Gibbs free energy

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Thermodynamic potentials
Internal energy	$U(S,V)$
Helmholtz free energy	$A(T,V)=U-TS$
Enthalpy	$H(S,P)=U+PV$
Gibbs free energy	$G(T,P)=U+PV-TS$
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In thermodynamics, the Gibbs free energy is the energy portion of a thermodynamic system available to do work. The Gibbs free energy is a thermodynamic potential and is therefore a state function of a thermodynamic system. It is defined as:

[G \equiv H-TS \,]

where (in SI units)

G is the Gibbs energy, (Joule)
H is the enthalpy (Joule)
T is the temperature (Kelvin)
S is the entropy (Joules per Kelvin)

Each quantity in the equation above can be divided by the amount of substance, measured in moles, to form molar Gibbs energy. The Gibbs energy is one of the most important thermodynamic functions for the characterization of a system. It is a factor in determining outcomes such as the voltage of an electrochemical cell, and the equilibrium constant for a reversible reaction. It is named after American physicist Josiah Willard Gibbs. The Gibbs free energy, in addition, goes by various names such as: Gibbs energy function, free energy, free enthalpy, thermodynamic potential at constant pressure, and others.

Contents

1 Overview
2 Why is the attachment ‘free’ so important?
3 Useful identities
4 Derivation of

4.1 Back to Entropy

5 See also
6 References
7 External links

Overview

In a simple manner, with respect to STP reacting systems, a general rule of thumb is:

Hence, out of this general natural tendency, a quantitative measure as to how near or far a potential reaction is from this minimum is when the calculated energetics of the process indicate that the change in Gibbs free energy ΔG is negative. Essentially, this means that such a reaction will be favored and will release energy. The energy released equals the maximum amount of work that can be performed as a result of the chemical reaction. Conversely, if conditions indicated a positive ΔG, then energy--in the form of work--would have to be added to the reacting system to make the reaction go.

Why is the attachment ‘free’ so important?

In the 18th and 19th centuries, the theory of heat, i.e. that heat is a form of energy having relation to vibratory motion, was beginning to supplant both the caloric theory, i.e. that heat is a fluid, and the four element theory in which heat was the lightest of the four elements. Many textbooks and teaching articles during these centuries presented these theories side by side. Similarly, during these years, heat was beginning to be distinguished into different classification categorize, such as “free heat”, “combined heat”, “radiant heat”, specific heat, heat capacity, “absolute heat”, “latent caloric”, “free” or “perceptible” caloric (calorique sensible), among others.

In 1780, for example, Laplace and Lavoisier stated: “In general, one can change the first hypothesis into the second by changing the words ‘free heat, combined heat, and heat released’ into ‘vis viva, loss of vis viva, and increase of vis viva.’” In this manner, the total mass of caloric in a body, called absolute heat, was regarded as a mixture of two components; the free or perceptible caloric could affect a thermometer while the other component, the latent caloric, could not. The use of the words “latent heat” implied a similarity to latent heat in the more usual sense; it was regarded as chemically bound to the molecules of the body. In the adiabatic compression of a gas, the absolute heat remained constant by the observed rise of temperature indicated that some latent caloric had become “free” or perceptible.

During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the French physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the German physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy’ for the expression E – TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions.

In modern use, we attach the term “free” to Gibbs free energy, i.e. for systems at constant pressure and temperature, or to Helmholtz free energy, i.e. for systems at constant volume and temperature, to mean ‘available in the form of useful work.’ With reference to the Gibbs free energy, we add the qualification that it is the energy free for non-volume work.

To note, some books do not include the attachment “free”, referring to G as simply Gibbs energy. This influence is the result of a 1988 IUPAC meeting designed to unified terminologies between the USA, Europe, and other countries, in which descriptive ‘free’ was supposedly banished. This ruling, however, is still far from accepted and the majority of published articles and books still use the descriptive ‘free’ for both historical, informative, and for clarification reasons.

