This presentation includes Gibbs Helmholtz equation and its application. Explanation of the term Chemical potential and Gibbs Duhem equation

1. Dr. Y. S. THAKARE
M.Sc. (CHE) Ph D, NET, SET
Assistant Professor in Chemistry,
Shri Shivaji Science College, Amravati
Email: yogitathakare_2007@rediffmail.com
B Sc- II Year
SEM-III
PAPER-III
PHYSICAL CHEMISTRY
UNIT- V -THERMODYNAMICS AND EQUILIBRIUM
Topic- Gibbs-Helmholtz equation and partial molar
free energy
31-August -20 1

2. VARIATION OF HELMHOLTZ FUNCTION / WORK FUNCTION WITH TEMPERATURE AND VOLUME
𝐴 = 𝐸 − 𝑇𝑆 … (1)
𝑑𝐴 = −𝑃𝑑𝑉 − 𝑆𝑑𝑇 … . . (8)
(a) If temperature is constant dT = 0. Then (8) becomes
(𝑑𝐴) 𝑇 = −(𝑃𝑑𝑉) 𝑇
𝜕𝐴
𝜕𝑉 𝑇
= −𝑃 ….(9)
(b) If volume is constant dV = 0. Then (8) becomes
(𝑑𝐴) 𝑉= (−𝑆𝑑𝑇) 𝑉
𝜕𝐴
𝜕𝑇 𝑉
= −𝑆 … (10)
Equation (9) and (10) gives the variations of work functions with temperature and volume.
VARIATION OF GIBB’S FUNCTION / FREE ENERGY WITH TEMPERATURE AND PREESSURE
𝐺 = 𝐻 − 𝑇𝑆 … (1)
𝑑𝐺 = 𝑉𝑑𝑃 − 𝑆𝑑𝑇 … (7)
(a) If temperature is constant dT = 0. Then (7) becomes
(𝑑𝐺) 𝑇 = (𝑉𝑑𝑃) 𝑇
𝜕𝐺
𝜕𝑃 𝑇
= 𝑉 ….(8)
(b) If pressure is constant dP = 0. Then (10) becomes
(𝑑𝐺) 𝑃= (−𝑆𝑑𝑇) 𝑃
𝜕𝐺
𝜕𝑇 𝑃
= −𝑆 … … (9)
Equation (8) and (9) gives the variations of work functions with temperature and pressure.17 -August -20
31-August -20
Dr. Yogita Sahebrao Thakare

3. SPONTANEITY IN TERMS OF FREE ENERGY
−𝑑𝐺 ≤ 𝑉𝑑𝑃 − 𝑆𝑑𝑇
At constant temperature and pressure, dT = 0 and dP = 0
𝑑𝐺 ≤ 0 … (8)
The equation (8) represents that, if in a chemical reaction change if free energy is negative i.e.
less than zero then the reaction will be spontaneous or feasible. And at equilibrium, free
energy change will be zero.
GIBB’S HELMHOLTZ EQUATION
The variation of free energy change with temperature at constant pressure is given by the
equation
𝜕𝐺
𝜕𝑇 𝑃
= −𝑆 … 1
i.e. 𝑑𝐺 = −𝑆𝑑𝑇
∴ 𝑑𝐺1= −𝑆1 𝑑𝑇 … . 2
𝑑𝐺2= −𝑆2 𝑑𝑇 … . 3
Consider equation (3) - (2) 𝑑𝐺2− 𝑑𝐺1 = −(𝑆2 − 𝑆1)𝑑𝑇
𝑑(∆𝐺) = −∆𝑆𝑑𝑇
At constant pressure,
𝑑(∆𝐺)
𝑑𝑇
= −∆𝑆 … (4)
We know that ∆𝐺 = ∆𝐻 − 𝑇∆𝑆
−𝑇∆𝑆 = ∆𝐺 − ∆𝐻
−∆𝑆 =
∆𝐺−∆𝐻
𝑇
… (5)31-August -20 Dr. Yogita Sahebrao Thakare

