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# What Is "free" energy

Free Energy is a thermodynamic quantity given by the subtraction between the internal energy of a system and the product of its absolute temperature and entropy. In Chemistry terms, it is also known as the difference between the enthalpy and the multiplication of the absolute temperature and entropy of a system, Gibbs Free Energy.

### Introduction

Free Energy refers to a very broad topic of energy. As a part of science we have Thermodynamic Free Energy, which is energy found in a physical system that can be made into work. We have Helmholtz Free Energy, which is energy that can be turned into work at a constant temperature and volume. And we have Gibbs Free-Energy, the measure of total entropy changes along with the surroundings with constant pressure. Free energy determines whether a chemical reaction is spontaneous or not spontaneous, in the Chemistry 2C course, the main focus is Gibbs Free Energy.

### Thermodynamic Free Energy

Thermodynamic free energy is the sum of work that a thermodynamic system can work. The theory is useful in the thermodynamics of chemical and/or thermal processes in research fields of engineering and many scientific studies. The free energy is the internal energy input of a system minus the the energy that cannot be used to create work. The unusable energy is known as entropy (S), which is multiplied by the temperature (T) of the system.

Table 1: Formulas of Thermodynamic Free Energy and the Four Represented Potentials of Free Energy

 Internal Energy ΔU(S,V) Enthalpy ΔH(S,P) = ΔU + PV Helmholtz Free Energy ΔF(T,V) = ΔU − TΔS Gibbs Free Energy ΔG(T,P) = ΔH − TΔS

Broken down, Thermodynamic free energy is very broad topic that is situated into different components. There are two Free Energy formulas and concepts that are wide used. They are: Helmoltz Free Energy which is widely used in Physics and Gibbs' Free Energy wide used by Chemistry standards. Helmholtz Free Energy depends on the Internal Energy, which is a function of entropy and volume. Gibbs Free Energy depends on Enthalpy, which comprises of a function of entropy and pressure.

### Helmholtz Free Energy

In thermodynamics, 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:

ΔF = ΔE – TΔS

• F = 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:

ΔF = ΔU –TΔS

• F = 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.

Example 1

In a dynamite explosion, the Absolute Temperature remained constant at 100C. Initially the internal energy began at 0J and during the explosion went up to 28,000J. The entropy began at 10KJ/K and changed to 18KJ/K. Calculate and solve for ΔF, Helmholtz Free Energy.

TK = 100oC + 273.15 = 127.315 K

ΔU = 28000J - 0j

= 28,000 J

ΔS = 18 KJ/K - 10 KJ/K

= 8 KJ/K

ΔF = ΔU –TΔS

= 28000J - (127.315 K x 8 KJ/K)

= 28000J - (127.315 K x 8000 J/K)

= -990520 J

### Gibbs' Free Energy

"Gibbs' Free Energy" was first introduced by Yale Professor, Josiah Willard Gibbs who was responsible for using the laws of thermodynamics to construct what is now known as Gibbs' Free Energy. He created a formula that involved, entropy, enthalpy, and temperature.

ΔG=ΔH-TΔS

Free Energy is demonstrated by the formula ΔG=ΔH-TΔS. As temperature increases, (TΔS) entropy becomes more important in determining free energy.

The internal energy, U, may be thought of as the energy required to create a system in the absence of changes in temperature or volume. As discussed when defining enthalpy, an additional amount of work, PV, needs to be done for when the system is created from a very small volume to make room for the system. Taken from Helmholtz free energy, an environment at constant temperature, T, will contribute an amount TΔS to the system, reducing the overall amount needed to create the system. This net energy contribution for a system made in environment temperature, T, from a negligible initial volume is called Gibbs free energy.

ΔG represents the free energy change. Free Energy change is a combination of enthalpy change, entropy change, and temperature that can be used to determine whether a process is spontaneous or nonspontaneous. It is used to predict reaction's spontaneity of a reaction with constant temperature and pressure, when ΔG is negative, the reaction is possibly spontaneous. When ΔG is positive, the reaction is possibly not spontaneous.  When ΔG is zero, there is no equilibrium and therefore, there is no reaction.  It is also represented as the energy difference between reactants and products.

