C8863 Free Energy Calculations -1C8863
Free Energy Calculations
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
JS/2022 Present Form of Teaching: Rev2
Lesson 1
Phenomenological thermodynamics
(Overview)
C8863 Free Energy Calculations -2-
Overview
C8863 Free Energy Calculations -3-
Thermodynamics
Or what you should already know….
C8863 Free Energy Calculations -4The
system and its environment
system
environment
system - the part of space and its material contents, which is the subject of
thermodynamic consideration
the system is separated from the
environment by real or fictional walls
System types Description
isolated system walls protects exchange of matter and energy with the
environment
closed system walls protects exchange of matter to the environment, but it can
exchange energy with it
open system it can exchange matter and energy with the environment
C8863 Free Energy Calculations -5System
state and its properties
System state can be described by properties (mass, volume, temperature, pressure,
etc.), which are needed for the full state description.
Thermodynamic properties (state variables or state quantities) are state functions. The
state functions ​​do not depend on the way how the system got into the
given state.
▪ Mass (m)
▪ Energy (E)
▪ Enthalpy (H)
▪ Internal energy (U)
▪ Gibbs free energy (G)
▪ Helmholtz free energy (F)
▪ Exergy (B)
▪ Entropy (S)
▪ Pressure (p)
▪ Temperature (T)
▪ Volume (V)
▪ Particle number (ni)
List of selected state functions:
Heat and work are NOT state functions!
C8863 Free Energy Calculations -6Thermodynamic
process and equilibrium
Thermodynamic process corresponds to system state change. It can represent a change in
volume, temperature, pressure, or change in composition as a result of
chemical reaction, phase separation, phase transition, etc.
Thermodynamic equilibrium is a state in which no state function changes over time.
(Chemical or other transformations may still take place in the system.
However, these must take place in conjunction so that they do not affect
the state of the system as a result.)
Thermodynamic laws:
➢ 0th law about thermodynamic equilibrium of multiple systems
➢ 1st law energy conservation law
➢ 2nd law about the spontaneity of events
➢ 3rd law about absolute entropy
https://en.wikipedia.org/wiki/Laws_of_thermodynamics
C8863 Free Energy Calculations -7The
first law
WdQddU +=
change of internal energy
of the system
heat exchanged with the environment
(form of energy)
work done
(form of energy)
It is a generalization of the energy conservation law to dissipative systems, i.e., such
systems that exchange heat and work with their surroundings.
The first law postulates internal energy as a state variable.
For closed systems with no change in chemical composition, the change of internal energy
is sum of exchanged heat and work done:
Sign convention for energy change:
+ (positive) - the system receives energy
- (negative) - the system releases energy
complete differential (U is a function of
system properties, a state function)
d
incomplete differential (Q and W are not
state functions)
d
Since U and W are well defined, the first law can also be seen as a definition of heat.
C8863 Free Energy Calculations -8The
first law - two notations
IUPAC (Chemists) Physicists
𝑑𝑈 = 𝑑𝑄 − 𝑑𝑊𝑑𝑈 = 𝑑𝑄 + 𝑑𝑊
W is the work done by the systemW is the work done on the system
https://goldbook.iupac.org/terms/view/I03103
see IUPAC Gold Book
In this course, we will use sign notation recommended by IUPAC.
C8863 Free Energy Calculations -9-
Heat
In thermodynamics, heat (Q) is energy in transfer to or from a thermodynamic system, by
mechanisms other than thermodynamic work or transfer of matter.
Quantity of heat transferred can be measured by calorimetry or determined
through calculations based on other quantities.
Heat capacity or thermal capacity (C) is a physical property of
matter, defined as the amount of heat to be supplied to an
object to produce a unit change in its temperature.
𝐶 = lim
Δ𝑇→0
Δ𝑄
Δ𝑇
At constant volume (isochoric process, dV = 0):At constant pressure (isobaric process):
The heat supplied to the system contributes
to both the work done and the change in
internal energy.
