# What are Free Energies

- Page ID
- 96640

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Free energy is a composite function that balances the influence of energy vs. entropy. To first define "free" energy, we shall examine the backgrounds of this term, what definitions carry it, and which specific definitions we, as chemists, will choose to refer to:

- Fossil fuels, global warming, and the usual popular controversies in popular science have led people to cry out for the pursuit of clean and renewable "free" energy (no doubt referring to long-term monetary cost). A most valiant cause; absolutely disregard it here. Forget you even read this definition. We'll even start at listing at 1. again!
- If a system is isothermal and closed, with constant pressure, it is describable by the Gibbs Energy, known also by a plethora of nicknames such as "free energy", "Gibbs free energy", "Gibbs function", and "free enthalpy". Because this module is located under "Gibbs Energy", we'll focus on this energy; the Helmholtz will be briefly mentioned.
- If a system is isothermal and closed, with constant volume, it is describable by the Helmholtz Energy, known also by an unnecessary amount of aliases such as "Helmholtz function", "work function", "Helmholtz free energy", and our favorite, "free energy". It will be mentioned in passing.
- Having two energies both called "free energy" is like having two brothers named Jack. More specifically, they'd be
*twin*brothers; the Gibbs and Helmholtz Energies describe situations with equations easily confused with each other. It's no wonder the IUPAC (the International Union of Pure and Applied Chemistry) officially refers to the two as Gibbs Energy and Helmholtz Energy, respectively. This should not be a surprise, because that's what they were originally named in the first place! Just keep in mind that some outdated or unsophisticated texts might still use the pseudonyms mentioned above (guised as, say, the title of a module).

## Gibbs Energy

The Gibbs Energy is named after a Josiah William Gibbs, an American physicist in the late 19^{th} century who greatly advanced thermodynamics; his work now serves as a foundation for this branch of science. This energy can be said to be the greatest amount of work (other than expansion work) a system can do on its surroundings, when it operates at a constant pressure and temperature.

First, a modeling of the Gibbs Energy by way of equation:

\[G = U + PV - TS\]

Where:

- \(U\)= Internal Energy
- \(TS\) = absolute temperature x final entropy
- \(PV\) = pressure x volume

Of course, we know that \(U + PV\) can also be defined as:

\[U + PV = H\]

Where:

- \(H\) is enthalpy

Which leads us to a form of how the Gibbs Energy is related to enthalpy:

\[G = H - TS\]

All of the members on the right side of this equation are state functions, so G is a state function as well. The change in G is simply:

\[\Delta{G} = \Delta{H} - T\Delta{S}\]

## How This Equation Was Reached

We will start with an equation for the total entropy change of the universe. Our goal is to whittle it down to a practical form, like a caveman shaping a unwieldy block of stone into a useful hand held tool!

\[\Delta{S_{universe}} = \Delta{S_{system}} + \Delta{S_{surroundings}}\]

An equation with variables of such scope is difficult to work with. We want to do away with the vagueness, and rewrite a more focused equation. We'll consider the case where temperature and pressure is constant. Here we go:

- \(\Delta{S_{surroundings}}\) can be rewritten as \( \dfrac{\Delta{H}}{T} \).

*The heat, q _{p}, that the system affects the surroundings with is the negative of the *\[(Delta{H}\)

*for the system. Because*\(-q_p = -\Delta{H}_{sys}\),

*the change in the entropy of the surroundings will be*\(\Delta{S}_{surroundings} = \Delta{H_{sys}}/T\).

- The equation becomes \[\Delta{S_{univ}} = \Delta{S_{sys}} + (-\Delta{H_{surr}}/T)\]

*A simple substitution.*

- The equation becomes \[T\Delta{S_{univ}} = \Delta{H_{surr}} - T\Delta{S_{sys}}\]

*Multiply both sides by T.*

- With the mighty powers of whoever discovering stuff getting to name it, we set \[-T\Delta{S_{univ}}\] equal to a great big \[\Delta{G}\] for the almighty Gibbs. Finally, we achieve the equation, \[\Delta{G} = \Delta{H} - T\Delta{S}\].

## Reasoning Behind the Equation

As a quick note, let it be said that the name "free energy", other than being confused with another energy exactly termed, is also somewhat of a misnomer. The multiple meanings of the word "free" can make it seem as if energy can be transferred at no cost; in fact, the word "free" was used to refer to what cost the system was free to pay, in the form of turning energy into work. \(\Delta{G}\) is useful because it can tell us how a system, when we're given only information on it, will act.

\[\Delta{G} < 0\]

indicates a spontaneous***** change to occur.

\[\Delta{G} > 0\]

indicates an absence of spontaneousness.

\[\Delta{G} = 0\]

indicates a system at equilibrium.

