Entropy and The Second Law of Thermodynamics

For a chemist, one of the main reasons for studying thermodynamics is to be able to predict whether or not a reaction will occur when reactants are added together under a given set of conditions (temperature, pressure and concentration).  A reaction that does  occur under the given set of conditions is called a spontaneous reaction.  If a reaction does not occur under specified conditions, it is said to be nonspontaneous.  The following few examples are spontaneous chemical and physical processes that can be observes in everyday life:

Water runs downhill, but never up, spontaneously.

A lump of sugar spontaneously dissolves in a cup of coffee, but dissolved sugar does not spontaneously reappear in its original form.

Water freezes spontaneously below 0^oC, and ice spontaneously melts above 0^oC.

Heat flows from a hotter object to a cooler one, but the reverse never happens spontaneously.

Iron exposed to water and oxygen forms rust, but rust does not spontaneously change back to iron.

These examples show that processes that occur spontaneously in one direction cannot, under the same conditions, also take place spontaneously in the opposite direction.

From a study of these examples and many, many others we conclude that being exothermic favors a reaction or process being spontaneous, but does not guarantee it.  In other words, we cannot decide whether or not a chemical reaction will occur spontaneously solely on the basis of energy changes in the system. 

In order to predict the spontaneity of a process, we need to know two things about the system.  One is the change in enthalpy, the other is entropy (S), which is a measure of the randomness or disorder of a system.  The more disordered a a system, the larger its entropy.  Like enthalpy and internal energy, entropy is a state function.  The change in entropy of a system, ΔS = S_final - S_initial, depends only on the initial and final states of the system and not on the particular pathway by which the system changes.  A positive ΔS indicates an increase in randomness or disorder.  The entropy change of a system is defined in terms of the heat transferred to the system during a reversible path from initial state to final state, regardless of how the process actually takes place.  We refer to this heat transferred as q_rev where the subscript 'rev' means that the path between the states is reversible.  If a process occurs at constant temperature, the entropy change is defined as q_rev divided by the absolute temperature:

ΔS = q_rev / T  (Constant T)

An example of a reversible process that occurs at a constant temperature is a phase change at the temperature at which the two phases are in equilibrium, such as the boiling of water at 100^oC.

 Example 1

Calculate the entropy change when 1 mol of water is converted to 1 mol of steam at 1 atm pressure.

The amount of heat transferred to the system during this process is the heat of vaporization, ΔH_vap, and the temperature at which the process is reversible is the normal boiling point.  For water  ΔH_vap = +40.67 kJ/mol and T_b = 100^oC = 373.2 K.

        ΔHvap   (1 mol)(+40.67 kJ/mol)(1000 J/1 kJ)
ΔSvap = ----- = ----------------------------------- = 109.0 J/K
         Tb                 (373.2 K)

We are now in a position to discuss why certain processes are spontaneous.  The connection between entropy and spontaneity is expressed by the second law of thermodynamics:  The entropy of the universe increases in a spontaneous process and remains unchanged in an equilibrium process.  Since the universe is made up of the system and the surroundings, the entropy change in the universe for any process is the sum of the entropy change in the system and the entropy change in the surroundings.

For a spontaneous process:    ΔS_univ = ΔS_sys + ΔS_surr > 0

For a equilibrium process:      ΔS_univ = ΔS_sys + ΔS_surr = 0

Entropy Changes in the System

 To calculate   ΔS_univ we need to know both ΔS_sys and ΔS_surr.  First lets focus on  ΔS_sys.  The standard entropy values of a large number of compounds have been measured in J / K · mol.  To calculate ΔS^o_rxn (which is ΔS_sys), we look up their values in a table and use the equation:

 ΔS^o_rxn = n S^o_(products) -  m S^o_(reactants) 

This equation is similar to the one we have used to determine the enthalpy of a reaction.  Now we will use the second law to decide if the reaction for the synthesis of ammonia is spontaneous at 25^oC.

N₂(g)  +  3 H₂(g) ==>  2 NH₃(g)       ΔH^o = -92.6 kJ

use the equation above to get ΔS^o_rxn or ΔS_sys  

ΔS^o_rxn = [2 S^o(NH₃)] - [S^o(N₂) + 3 S^o(H₂)]

            = (2 mol)(193 J/K · mol) - [(1 mol)(192 J/K · mol) + (3 mol)(131J/K · mol)]

           = 199 J/K

We can calculate ΔS_surr from the following relationship.

            -ΔHsys      -(-92.6 x 1000)J
ΔSsurr = -------- = ---------------- = 311 J/K 
               T          298.2 K  

The change in entropy of the universe is 

ΔS_univ = ΔS_sys + ΔS_surr

            = - 199 J/K  + 311 J/K = 112 J/K

Since ΔS_univ is positive, we predict that the reaction is spontaneous.  We should note however that just because a reaction is spontaneous, does not mean that it will have a rate that makes it practical.