs (1.19a) and (1.20a) in graphical form.

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