The entropy, S, of a system quantifies the degree of disorder or randomness in the system; larger the number of arrangements available to the system, larger is the entropy of the system.

Entropy is a state function; the change in entropy during a process, S, depends on the initial and final entropies of the system

S = Sfinal - Sinitial

If S > 0 => Sfinal > Sinitial

If S < 0 => Sfinal < Sinitial

Since entropy is related to the number of possible states a system can occupy, all states being equal in energy, allows a determination of an absolute value of entropy

S = k ln W

W is the number of ways, or microstates, that the atoms or molecules of a sample can be arranged, each microstate being equal in energy.

k is Boltzmann constant (1.3807 x 10-23 J K-1)

Arrangement of four CO molecules

S = k ln W; W = 1 => S = 0 J K-1

Third Law of Thermodynamics: The entropy of a crystalline substance at equilibrium approaches zero as the absolute zero of temperature is approached.

16 ways of arranging four CO molecules, each being equal in energy

S = k ln W; S = 3.8281 x 10-23 J K-1

Molecules can translate, vibrate and rotate.

Each of these motions contribute to the molecule’s degree of freedom.

Molecules have three translational degrees of freedom, two or three rotational degrees of freedom; the number of vibrational degrees of freedom depends on the number of atoms.

Larger the degrees of freedom that the system can exhibit, higher is the entropy of the system.

For example, H2 has fewer vibrational degrees of freedom than CCl4.

Hence the entropy of H2(g) < entropy of CCl4(g)

For the same molecule: Sgas > Sliquid > Ssolid

Expression for Change in Entropy

A process accompanied by a large amount of heat, correlates to a larger degree of disorder.

S q

A greater change in disorder occurs if the temperature at which process is carried out is low than when it is high.

S 1/T

S =qrev

T

T is the temperature

qrev is the energy transferred as heat during the process when the system follows a reversible path.

The change in entropy of the system is:

A reversible path is one that can be reversed by an infinitesimal change in a variable.

In an irreversible process an infinitesimal change in a variable does not reverse the process

Reversible expansion: external pressure is matched to the pressure of the gas at every stage of the expansion; an infinitesimal change in the external pressure reverses the direction of the process

Irreversible expansion: expansion against an external pressure that differs by a finite amount from the pressure of the system; an infinitesimal change in the external pressure does not reverse the direction of the process.

Entropy Changes Accompanying Physical Changes

At the transition point, temperature remains constant.

At the transition temperature, the transfer of heat is reversible. Hence q = qrev

Since the transition takes place at constant pressure,

H = qrev

Svap =Hvap

Tb

Sfus =Hfus

Tf

If the liquid, solid, gas are in their standard states, defines the standard entropy, So

Entropy of vaporization

Entropy of fusion

Calculate the standard entropy of vaporization of acetone at its boiling point of 329.4 K? The standard heat of vaporization at its boiling point is 29.1 kJ/mol

Sovap = 2.91 x 104 J mol-1/329.4 K

= 88.3 J K-1 mol-1

Entropies Accompanying Chemical Change

The standard molar entropy, So, is the absolute entropy of one mole of a substance in its standard state.

Units of molar entropy - J K-1mol-1

Standard Reaction Enthalpies

So = nSo (products) - nSo (reactants)

N2(g) + 3H2 (g) --> 2NH3(g)

So = 2So (NH3(g)) - So (N2(g)) - 3So (H2(g))

So = -198.3 J/K

As expected So is negative for this reaction

Reactions that increase the number of gas phase molecules tend to be accompanied by positive changes in entropy.

Dissolution reactions are typically accompanied by an increase in entropy.

However, there are examples where the opposite is true:

MgCl2(s) --> Mg2+(aq) + 2Cl-(aq)

is accompanied by a negative change in entropy.

Conditions for spontaneous processes

While it is true that most spontaneous processes occur with an increase in entropy of the system, there are examples of spontaneous processes that appear to occur with decreasing entropy.

For example, below 0oC, water spontaneously freezes even though the process is accompanied by a decrease in entropy.

When water freezes the heat liberated is taken up by the surroundings, whose entropy increases.

Change in entropy of the universe, Suniverse:

Suniverse = Ssystem+ Ssurrounding

The second law of thermodynamics states:

For a process to be spontaneous, the entropy of the universe must increase.Suniverse = Ssystem+ Ssurrounding > 0 for a spontaneous process

The entropy of an isolated system increases in any spontaneous process.

Processes in (a) and (b) are exothermic, and spontaneous since Suniverse > 0, even though in (b) Ssyst< 0

Process is endothermic and spontaneous

Ssyst > 0; Ssurr < 0, however Suniverse > 0.

If Suniverse < 0 => non-spontaneous process

If Suniverse > 0 => spontaneous process

If Suniverse = 0 => the process is at equilibrium

No process that produces order (a decrease in entropy) in a system can proceed spontaneously without producing an equal or greater disorder (increase in entropy) in its surroundings.

Unlike energy, entropy is not conserved; Suniverse is continually increasing.

The Gibbs Free Energy Function

To determine the entropy change of the surroundings is usually hard.

To determine if a process is spontaneous or not, we need to a way to account for the change in entropy that the system undergoes.

The Gibbs free energy function, G, allows us to focus on the changes of the thermodynamic properties of the system.

At constant pressure and temperature:

G = H - T S

Units of G - J

G is a state function, like H and S

The change in the free energy function accompanying a process undergone by a system at constant P and T is

Gsyst = Hsyst - TSsyst

Can Gsyst provide a criterion for spontaneity?

Assume that surroundings are so large that the temperature and pressure remains constant.

If enthalpy change of system is Hsyst, then qsurr = -Hsyst

At constant pressure and temperature:

Ssurr= - Hsyst

T

Suniverse = Ssyst+ Ssurrounding > 0 for a spontaneous process

- Hsyst

TSuniverse = Ssyst > 0

Mutliplying by T

T Ssyst - Hsyst > 0 for a spontaneous reaction

=> Hsyst - T Ssyst < 0 for a spontaneous reaction

=> Gsyst < 0 for a spontaneous reaction

For a process at constant P and T

Gsyst < 0 => spontaneous

Gsyst > 0 => non spontaneous

Gsyst = 0 => equilibrium

The sign of Gsyst is determined by the relative magnitudes of Hsyst and TSsyst accompanying the process

Gsyst = Hsyst - TSsyst

Or simply: G = H - TS

Equilibrium state corresponds to the lowest point