Thermochemistry - Enthalpy, Entropy, and Gibbs Free Energy

Enthalpy

The enthalpy of a reaction is a measure of how much heat is absorbed or given off when a chemical reaction takes place.  It is represented by ΔHrxn and is found by subtracting the enthalpy of the reactants from the enthalpy of the products: 

ΔHrxn = ΣΔHf products - ΣΔHf reactants

The Greek letter Σ, may be new to you. In mathematics, it is used to represent the phrase "to sum." Therefore, this equation is telling us to sum the enthalpy of the products and subtract the sum of the enthalpy of the reactants.  Using a table of Standard Thermodynamic Values at 25°C, you may notice that the table, which covers many pages, has five columns.  The first column is the formula of an element or compound you are looking up.  The second column is its state of matter - which is very important.  The third column lists Hformation values, or the enthalpy of formation.  This is the amount of energy needed to form one mole of that compound.  Most values as you can see are negative because releasing energy (exothermic) is a more common process in nature.

Find sodium sulfide, or Na2S.  As you can see, its enthalpy of formation is  -373.21 kJ/mol.  This means that when one mole of sodium sulfide is formed from its constituent elements (sodium and sulfur), -373.21 kilojoules of energy is released.  Elements in their free state at their state of matter at 25°C (this is called the "standard state") are assigned a value of 0.0.  This is because elements are not formed from anything more basic, therefore no energy must be absorbed or released to create them.  When the enthalpy of reaction is calculated, a negative value indicates the reaction is exothermic.  A positive value indicates the reaction is endothermic.

Entropy

The entropy change from a reaction, or Srxn, is a measure of the dispersal of energy and matter that takes place during a reaction.  As far as identifying an increase in dispersal of matter, there are two things that indicate an increase in entropy:
• Have more total moles of products than total moles of reactants.
• Have products that are in states of matter that exhibit high amounts of freedom for their particles, namely gases and aqueous compounds.

The entropy of a reaction can be calculated using a formula similar to the enthalpy of reaction:
ΔSrxn = ΣΔSproducts - ΔΣSreactants

Gibbs Free Energy

Gibbs Free Energy is a quantity used to measure the amount of available energy (to do work) that a chemical reaction provides.  Furthermore, it can be used to determine whether or not a reaction is spontaneous (works) at a given Kelvin temperature.  Reactions are very temperature dependent, and sometimes work significantly better at some temperatures than others.  The ΔGf° values provided in the table are only viable at 25°C (298.15 K). Similar to the equations for ΔHrxn and ΔSrxn, ΔGrxn is the difference between the sum of the free energy of formation values of the products and reactants:
ΔGrxn = ΣΔGf products - ΔΣGf reactants
A positive ΔGrxn indicates the reaction is nonspontaneous, a negative ΔGrxn indicates the reaction is spontaneous, and a value close to zero indicates an equilibrium. It's important to note that spontaneous does not necessarily mean fast. A spontaneous reaction is immediate, but like the rusting of metal, may be slow. Reaction rate is governed by other factors that are not related to the thermochemical quantities discussed here.

For all temperatures, including 25°C, the following equation can be used to determine spontaneity: 
ΔGrxn = ΔHrxn - TΔSrxn
In order to use this equation properly, keep these thoughts in mind:
• The temperature must be Kelvin, which is done by adding 273.15 to the Celsius temperature.
• Srxn must be converted to kJ/K.

The value calculated for ΔGrxn should be considered an approximate, particularly as the temperature moves further away from 25°C. Both ΔHrxn and ΔSrxn will vary with temperature. Although ΔSrxn tends to vary more, its impact on ΔGrxn tends to be less. This is because ΔSrxn is measured in units of J/K, and when converted to kJ/K (to agree with the units for ΔHrxn and ΔGrxn - kilojoules), it is numerically small. ΔHrxn tends to vary less than ΔSrxn, but because its value is usually several orders of magnitude greater than a kJ/K value for ΔSrxn, it affects ΔGrxn greatly. Nevertheless, there are some reactions for which the above equation can give a reliable value over a large temperature range.

The Relationship between Spontaneity and the Sign of Enthalpy and Entropy Values

Consider the following relationships:
When ΔHrxn is negative and ΔSrxn is positive, ΔGrxn will be negative (favorable) over all temperatures.
When ΔHrxn is negative and ΔSrxn is negative, ΔGrxn will be negative (favorable) at high temperatures.
When ΔHrxn is positive and ΔSrxn is negative, ΔGrxn will not be negative (favorable) at any temperature.
When ΔHrxn is posttive and ΔSrxn is positive, ΔGrxn will be negative (favorable) at low temperatures.

Graphing Free Energy as a Function

Upon inspection, the equation ΔGrxn = ΔHrxn - TΔSrxn can be proven to represent a linear function, where ΔGrxn is calculated over a series of temperatures while Hrxn and ΔSrxn remain constant.  Recall the equation y = mx + b represents a linear equation, where each variable corresponds to a variable in ΔGrxn = ΔHrxn - TΔSrxn.  Rewriting the free energy equation as ΔGrxn = - TΔSrxn + ΔHrxn makes it easier to see the parallel.  Despite the position of T, it is not the slope of the equation.  -ΔSrxn will represent m or the slope. This is because slope, like ΔSrxn, is a constant for a particular reaction. The negative sign in fron of ΔSrxn is a consequence of the -TΔSrxn term in the original equation. Since temperature cannot be negative, the sign is applied to ΔSrxn.Since x is allowed to fluctuate (as is the temperature) T corresponds to x.  This leaves ΔHrxn, which must correspond to b or the y-intercept of the equation.  This can be used for determining a range of temperatures for which a reaction will be spontaneous or not. As mentioned earlier, this will work over a small temperature range since ΔHrxn and ΔSrxn will change considerably over a large temperature range.

Relationship Among ΔGrxn, K, and Ecell

ΔGrxn can be calculated using one of three equations:
ΔGrxn = ΔHrxn - TΔSrxn
ΔGrxn = -RTlnK
ΔGrxn = -nFEcell

These equations can be substituted for one another, and from that the following relationships can be ascertained:

If a reaction is spontaneous, then ΔGrxn is negative, K approaches infinity, and Ecell is positive.
If a reaction is nonspontaneous, then ΔGrxn is positive, K approaches zero, and Ecell is negative.
If a reaction is at equilibrium, then ΔGrxn is zero, K equals 1, and Ecell is zero.

Further Reading

(1) Raizen, Mark G. Demons, Entropy and the Quest for Absolute Zero. Scientific American, March 2001, pp 54-59.
(2) Sanders, Laura. Molecules Get Superchilly Reaction. Science News, April 10, 2010, p 11.
(3) Witze, Alexandra. Quantum Rules Get Mechanical. Science News, April 10, 2010, p 10.