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# Thermodynamic Potentials

Most natural thermodynamic systems are open systems that exchange heat and work with their environment, rather than the closed systems described so far. For example, living systems can reduce their entropy as they expand and develop; they create structures of greater internal energy (i.e. lower entropy) out of the nutrients they absorb. It does not represent a violation of the second law of thermodynamics because a living organism does not constitute a closed system.
To simplify applying the laws of thermodynamics to open systems, criteria with the dimensions of energy, referred to as thermodynamic potentials, are introduced to describe the system. The resulting formulas are expressed in regards to the Helmholtz Free Energy (F) and the Gibbs Free Energy (G), named after the 19th-century German Physiologist and Physicist Hermann von Helmholtz also the simultaneous American Physicist Josiah Willard Gibbs. The key conceptual step is to separate a system from its heat reservoir. A system is held at a constant temperature level of T by a heat reservoir (i.e., the environment). Yet, the heat reservoir is no longer considered to be part of the system. Remember that the internal energy change (ΔU) of a system is given by

ΔU = Q − W

where Q is the heat absorbed, and W is the work done. Generally, Q and W separately are not state functions because they are path-dependent. However, if the path is specified as any reversible isothermal process, the heat associated with the optimum work (Wmax) is Qmax = TΔS. With this substitution,
we can rearrange the above equation as

− Wmax = ΔU − TΔS

Note that right here, ΔS is the entropy change just of the system being held at a constant temperature level, such as a battery. Unlike the case of an isolated system, as taken into consideration previously, it does not include the entropy change of the heat reservoir (i.e., the environments) required to maintain the temperature level constant. If these additional entropy changes in the reservoir, the total entropy change would certainly be zero, as in the case of an isolated system. Since the quantities U, T, and S on the right-hand side are all state functions, it follows that − Wmax must also be a state function. It leads to the definition of Helmholtz Free Energy.

F = U − TS

such that, for any isothermal change of the system,

ΔF = ΔU − TΔS

is the negative of the optimum work extracted from the system. The actual work extracted might be smaller than the ideal optimum, or perhaps zero, which indicates that W ≤ − ΔF, with equality applying in the ideal limiting case of a reversible process. When the Helmholtz Free Energy reaches its minimum value, the system has reached its equilibrium state and can extract no further work. Therefore, the equilibrium condition of optimum entropy for isolated systems is the condition of minimum Helmholtz Free Energy for open systems held at a constant temperature level. The one additional precaution required is that work done against the environment be included if the system expands or contracts throughout the process is taken into consideration. Generally, processes are specified as taking place at constant volume and the temperature level so that no correction is needed.
Although the Helmholtz Free Energy is useful in describing processes inside a container with rigid wall surfaces, most processes in real-life take place under constant pressure rather than constant volume. For example, chemical reactions in an open test tube or a tomato’s growth in a garden occur under conditions of nearly constant atmospheric pressure. It is for the description of these cases that introduced Gibbs Free Energy. As previously established, the quantity

− Wmax = ΔU − TΔS

is a state function equal to the change in the Helmholtz Free Energy. Suppose that the process involves a large change in volume (ΔV), such as when water boils to form a vapor. The work done by the expanding water vapor as it pushes back the surrounding air at pressure P is PΔV. It is the amount of work that is now split out from Wmax by writing it in the form

Wmax = W ′ max + PΔV

where W ′ max is the optimum work that can extract from the process at constant temperature level T and pressure P, other than the atmospheric work (PΔV). Substituting this partition right into the above equation for − Wmax and also moving the PΔV term to the right-hand side then yields

− W ′ max = ΔU + PΔV − TΔS

It leads to the definition of the Gibbs Free Energy

G = U + PV − TS

such that, for any isothermal change of the system at constant pressure,

ΔG = ΔU + PΔV − TΔS

The negative of the optimum work W ′ max can extract from the system, other than atmospheric work. The actual work extracted could be smaller than the ideal optimum, or even zero, which implies that W ′ ≤ − ΔG, with equality applying in the ideal limiting case of a reversible process. Like in the Helmholtz case, when the Gibbs Free Energy reaches its minimal value, the system has reached its equilibrium state and can extract no further work from it. Therefore, the equilibrium condition ends up being the condition of minimal Gibbs Free Energy for open systems held at constant temperature level and pressure, and also the direction of spontaneous change is always toward a state of lower free energy for the system (like a ball rolling downhill right into a valley). Notice specifically that the entropy can now spontaneously decrease (i.e., TΔS can be negative), provided that this decrease is more than balanced out by the ΔU + PΔV terms in the definition of ΔG. As further discussed, an easy example is the spontaneous condensation of vapor into the water. Although the entropy of water is much less than the entropy of vapor, the process occurs spontaneously, provided that adequate heat energy is taken away from the system to keep the temperature level from rising as the vapor condenses.
An automobile battery provides a familiar example of free energy changes. When the battery is fully charged, its Gibbs Free Energy is at an optimum, and also, when it is fully discharged (i.e., dead), its Gibbs Free Energy is at a minimum. The change between these two states is the optimum amount of electrical work extracted from the battery at a constant temperature level and pressure. The amount of heat absorbed from the environment to maintain the temperature level of the battery constant (represented by the TΔS term) and any work done against the environment (represented by the PΔV term) are automatically taken into consideration in the energy equilibrium.