4. Substituting the value of ∆S from equation (5) to (4)
∴
𝒅(∆𝑮)
𝒅𝑻
=
∆𝑮 − ∆𝑯
𝑻
𝑻
𝒅(∆𝑮)
𝒅𝑻
= ∆𝑮 − ∆𝑯
i.e. ∆𝑮 = ∆𝑯 + 𝑻
𝒅(∆𝑮)
𝒅𝑻 𝑷
… 6
This equation is known as Gibb’s–Helmholtz equation in terms of free energy and enthalpy.
By analogy Gibb’s-Helmholtz equation in terms of work function and internal energy can be
derive at constant volume which can be derived at constant volume which can be given as
∆𝑨 = ∆𝑬 + 𝑻
𝒅(∆𝑨)
𝒅𝑻 𝑽
… 7
31-August -20 Dr. Yogita Sahebrao Thakare

5. APPLICATIONS OF GIBB’S-HELMHOLTZ’S EQUATION
(a) Calculations of E.M.F. of a reversible cell:
The decrease in free energy produced by the passage of nF coulombs of electricity through a
reversible cell is given by
−∆𝐺 = 𝑛𝐸𝐹
Where, n = number of electrons lost or gained
E=E.M.F. of the reversible cell
F= Faraday of electricity (96500 coulomb)
Putting the values of ∆𝐺 in Gibb’s Helmholtz equations, we get
−𝑛𝐸𝐹 = ∆𝐻 + 𝑇
𝑑(−𝑛𝐸𝐹)
𝑑𝑇 𝑃
−𝑛𝐸𝐹 = ∆𝐻 − 𝑛𝐹 𝑇
𝑑𝐸)
𝑑𝑇 𝑃
𝐸 = −
∆𝐻
𝑛𝐹
+ 𝑇
𝑑𝐸
𝑑𝑇 𝑃
Here E gives E.M.F. of a reversible cell
(b) Calculations of enthalpy change
The Gibb’s-Helmholtz equation can be used for calculating the enthalpy changes
occurring in isothermal reaction.
31-August -20 Dr. Yogita Sahebrao Thakare

6. STANDRED FREE ENERGY (∆𝐆) 𝟎
It is defined as “The change in free energy when all the reactant are in their standard
state of unit pressure or unit concentration are converted at 250C in to the products, which are
also in their standard states”.
The ∆𝐺0 for a reaction can be written as
∆𝐺0
= ∆𝐺 𝑝𝑟𝑜𝑑𝑢𝑐𝑡
0
− ∆𝐺 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡
0
∆𝐺0
= ∆𝐻0
− 𝑇∆𝑆0
Where, ∆𝐻0
and ∆𝑆0
are the standard change in the enthalpy and standard change in entropy
respectively.
31-August -20 Dr. Yogita Sahebrao Thakare

7. CHEMICAL POTENTIAL OR PARTIAL MOLAR FREE ENERGY OR THERMODYNAMICS OF OPEN
SYSTEM
The thermodynamics properties U, H, S, A and G are extensive properties because their values
depends on the number of moles of the system.
Consider the extensive property, free energy. Let it be
𝐺 = 𝑓(𝑇, 𝑃, 𝑛1, 𝑛2, 𝑛3, … . . ) ….(i)
𝑑𝐺 =
𝜕𝐺
𝑑𝑇 𝑃,𝑛1,𝑛2,𝑛3 ,
𝑑𝑇 +
𝜕𝐺
𝑑𝑃 𝑇,𝑛1,𝑛2,𝑛3 ,
𝑑𝑃 +
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,
𝑑𝑛1 +
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,
𝑑𝑛2+ . . . . (𝑖𝑖)
If the temperature and pressure of the system kept constant, then dT = 0 and dP = 0
(𝑑𝐺) 𝑇,𝑃=
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,
𝑑𝑛1 +
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,
𝑑𝑛2 + ⋯ (𝑖𝑖𝑖)
Each derivative on the right hand side is called partial molar property and it is represented by
putting the bar over the symbol of that particular property i.e 𝐺1, 𝐺2, 𝐺3, for the 1st , 2nd, 3rd,
etc. respectively. Thus,
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,
= 𝐺1
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,
= 𝐺2 , etc. In general ith component
𝜕𝐺
𝑑𝑛𝑖 𝑇,𝑃,𝑛1,𝑛2,𝑛3 ,….
= Gi … . (𝑖𝑣)
31-August -20 Dr. Yogita Sahebrao Thakare