Table 2:

 ΔH ΔS spontaneous? Negative Positive Yes Negative Negative possibly (at low temperature) Positive Positive possibly (at high temperature) Positive Negative No

Table 3:

 Name Symbol Units Definition Entropy ΔS J/mol•K measure of randomness (change of disorder) Enthalpy ΔH kJ/mol Change in heat Temperature T Kelvins Measure of heat
Example 2

Calculate ΔH, ΔS, andΔG for the below reaction to determine whether the reaction is spontaneous or not at STP.

CO(g) + H20(g) CO2(g) + H2 (g)

Compound       Hfo(kJ/mol)  S°(J/mol-K)

CO(g)                   -110.5             197.7

H20(g)                  -241.8             188.8

CO2(g)                  -393.5             213.7

H2 (g)                       0                 130.7

ΔHo = Hfo(products) - Hfo(reactants)

= [1 mol H2 x 0 kJ/mol + 1 mol CO2 x -393.5 kJ/mol] - [1 mol H20 x -241.8 kJ/mol + 1 mol CO x -110.5 KJ/mol]

= -41.2 kJ

ΔSo = So(products) - So(reactants)

= [1 mol H2 x 130.7 kJ/mol + 1 mol CO2 x 213.7 kJ/mol] - [1 mol H20 x 188.8 kJ/mol + 1 mol CO x 197.7 KJ/mol]

= -42.1 J/K

TK = 25o C + 273.15 = 298.15 K

ΔGo = ΔHo - TΔSo

= -41.2 kJ - (298.15 K x -42.1 J/K)

= -41,200 J + 12,552.12 J

= -28,647.88 J

This reaction is spontaneous.

### Law of Thermodynamics

The Second Law of Thermodynamics states that even though the total energy is unchanged, after every energy transformation, the amount of free energy decreases. Entropy tends to increase. The energy is usually lost by an energy form such as heat.

### References

1. Atkins, Peter, and Loretta Jones. Chemistry: Molecules, Matter, and Change. New York: W.H. Freeman and Company, 1997.
2. Brady, James E.. General Chemistry: Principles & Structure. Jamaica, New York: John Wiley & Sons, 1990.
3. Gerhart, Karen. Introduction To Biology. Dubuque, IA: Kendall Hunt Publishing Company, 2009.
4. Petrucci, Ralph H., Herring, Goeffrey F., Madura, Jeffrey D., and Bissonnette, Carey. General Chemistry: Principles and Modern Applications. 9th ed. New Jersey: Pearson Canada, 2007.
5. Raymond, Kenneth W.. General, Organic, and Biological Chemistry And integrated Approach. Massachussets: John Wiley and Sons, 2008.

### Problems

1.  When enthalpy is negative and entropy is positive,  what is free energy?

Solution:  When enthalpy is negative and entropy is positive, the free energy is always negative.

2. When enthalpy is negative and entropy is negative, free energy is?

Solution:  negative at low temperatures.

4.

H2O(l) + SO3(g) --> H2SO4(aq)

T 256 K

H  -300       -400          -900

S   70           250           20.0

ΔH: (-900)-(-400+-300)=-200kJ/mol

ΔS: (20)-(250+20)=-250J/mol K=.250J/mol K

ΔG=-200-(256*.250)

ΔG=-136

5. Challenge, find the answers without any solutions.

CH4(g) + 2O2--> CO2(g)+2H2O(l)

H -150 J/Kmol  -200 J/Kmol   -314.5 J/Kmol    -70J/Kmol

S  200                300              400                 20

### Contributors

• Yalda Tarin (UCD), Ryan Cheung (UCD)

13:03, 21 Jan 2014

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### Textbook Maps

This material is based upon work supported by the National Science Foundation under Grant Number 1246120