BiosciencesLecture6

Lecture 6: Entropy and Gibbs Free Energy Concept
Review:
o Concept of efficiency of engine
o The concept of entropy
o Entropy change during reversible and irreversible
paths.
Today:
o Entropy variation with
o Temperature
o Pressure
o Composition
o Reaction
The Carnot’s concept of engine revisited
Reservoir at T1
q1
w
q2
Reservoir at T2
Efficiency
q1  q2
T2

 1
q1
T1
Entropy
dq
S  
 0 for reversible cyclic path
T
S  0 for irreversib le path
S  k ln( W )
Entropy is a state variable!
Temperature dependence of Entropy
Using the usual conditions such as isobaric or isochoric
paths we can see that:
T 
C dT
dT
 dq 
S       v
 Cv 
 C v ln  2  isochroic path
T
T
T 
 T1 
T 
C dT
dT
 dq 
S       P
 CP 
 C P ln  2  isobaric path
T
T
T 
 T1 
Just as in case of ΔH the above formulae apply as long as
system remains in single phase. On the other hand if
system undergoes a phase transition, at constant
temperature and pressure.
qP , L  H Transition  Stran 
H Tr
TTr
Thus overall temperature dependence of the entropy will
include the changes involved at the phase transition.
T
dT H m
dT
S (T )  S (0 K )   C P ( s)

  C P (l )
T
Tm
T
o
TM
Tm
0
Such an approach is valuable for calculating entropy at
arbitrary temperature for a single component system.
Pressure Dependence of Entropy
For solids and liquids entropy change with respect to pressure
is negligible on an isothermal path. This is because the work
done by the surroundings on liquids and solids is miniscule
owing to very small change in volume. For ideal gas we can
readily calculate the entropy dependence on the pressure as
follows:
dS 
dq rev  dw P.dV


T
T
T
 E  0
d ( PV )  0  PdV  VdP  dV  
V
dP
P
V .dP
dP
 nR
T
P
P 
S  nR ln  2 
 P1 
dS  
Entropy of Mixing
Consider two non-reacting gases A and B initially at identical
pressure, P separated from each other. If we allow them to
mix they will do so spontaneously and it is difficult to separate
them. The mixture will have identical pressure P, but we can
think of it as if it’s made up sum of partial pressures of A and
B. We can calculate the net change in the entropy of this
system.
 X P
 X P
S A  n A R ln  A   n A R ln( X A ) S B  nB R ln  B   nB R ln( X B )
 P 
 P 
ST   R.(n A  nB )X A ln( X A )  X B ln( X B )
This is a very important relation used frequently in problems
involving solutions.
Entropy Changes involved in chemical reactions
Consider a general reaction
n A A  nB B  nC C  nD D
S  S Products  S Re ac tan ts
S  nC SC  nD S D  n A S A  nB S B
o S is the standard molar entropy of formation at
25ºC. Tables of heat of formation are in appendix
A5-A7. Unlike enthalpy of formation, entropy of
formation of elements in not zero at 25 C.
o Entropy increases in a reaction that leads to
increased number of particles and vice versa
o If the reactants are charged and the reaction takes
place in aqueous medium resulting in decreased net
charge, then it leads to an increase in entropy. This
is because polar water molecules are ordered in a
specific way around charged species.
o Whether, a reaction occurs spontaneously or not
cannot be predicted solely based on the change in
entropy.
Gibbs Free energy
Gibbs developed the precise criterion for a spontaneous
reaction or process, in general. Gibbs Free Energy is a
new measure of energy:
G  H  TS
Since it is made up of state variables, it is also a state
variable and is extensive in nature. In general this free
energy is especially valuable when we are considering
processes that are carried out at constant temperature
and pressure. Consider the following
dG  dH  TdS  S .dT
u sin g definition of H
dG  dE  P.dV  V .dP  TdS  S .dT
according to1st law and for reversible paths
dG  dq rev  dw  P.dV  V .dP  TdS  S .dT
according to 2nd law
dG  TdS  dw  P.dV  V .dP  TdS  S .dT
for dw which is comb int ion of mechanical and other type of work .
dG  dwrev  dwPV  P.dV  V .dP  S .dT
 dwrev  V .dP  S .dT
but at cons tan t T and P
dG  dwrev
Thus ΔG at constant temperature and pressure equals
the ability of system to perform useful work other than
the simple PV work.
Gibb’s criterion for Spontaneous Process.
Thus, at constant T and P, the system can do useful
work on the surrounding if the sign of wrev is negative.
This means that the sign of ΔG must be negative for the
process to take to place. On the other hand, if the
process is directed in the opposite direction, that is,
surrounding does work on the system then sign of ΔG is
positive. While if the system is in equilibrium with
surrounding then the maximum reversible work that
either surrounding or system can do is zero. Then ΔG is
zero. These are the three fundamental criteria.
G  0  Spon tan eous process at cons tan t T and P
G  0  non  spon tan eous process at cons tan t T and P
G  0  Equilibriu m at con tan t T and P