Gibbs free energy

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In thermodynamics the Gibbs free energy (sometimes also known as free enthalpy) is a thermodynamic potential and is therefore a state function of a thermodynamic system. It is defined as

<math>G \equiv U+PV-TS \,</math>

where (in SI units)

• G is the Gibbs free energy, (joule)
• U is the internal energy (joule)
• P is the pressure (pascal)
• V is the Volume (cubic meter)
• 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 free energy. The Gibbs free 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.

Contents 1 Overview 2 Useful identities 3 Derivation of Gibbs Free Energy 3.1 Back to Entropy 4 See also

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 release energy in the form of work. 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.

Useful identities

<math>\Delta G = \Delta H - T \Delta S \,</math> for constant temperature

<math>\Delta G^\circ = -R T \ln K \,</math>

<math>\Delta G = \Delta G^\circ + R T \ln Q \,</math>

<math>\Delta G = -nF \Delta E \,</math>

and rearranging gives

<math>nF\Delta E = RT \ln K \,</math>

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

where

ΔG = change in Gibbs free 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:

<math>K_{eq}=e^{- \frac{\Delta G^\circ}{RT}}</math>

<math>\Delta G^\circ = -RT(\ln K_{eq}) = -2.303RT(\log K_{eq})</math>

which relates the equilibrium constant with Gibbs free energy.

Derivation of Gibbs Free Energy

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

<math> \Delta S_{tot} \ge 0 \,</math>

and if <math> \Delta S_{tot} = 0 \,</math> 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 <math> \left( {\Delta q\over T} = \Delta S = 0 \right) \,</math>.

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

<math> \Delta S_{int} + \Delta S_{ext} \ge 0 \,</math>.

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

<math> \Delta S_{ext} = - {\Delta q\over T} \,</math>

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:

<math> \Delta S_{int} - {\Delta q\over T} \ge 0 \,</math>

Multiply both sides by T:

<math> T \Delta S_{int} - \Delta q\ge 0 \,</math>

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

<math> T \Delta S_{int} - \Delta H \ge 0\, </math>

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

<math> \Delta H - T \Delta S_{int} \le 0 \,</math>

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

<math> \Delta G = \Delta H - T \Delta S_{int} \,</math> (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:

<math> \Delta G < 0 \,</math> favored reaction

<math> \Delta G = 0 \,</math> reversible reaction

<math> \Delta G > 0 \,</math> disfavored reaction

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

<math> \Delta G < 0 \,</math> Spontaneous

<math> \Delta G = 0 \,</math> Equilibrium

<math> \Delta G > 0 \,</math> Nonspontaneous

Gibbs free energy G itself is defined as

<math> G = H - T S_{int} \,</math> (2)

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

Thus, Gibbs free 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

<math> \Delta H = 0 \,</math>

Therefore the Gibbs free energy of a closed system is:

<math> \Delta G = -T \Delta S \,</math>

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

See also

• Free energy
• Helmholtz free energy
• Enthalpy
• Thermodynamics
• Willard Gibbs
• Gibbde:Gibbs-Energie

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