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, DS, depends on the initial and final entropies of the
system
DS = Sfinal - Sinitial
If DS > 0 => Sfinal > Sinitial
If DS < 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.
DS a q
A greater change in disorder occurs if the temperature at
which process is carried out is low than when it is high.
DS a 1/T
The change in entropy of the system is:
qrev
DS =
T
T is the temperature
qrev is the energy transferred as heat during the process
when the system follows a reversible path.
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,
DH = qrev
Entropy of vaporization
DHvap
DSvap =
Tb
DHfus
DSfus =
Tf
If the liquid, solid, gas are in their standard states, defines
the standard entropy, DSo
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
DSovap = 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
DSo = S nSo (products) - S nSo (reactants)
N2(g) + 3H2 (g) --> 2NH3(g)
DSo = 2So (NH3(g)) - So (N2(g)) - 3So (H2(g))
DSo = -198.3 J/K
As expected DSo 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, DSuniverse:
DSuniverse = DSsystem+ DSsurrounding
The second law of thermodynamics states:
For a process to be spontaneous, the entropy of the universe
must increase.
DSuniverse = DSsystem+ DSsurrounding > 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 DSuniverse > 0, even though in (b) DSsyst< 0
Process is endothermic and spontaneous
DSsyst > 0; DSsurr < 0, however DSuniverse > 0.
If DSuniverse < 0 => non-spontaneous process
If DSuniverse > 0 => spontaneous process
If DSuniverse = 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; DSuniverse 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-TS
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
DGsyst = DHsyst - TDSsyst
Can DGsyst provide a criterion for spontaneity?
Assume that surroundings are so large that the temperature
and pressure remains constant.
If enthalpy change of system is DHsyst, then qsurr = -DHsyst
At constant pressure and temperature:
DHsyst
T
DSsurr= -
DSuniverse = DSsyst+ DSsurrounding > 0 for a spontaneous process
DSuniverse = DSsyst
-
DHsyst
T
>0
Mutliplying by T
T DSsyst - DHsyst > 0 for a spontaneous reaction
=> DHsyst - T DSsyst < 0 for a spontaneous reaction
=> DGsyst < 0 for a spontaneous reaction
For a process at constant P and T
DGsyst < 0 => spontaneous
DGsyst > 0 => non spontaneous
DGsyst = 0 => equilibrium
The sign of DGsyst is determined by the relative magnitudes
of DHsyst and TDSsyst accompanying the process
DGsyst = DHsyst - TDSsyst
Or simply: DG = DH - TDS
Equilibrium state corresponds to the lowest point