The Gibbs Energy reaches the minimum value when equilibrium is reached. Here, it is represented as a graph, where x represents the extent of how far the reaction has occurred. The minimum of the function has to be smooth, because \(G\) must be differentiable (its first derivative has to exist at the minimum).

It was briefly mentioned that \[\Delta{G}\] is the energy available to be converted to work. The definition is self evident from the equation.

Look at \[\Delta{G} = \Delta{H} - T\Delta{S}\]. Recall that \(\Delta{H}\) is the total energy that can be made into heat. \[T\Delta{S}\] is the energy not available to be converted to work. By a reordering of the Gibbs Energy equation:

\[\Delta{H} = \Delta{G} - T\Delta{S}\]

Expressed in words:

*the energy available to be turned into heat = \[\Delta{G}\] - the energy that is not free to do work. This lets us see that \(\Delta{G}\) MUST be the energy free to do work.*

Why constant temperature and pressure? It just so happens that these are regularly occurring factors in the laboratory, making this equation practical to use, and useful as well, for chemists. An example of Gibbs Energy in the real world is the oxidation of glucose; \[\Delta{G}\] in this case is equal to 2870 kJ, or 686 Calories. For living cells, this is the primary energy reaction.

## Helmholtz Free Energy

The **Helmholtz free energy** is deemed as a thermodynamic potential which calculates the “useful” work retrievable from a closed thermodynamic system at a constant temperature and volume. For such a system, the negative of the difference in the Helmholtz energy is equal to the maximum amount of work extractable from a thermodynamic process in which both temperature and volume are kept constant. In these conditions, it is minimized and held constant at equilibrium. The Helmholtz free energy was originally developed by Hermann von Helmotz and is generally denoted by the letter *A*, or the letter *F* . In physics, the letter *F* is mostly used to denote the Helmholtz energy, which is often called the *Helmholtz function* or simple term “free energy."

Introduced by German physicist Hermann Helmholtz in 1882, **Helmholtz free energy **is the thermodynamic potential found in a system of constant species with constant temperautre and constant volume, given by the formula:

\[ΔA = ΔE – TΔS\]

- A = Helmholtz Free Energy in Joule
- E = Energy of the System in Joule
- T = Absolute Temperature in KElvin
- S = Entropy in Joule/Kelvin

Summarily, the Helmholtz free energy is also the measure of an isothermal-isochoric closed system’s ability to do work. If any external field is missing, the Helmholtz free energy formula becomes:

\[ΔA = ΔU –TΔS\]

- A = Helmholtz Free Energy in Joule
- U = Internal Energy in Joule
- T = Absolute Temperature in Kelvin
- S = Entropy in Joule/Kelvin

The internal energy (U) can be said to be the amount of energy required to create a system in the nonexistant changes of temperature (T) or volume (V). However, if the system is created in an environment of temperature, T, then some of the energy can be captured by spontaneous heat transfer between the environment and system. The amount of this spontaneous energy transfer is TΔS where S is the final entropy of the system. In that case, you don't have to put in as much energy. Note that if a more disordered, resulting in higher entropy, the final state is created, where less work is required to create the system. The Helmholtz free energy becomes a measure of the sum of energy you have to put in to generate a system once the spontaneous energy transfer of the system from the environment is taken into account.

Helmholtz Free Energy is generally used in Physics, denoted with the leter *F*, while Chemistry uses,* G*, Gibbs' Free Energy.

## Relating Helmholtz Energy to Gibbs Energy

The Helmholtz Energy is given by the equation:

\[A = U - TS\]

It is comparable to Gibbs Energy in this way:

\[G = A + PV\]

The Helmholtz Energy is used when having a constant pressure is not feasible.

Along with internal energy and enthalpy, the Helmholtz Energy and Gibbs Energy make up the quad group called the **thermodynamic potentials**; these potentials are useful for describing various thermodynamic events.

TS represents energy from surroundings, and PV represents work in expansion. If needed, refer to the links above to refresh your memory on enthalpy and internal energy.

## References

- Petrucci, et al. General Chemistry: Principles & Modern Applications; Ninth Edition. Pearson/Prentice Hall; Upper Saddle River, New Jersey 07.
- Mortimer, Robert G. Physical Chemistry; Third Edition. Elsevier Inc.; 2008.

## Contributors and Attributions

- Alexander Shei