## Gibbs Free Energy and Chemical Reactions

All batteries depend on some chemical reaction of the form

reactants → products

for the generation of electricity or reverse reaction as the battery is recharged. The change in free energy (− ΔG) for a reaction might be determined by measuring directly the amount of electrical work that the battery might do and afterwards using the equation Wmax = − ΔG. However, the power of thermodynamics is that − ΔG can be calculated without building every feasible battery and measuring its efficiency. If the Gibbs Free Energies of the individual substances making up a battery are known, then the total free energies of the reactants can be subtracted from the total free energies of the products to find the change in Gibbs Free Energy for the reaction,

ΔG = Gproducts − Greactants

Once the free energies are known for various substances, one can quickly recognize the very best prospects for actual batteries. A good part of the practice of thermodynamics is worried about determining the free energies and various other thermodynamic properties of individual substances so that ΔG for reactions can be calculated under different conditions of temperature level and also pressure.
The above discussion can interpret the term reaction in the widest possible sense as any change in matter from one form to another. Along with chemical reactions, a reaction might be something as simple as ice (reactants) transforming to liquid water (products), the nuclear reactions in the interior of stars, or fundamental particle reactions in the very early universe. Whatever the process, the direction of spontaneous change (at constant temperature level and pressure) is always decreasing free energy.

## Enthalpy and the Heat of Reaction

As discussed above, the free energy change Wmax = − ΔG represents the optimum feasible useful work that can extract from a reaction, such as in an electrochemical battery. It represents one extreme limit of a continuous range of possibilities. At the various other extremes, for example, battery terminals can be connected directly by a wire and also, the reaction is allowed to proceed freely without doing any useful work. In this case, W ′ = 0, and also the first law of thermodynamics for the reaction becomes

ΔU = Q0 − PΔV

where Q0 is the heat absorbed when the reaction does no useful work and also, as previously, PΔV is the atmospheric work term. The key point is that the quantities ΔU and also PΔV are precisely the same as in the various other limiting cases in which the reaction does optimum work. It follows since these quantities are state functions, which depend only on the initial and final states of a system and not on any path connecting the states. The amount of useful work done represents different paths connecting the same initial and final states. It leads to the definition of enthalpy (H) or heat content, as

H = U + PV

Its importance is that, for a reaction occurring freely (i.e., doing no useful work) at a constant temperature level and also pressure, the heat absorbed is

Q0 = ΔU + PΔV = ΔH

where ΔH is called the Heat of Reaction. The heat of reaction is very easy to measure because it simply represents the amount of heat emitted if the reactants are mixed in a beaker and allowed to react freely without doing any useful work.
The above definition for enthalpy and its physical significance allows the equation for ΔG to be written in a particularly illuminating and instructive form.

ΔG = ΔH − TΔS

Both terms on the right-hand side represent heats of reaction yet under different circumstances. ΔH is the heat of reaction (i.e., the amount of heat absorbed from the environment to hold the temperature level constant) when the reaction does no useful work. Also, TΔS is the heat of reaction when the reaction does optimum useful work in an electrochemical cell. The (negative) difference between these two heats is exactly the optimum useful work ΔG that can be extracted from the reaction. Therefore, it can obtain useful work by contriving for a system to extract additional heat from the environment and convert it right into work. The difference ΔH − TΔS represents the fundamental limitation imposed by the second law of thermodynamics on how much additional heat can be extracted from the environment and converted into useful work for a given reaction mechanism. An electrochemical cell (such as an automobile battery) is a creation utilizing which can make a reaction to do the optimum feasible work against an opposing electromotive force. Also, the reaction ends up being reversible because a slight increase in the negative voltage will certainly cause the direction of the reaction to reverse and the cell to start charging up instead of discharging.
As a simple example, consider a reaction in which water turns reversibly right into vapor by boiling. To make the reaction reversible, suppose that the mixture of water and vapor is contained in a cylindrical tube with a movable piston and held at the boiling point of 373 K (100 ° C) at one atmospheric pressure by a heat reservoir. The enthalpy change is ΔH = 40.65 kilojoules per mole, which is the latent heat of vaporization. The entropy change is

ΔS =40.65/ 373 = 0.109 K (kilojoules per mole)

It represent the higher degree of disorder when water evaporates and turns to vapor. The Gibbs Free Energy change is ΔG = ΔH − TΔS. In this case, the Gibbs Free Energy change is zero because the water and vapor are in equilibrium and can extract no useful work from the system (apart from work done against the environment). In other words, the Gibbs Free Energy per molecule of water (also called the Chemical Potential) is the same for both liquid water and vapor. Therefore, water molecules can pass freely from one phase to the other without any change in the total free energy of the system.