DOE notes

1. Part-9
The Gibbs Free Energy
and equilibria

2. • “Gibbs Free Energy” is energy that is still useful.
• A chemical reaction will occur if the Gibbs would decrease.
G = H - TS
Gibbs free energy is a measure of chemical energy
All chemical systems tend naturally toward states of minimum
Gibbs free energy
G = Gibbs Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as
randomness)
Gibbs free energy also known as the free enthalpy
Is a thermodynamic potential that measures the maximum or reversible work
that may be performed by a system at a constant temperature and pressure
(Isothermal, Isobaric)

3. Spontaneity and Gibbs Free Energy
• Gibbs Free energy is a measure of the spontaneity of a process
• ΔG is the free energy change for a reaction under standard state
conditions
• At constant temperature and pressure: ΔG = ΔH – TΔS
– an increase in ΔS leads to a decrease in ΔG
– if ΔG < 0, the forward reaction is spontaneous
– if ΔG > 0, the forward reaction is nonspontaneous
– if ΔG = 0, the process is in equilibrium

4. • The Gibbs Free Energy is generally agreed to be the
“weapon of choice” for describing (a) chemical reactions
and (b) equilibria between phases. It is defined as:
• G = H – TS = U + PV – TS (1)
Where H = Enthalpy
• U = Total internal energy
• T = [Absolute] Temperature
• S = Entropy
• Obviously dG = dU + PdV +VdP – TdS – SdT
The Gibbs Free Energy and equilibria

5. • Remember that thermodynamic variables come in pairs
One is “intrinsic” (does not depend on system size)
The other is “extrinsic” (depends on system size)
• Examples: P and V, T and S…
• Also G and n, the number of moles of stuff in the system.
• Hence G is the appropriate variable when material is moving between
phases
Note:

6. From the First Law of Thermodynamics
• dU = TdS – PdV
since dS = dQ/T and the mechanical work done on a system
when it expands is –PdV.
• Substituting into
• dG = dU + PdV +VdP – TdS – SdT
• Leaves: dG = -SdT + VdP
Clapeyron’s Equation

7. Closed System
• Closed system contains pure substance
– vapor
– condensed phase
• Phases co-exist in equilibrium.
Write the Free Energy Equation twice
• Once for each phase
• dGc = -ScdT + VcdP c refers to the condensed phase
• dGv = -SvdT + VvdP v refers to the vapor phase

8. Definition of chemical equilibrium between two phases
• Free energy is the same in both phases Gc = Gv
• Changes in free energy when some independent variable is
changed must be the same if they are to remain in equilibrium
dGc = dGv
-ScdT + VcdP = -SvdT + VvdP
(Sv - Sc )dT = (Vv- Vc)dP
• (Sv - Sc ) is the entropy change that takes place when material moves from
the condensed phase to the vapor
• ΔS = ΔQ/T where ΔQ is the amount of heat required per mole of material
moved between the phases
• ΔQ is just the heat of vaporization!

9. • dP/dT = (Sv – Sc)/(Vv – Vc) = ΔHv/(TΔV)
This is the Clapeyron equation
• It relates the change in pressure of a vapor to the temperature
in a closed, mono-component system to the heat of
vaporization, system temperature and molar volume change of
the material on vaporization.
dP S
or
dT V



From the Clapeyron’s Equation we can calculate phase diagrams.
H=U+PV=Q

10. Creating of an Ideal Gas
• For lack of a better model, we treat most vapors as ideal gases, whose
molar volume is given by:
• V/n = RT/P
• Alternatively, equation of state is needed
• Molar volume of gas is typically factor of 500 larger than condensed phase
• Hence Vc is negligible in comparison
Substituting and Integrating
dP = (ΔHv/Vv)dT/T = (PΔHv/RT)dT/T
dP/P = ΔHv/R)dT/T2
ln(P(T)/ P0) = -(ΔHv/R)(1/T – 1/T0)
P(T) = P0 exp(-ΔHv/R(1/T – 1/T0))
Integrating

11. • The vapor pressure in equilibrium with a condensed phase
increases exponentially (sort of: exp(-1/T) isn’t exactly an
exponential!) with temperature from zero up to the critical
temperature.
• Deviations from linearity on the log-log plot
– Temperature dependence of the heat of vaporization
– exp (-1/T) isn’t really linear in the exponent.

12. Heat of Vaporization from CRC Data
Log10p(Torr) = -0.2185*A/T + B
Vapor Pressure of Water
Temperature (C)
-20 0 20 40 60 80 100 120
Vapor
Pressure
(Torr)
0.1
1
10
100
1000
10000
"Normal boiling point"

13. 1. Determine the vapor pressure at 77 K for
a. Water
b. Carbon monoxide
2. What is the boiling point of water in a vacuum system at 10-6 Torr?
HW
3. In the chemical equation G = H - TS, the term G stands for
A) entropy
B) the reactants
C) enthalpy
D) free energy
E) the products