8. This quantity called as partial molar free energy or chemical potential and is usually represented
by symbol µ. Thus
µi
= Gi =
𝜕𝐺
𝑑𝑛𝑖 𝑇,𝑃,𝑛1,𝑛2,𝑛3 ,….
…(𝑣)
If dni = 1 mole, Then, µ = ( 𝑑𝐺) 𝑇,𝑝,𝑛1,𝑛2,𝑛3 ,….
The chemical potential of a constituents in a mixture is the increase in free energy which takes
place at constant temperature and pressure when one mole of that constituent is added to the
system, keeping the amount of all other constituent constant.
Equation (iii) can be written as
( 𝑑𝐺) 𝑇,𝑃= µ1
dn1 + µ2
dn2 + µ3
dn3 + ⋯ …(𝑣𝑖)
By the definition of definite composition represented by the number of moles 𝑛1, 𝑛2, 𝑛3, etc.
equation (iv) on integration gives
𝐺 𝑇,𝑃,𝑁 = n1µ1
+ n2µ2
+ n3µ3
+ ⋯ ….(vii)
The chemical potential of a constituents in a mixture is its contribution per mole to the total free
energy of the system of a constant composition at constant temperature and pressure. It may be
noted that whereas free energy is an extensive property the chemical potential is an intensive
property because it refers to one mole of substance.
Where, subscript N stands for constant composition.
31-August -20 Dr. Yogita Sahebrao Thakare

9. GIBB’S DUHEM EQUATION
We know that for an open system,
𝐺 = 𝑓(𝑇, 𝑃, 𝑛1, 𝑛2, 𝑛3, … . . ) ….(i)
Now, if there is small change in the temperature, pressure and the amounts of the constituents,
then the change in the property G is given by,
𝑑𝐺 =
𝜕𝐺
𝑑𝑇 𝑃,𝑛1,𝑛2,𝑛3 ,
𝑑𝑇 +
𝜕𝐺
𝑑𝑃 𝑇,𝑛1,𝑛2,𝑛3 ,
𝑑𝑃 +
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,
𝑑𝑛1 +
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,
𝑑𝑛2+. . . . (𝑖𝑖)
If the temperature and pressure of the system kept constant, then dT = 0 and dP = 0, so that
equation (ii) becomes
(𝑑𝐺) 𝑇,𝑃 =
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,
𝑑𝑛1 +
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,
𝑑𝑛2 + ⋯ (𝑖𝑖𝑖)
Putting
𝜕𝐺
𝑑𝑛1 𝑇,𝑃,𝑛2,𝑛3 ,….
= µ1
and
𝜕𝐺
𝑑𝑛2 𝑇,𝑃,𝑛1,𝑛3 ,….
= µ2
and so on in equation (iii), we get
(𝑑𝐺) 𝑇,𝑃
= µ1
𝑑𝑛1 + µ2
𝑑𝑛2 + ⋯ … (𝑖𝑣)
By the definition of definite composition represented by the number of moles 𝑛1, 𝑛2, 𝑛3, etc.
equation (iv) on integration gives
𝐺 𝑇,𝑃,𝑁 = n1µ1
+ n2µ2
+ n3µ3
+ ⋯
Differentiating this equation under the condition of constant temperature and pressure but
varying composition, we get,
31-August -20 Dr. Yogita Sahebrao Thakare

10. ( 𝑑𝐺) 𝑇,𝑃 = n1dµ1
+ µ1
𝑑𝑛1 + n2dµ2
+ µ2
𝑑𝑛2 + n3dµ3
+ µ3
𝑑𝑛3 + ⋯
(𝑑𝐺) 𝑇,𝑃 = (n1dµ1
+ n2dµ2
+n3dµ3
+ ⋯ ) + (µ
1
𝑑𝑛1 + µ2
𝑑𝑛2 + µ3
𝑑𝑛3 + ⋯ ) …...(v)
Substituting the value of (iv) in (v), we get
(𝑑𝐺) 𝑇,𝑃 = (n1dµ1
+ n2dµ2
+ n3dµ3
+ ⋯) + (𝑑𝐺) 𝑇,𝑃
OR n1dµ1
+ n2dµ2
+ n3dµ3
+ ⋯ = 0
OR ni 𝑑µ𝑖
= 0
This equation which is applicable to a system under constant temperature and pressure is called
Gibb’s-Duhem equation.
31-August -20 Dr. Yogita Sahebrao Thakare