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 Hrxn and is found by subtracting the enthalpy of the reactants from the enthalpy of the products: 

Hrxn = ΣHproducts –  ΣHreactants

The Greek letter sigma, 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.  You should turn use your handbook to find the table labeled “Standard Thermodynamic Values at 25°C.”  You will 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. 

The reaction between sodium peroxide and zinc, using water as a catalyst, is highly exothermic.
Find sodium sulfide, or Na2S.  As you can see, its enthalpy of formation is  -373.2128 kJ/mol.  This means that when one mole of sodium sulfide is formed from its constituent elements (sodium and sulfur), -373.2128 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:

Srxn = ΣSproducts –  ΣSreactants

The decomposition of mercury(II) oxide creates positive entropy, because there are more moles of product than reactant and a liquid and gas are produced from a solid.
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 Grxn values provided in the table are only viable at 25ºC (298.15 K).  For all temperatures, including 25ºC, the following equation can be used: 

Grxn = Hrxn – TSrxn

 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.
  • A positive Grxn indicates the reaction is nonspontaneous, a negative Grxn is spontaneous, and a value close to zero indicates an equilibrium.
The Relationship between Spontaneity and the Sign of Enthalpy and Entropy Values
Consider the following table below:
  +S -S
+H G could be positive or negative.  However, the reaction will be most favorable at lower temperatures. G is always negative.  The reaction will not work at any temperature.
-H G is always positive.  The reaction will work at any temperature. G could be positive or negative.  However, the reaction will be most favorable at higher temperatures.
Graphing Free Energy as a Function
Upon inspection, the equation Grxn = Hrxn – T Srxn can be proven to represent a linear function, where Grxn is calculated over a series of temperatures while Hrxn and Srxn remain constant.  Recall the equation y = mx + b represents a linear equation, where each variable corresponds to a variable in Grxn = Hrxn – T Srxn.  Rewriting the free energy equation as Grxn = – T Srxn + Hrxn makes it easier to see the parallel.  Despite the position of T, it is not the slope of the equation.  Since the slope must be constant, ΔSrxn will represent m or the slope.  Since x is allowed to fluctuate (as is the temperature) T corresponds to x.  This leaves Hrxn, 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. Realize this will work over a small temperature range since S will change considerably over a large temperature range.
Example Problem

2 C2H6 (g) + 7 O2 (g) 4 CO2 (g) + 6 H2O (g) at 300 degrees Celsius

 Hrxn = ΣHproducts –  ΣHreactants

 

[4(-393.5) + 6(-241.8)] – [-84.0 + 7(0.0)]

-3024.8 – -84.0

-2940.0 kJexothermic (because enthalpy is negative)

Note that the equation can now be rewritten showing the enthalpy value as a product, because heat is being produced:

2 C2H6 (g) + 7 O2 (g) 4 CO2 (g) + 6 H2O (g) + 2940.0 kJ

Srxn = ΣSproducts –  ΣSreactants

 

[4(213.8) + 6(188.8)] – [229.2 + 7(205.2)]

1988.0 – 1665.6

322.4 J/K creates disorder (because entropy is positive)

Grxn = Hrxn – TSrxn

-2940.0 – (573.15)(.3224)

-3124.8 kJ spontaneous at this temperature (because the free energy is negative)

Phase Changes
Enthalpy can be used to determine the energy needed to change the state of matter for a particular substance.  Specifically, the energy needed to cause a solid/liquid phase change is called the enthalpy of fusion, Hfus.  When a substance changes from a solid to liquid, energy must be absorbed in order to separate the particles.  Consequently, a solid to liquid phase change is endothermic.  Since the reverse process must occur for a liquid to solid phase change, that process is exothermic.  The value for Hfus does not change, only its sign changes.  A liquid to gas change, measured by the enthalpy of vaporization (Hvap), can be viewed similarly.  To change a liquid to a gas, energy must be absorbed and the process is endothermic, hence a positive enthalpy value.  Condensation is an exothermic process and the enthalpy value is negative.
Relationship Among ΔGrxn, K, and Ecell
Reaction is... ΔGrxn K Ecell
nonspontaneous positive very small (approaches zero) negative
equilibrium 0 1 0
spontaneous negative very large (approaches infinity) positive
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.