Useful identities

[\Delta G = \Delta H - T \Delta S \,] for constant temperature

[\Delta G^\circ = -R T \ln K \,]

[\Delta G = \Delta G^\circ + R T \ln Q \,]

[\Delta G = -nF \Delta E \,]

and rearranging gives

[nF\Delta E^\circ = RT \ln K \,]

[nF\Delta E = nF\Delta E^\circ - R T \ln Q \, \,]

which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction.

where

ΔG = change in Gibbs energy

ΔH = change in enthalpy

T = temperature

ΔS = change in entropy

R = gas constant

ln = natural logarithm

K = equilibrium constant

Q = reaction quotient

n = number of electrons/mole product

F = Faraday constant (coulombs/mole)

ΔE = electrical potential of the reaction

We also have:

[K_=e^}]

[\Delta G^\circ = -RT(\ln K_) = -2.303RT(\log K_)]

which relates the equilibrium constant with Gibbs energy.

Derivation of

Let S_tot be the total entropy of a thermally closed system. An isolated system cannot exchange heat with its surroundings. Total entropy is only defined for an isolated system, an open system has internal entropy instead.

The second law of thermodynamics states that if a process is possible, then

[ \Delta S_ \ge 0 \,]

and if [ \Delta S_ = 0 \,] then the process is reversible.

Since the heat transfer Δq vanishes for a closed system, then any reversible process will be adiabatic, and an adiabatic process is also isentropic [ \left( = \Delta S = 0 \right) \,].

Now consider an open system. It has internal entropy S_int, and the system is thermally connected to its surroundings, which have entropy S_ext.

The entropy form of the second law does not apply directly to the open system, it only applies to the closed system formed by both the system and its surroundings. Therefore a process is possible if

[ \Delta S_ + \Delta S_ \ge 0 \,].

We will try to express the left side of this equation entirely in terms of internal state functions. ΔS_ext is defined as:

[ \Delta S_ = - \,]

Temperature T is the same both internally and externally, since the system is thermally connected to its surroundings. Also, Δq_rev is heat transferred to the system, so -Δq_rev is heat transferred to the surroundings, and −ΔQ/T is entropy gained by the surroundings. We now have:

[ \Delta S_ - \ge 0 \,]

Multiply both sides by T:

[ T \Delta S_ - \Delta q\ge 0 \,]

ΔQ is heat transferred to the system; if the process is now assumed to be isobaric, then Δq_p = ΔH:

[ T \Delta S_ - \Delta H \ge 0\, ]

ΔH is the enthalpy change of reaction (for a chemical reaction at constant pressure and temperature). Then

[ \Delta H - T \Delta S_ \le 0 \,]

for a possible process. Let the change ΔG in Gibbs energy be defined as

[ \Delta G = \Delta H - T \Delta S_ \,] (1)

Notice that it is not defined in terms of any external state functions, such as ΔS_ext or ΔS_tot. Then the second law becomes:

[ \Delta G < 0 \,] favored reaction

[ \Delta G = 0 \,] reversible reaction

[ \Delta G > 0 \,] disfavored reaction

Also, the sign of Delta G tells us about the spontaneity of the reaction.

[ \Delta G < 0 \,] Spontaneous

[ \Delta G = 0 \,] Equilibrium

[ \Delta G > 0 \,] Nonspontaneous

Gibbs energy G itself is defined as

[ G = H - T S_ \,] (2)

but notice that to obtain equation (2) from equation (1) we must assume that T is constant.

Thus, Gibbs energy is most useful for thermochemical processes at constant temperature and pressure: both isothermal and isobaric. Such processes don't move on a P-V diagram; and therefore appear to be thermodynamically static. However, chemical reactions do undergo changes in chemical potential, which is a state function. Thus, thermodynamic processes are not confined to the two dimensional P-V diagram. There is a third dimension for n, the quantity of gas. Naturally for the study of explosive chemicals, the processes are not necessarily isothermal and isobaric. For these studies, Helmholtz free energy is used.

Back to Entropy

If a closed system (Δq_rev = 0) is at constant pressure (Δq_rev = ΔH), then

[ \Delta H = 0 \,]

Therefore the Gibbs energy of a closed system is:

[ \Delta G = -T \Delta S \,]

and if [ \Delta G \le 0 \,] then this implies that [ \Delta S \ge 0 \,], back to where we started the derivation of ΔG.

References

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