Introduction to free energy functions

INTRODUCTION TO FREE ENERGY
FUNCTIONS
Dr Bincy Joseph
ASSISTANT PROFESSOR
DEPARTMENT OF CHEMISTRY
ST. MARY’S COLLEGE,
THRISSUR
Introduction to Free energy Functions,Dr.Bincy Joseph,St.Mary's college,Thrissur.

Free Energy Functions - Helmholtz
energy and Gibbs energy
For expressing spontaneity criterion for a system alone, introduces new
thermodynamic function ,under two constraints
i. constant T & V – Helmholtz Free energy -A
ii. constant T & P – Gibbs Free energy-G

Work Function/Helmholtz free
energy /Helmholtz energy
A = U-TS
ΔA = A2-A1
= (U2 - TS2) –(U1- TS1)
= (U2- U1) – T (S2-S1)
ΔA = ΔU - TΔS

Physical significance of Helmholtz energy
ΔA = ΔU - TΔS
ΔA = ΔU- qrev
First law
Wrev = ΔU - qrev
ΔA = Wrev
- ΔA = -Wrev
-(ΔA)T,V = -Wrev

Free Energy /Gibbs free Energy
G= H-TS
ΔG = G2-G1
= (H2-TS2) – (H1-TS1)
= (H2-H1) – T(S2-S1)
ΔG = ΔH - TΔS

Physical significance of Free
energy (Gibbs Energy)
ΔU = q + w ( for a process carried out at constant P & T)
ΔU = q + wexpansion + w non-expansion
ΔU = q – PdV + wnon-expansion
q = ΔU + PdV - wnon-expansion
q = ΔH – wnon-expansion
For a process taking place reversibly at constant temperature,
qrev = TΔS
TΔS= ΔH – wnon-expansion
ΔH – TΔS = wnon-expansion
ΔG = wnon-expansion ΔG = wuseful - ΔG = - wuseful.
- (ΔG)T,P = - wuseful.

Relationship between G and A
G = H-TS
G = U + PV-TS
Since A = U-TS
G = A + PV

Variation of Helmholtz energy as a
function of T &V
A = U -TS
dA = dU – TdS - SdT
= (TdS - PdV) – TdS - SdT
dA = - SdT - PdV

Variation of Gibbs energy with temperature at
constant pressure -The Gibbs Helmholtz equation.
G= H -TS
H=U + PV
G= U + PV-TS
On differentiation
dG=dU + PdV + VdP – TdS - SdT
First Law dq = dU - dw
dq = dU + PdV
Second Law, dqrev = TdS

The Gibbs Helmholtz equation…..
TdS = dU + PdV
dG= TdS + VdP – TdS - SdT
dG= VdP - SdT
At constant P, dP=?
dG = -SdT
This can be represented as
[
𝜕𝐺
𝜕𝑇
]p = -S

The Gibbs Helmholtz equation…..
For initial state [
𝜕𝐺1
𝜕𝑇
]p= -S1
For final state [
𝜕𝐺2
𝜕𝑇
]p= -S2
[
𝜕𝐺2
𝜕𝑇
]p - [
𝜕𝐺1
𝜕𝑇
]p= - (S2 –S1)
[
𝜕(𝐺2−𝐺1)
𝜕𝑇
]p = -ΔS
[
𝜕(Δ𝐺)
𝜕𝑇
]p = -ΔS
ΔG = ΔH-TΔS
ΔG = ΔH + T [
𝜕(Δ𝐺)
𝜕𝑇
]p

The Gibbs Helmholtz equation…..Significance
 Calculate ΔH for a process taking place at constant pressure if ΔG at two
different temperatures are given.
 Calculate ΔG for a process at a certain temperature if ΔG at another
temperature and ΔH are given.

Other Applications…
 Calculation of the parameters related to the energies of
electrochemical cell.
ΔG = -nFEcell
 Derivation of other important equations of physical
chemistry.

Variation of Gibbs energy with P at constant T -
Gibbs energy change for an ideal gas in a reversible
isothermal process
For a system undergoing a reversible change
dG= VdP – SdT
At constant T, dT=0
dG = VdP
ΔG = 𝑝1
𝑝2
𝑉𝑑𝑃
ΔG = 𝑝1
𝑝2 𝑅𝑇
𝑃
𝑑𝑃
ΔG =RT ln P2/P1
ΔG =RT ln V1/V2

Gibbs energy change for an ideal gas in a
reversible isothermal process…
For n mole of an ideal gas
ΔG = nRT ln P2/P1
ΔG = nRT ln V1/V2
ΔG =2.303 nRT log P2/P1
ΔG =2.303 nRT log V1/V2

Maxwell’s relations
Combined form of first and second law
dU=TdS – PdV ….i (Clausius equation)
Relationship between enthalpy and entropy
dH= TdS+ VdP ……ii
Variation of the workfunction as a function of T & V is given by
dA= -SdT-PdV …..iii
Variation of Gibbs energy as a function of T & P is
dG= VdP-SdT …..iv

Maxwell’s relations….
Clausius equation, dU = TdS – PdV
Imposing the constraint of constant V ,dV=0
(𝜕U/𝜕S)v = T
Differentiating above eqn w.r.to V ,keeping S constant
(𝜕2U/𝜕S. 𝜕V) = (𝜕T/𝜕V)s
Imposing the constraint of constant S, dS = 0
(𝜕U/𝜕V)s = -P
(𝜕2U/𝜕V. 𝜕S) = -(𝜕P/𝜕S)v
(𝜕T/𝜕V)s = - (𝜕P/𝜕S)v

Maxwell’s relations….

Gibbs energy (Free energy) criteria for
spontaneity and thermodynamic equilibrium
Logical Approach
A spontaneous process occurs with a decrease in Gibbs energy (free energy).
For a spontaneous process (ΔG)T,P is negative.
(ΔG)T,P< 0 ,for a spontaneous process.
There is no change in Gibbs energy of the system ,if the process is at equilibrium.
(ΔG)T,P= 0 ,for equilibrium.
For a non spontaneous process, (ΔG)T,P is ….

Mathematical Approach
G = H-TS H = U + PV G = U+PV-TS
On differentiation
dG= dU + PdV + VdP – TdS - SdT
From first Law
dU= dq+ dw
=dq- PdV
dG= dq- PdV + PdV + VdP – TdS - SdT
dG= dq+ VdP – TdS – SdT, at constant T & P, dP=o, dT=0.Hence
(dG)T,P = dq – TdS

Gibbs energy (Free energy) criteria for
reversible process equilibrium state
When the change is reversible, dq = TdS
(dG)T,P= 0
(ΔG)T,P= 0

Conditions for a process to be spontaneous
A process would be spontaneous only if the Gibbs Free
energy change (ΔG) in the process is negative.
ΔG = ΔH-TΔS

Thank You

Introduction to free energy functions

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