The heat supplied contributes only to the
change in internal energy.
𝑑𝑄 = 𝑑𝑈 + 𝑝𝑑𝑉 = 𝑑𝐻 𝑑𝑄 = 𝑑𝑈
𝐶 𝑝 =
𝑑𝑄
𝑑𝑇 𝑝
=
𝑑𝐻
𝑑𝑇
𝐶 𝑉 =
𝑑𝑄
𝑑𝑇 𝑉
=
𝑑𝑈
𝑑𝑇
enthalpy
at constant pressure at constant volume
C8863 Free Energy Calculations -10The
second law
It postulates the entropy as a state function:
T
dQ
dS rev
=
T
Qd
dS 
reversible action irreversible action (spontaneous)
The most important postulate of thermodynamics. It speaks about time flow direction
(time arrow). The direction of time is determined by the irreversible events.
For an isolated system, the direction of time is the same as the increase in entropy.
Spontaneous events are accompanied by an increase in entropy.
In an isolated system, the entropy increases until equilibrium is reached. At equilibrium,
the value of entropy is maximal and constant in time.
C8863 Free Energy Calculations -11Reversible
Process
In thermodynamics, a reversible process is a process, involving a system and its
surroundings, whose direction can be reversed by infinitesimal changes in some
properties of the surroundings, such as pressure or temperature.
Throughout an entire reversible process, the system is in thermodynamic equilibrium,
both physical and chemical, and nearly in pressure and temperature equilibrium with its
surroundings. This prevents unbalanced forces and acceleration of moving system
boundaries, which in turn avoids friction and other dissipation. To maintain equilibrium,
reversible processes are extremely slow.
While processes in isolated systems are never reversible, cyclical processes can be
reversible or irreversible. Reversible processes are hypothetical or idealized but central to
the second law of thermodynamics.
Melting or freezing of ice in water is an example of a realistic process that is nearly
reversible.
https://en.wikipedia.org/wiki/Reversible_process_(thermodynamics)
C8863 Free Energy Calculations -12Combination
of first and second laws
For closed system and reversible process without change in chemical composition, it is
possible to combine the first and second laws:
𝑑𝑈 = 𝑑𝑄 + 𝑑𝑊
𝑑𝑄 = 𝑇𝑑𝑆 𝑑𝑊 = −𝑝𝑑𝑉
first law
second law
pressure–volume work
Reorganization leads to the fundamental thermodynamic relation, which is also valid for
irreversible processes because all variable are state functions:
𝑑𝑈 = 𝑇𝑑𝑆 − 𝑝𝑑𝑉
Internal energy is thus a state function U(S,V) depending on two state variables S and V.
𝑑𝑈 =
𝜕𝑈
𝜕𝑆 𝑉
𝑑𝑆 +
𝜕𝑈
𝜕𝑉 𝑆
𝑑𝑉
𝜕𝑈
𝜕𝑆 𝑉
=T
𝜕𝑈
𝜕𝑉 𝑆
= −𝑝
reformulating as a total differential https://en.wikipedia.org/wiki/Thermodynamic_potential
C8863 Free Energy Calculations -13Spontaneity
of Process
int
ext In an isolated system, the entropy increases
until equilibrium is reached.
𝑑𝑆 𝑎𝑙𝑙 > 0
irreversible action
(spontaneous process)all
Δ𝑆int + Δ𝑆 𝑒𝑥𝑡 = Δ𝑆all
C8863 Free Energy Calculations -14Spontaneity
of Process
int
ext In an isolated system, the entropy increases
until equilibrium is reached.
𝑑𝑆 𝑎𝑙𝑙 > 0
irreversible action
(spontaneous process)all
Δ𝑆int + Δ𝑆 𝑒𝑥𝑡 = Δ𝑆all
it is a property of the system
Q
it can be estimated from heat exchange with surroundings
Δ𝑆 𝑒𝑥𝑡 =
Δ𝑄 𝑟𝑒𝑣,𝑒𝑥𝑡
𝑇
=
−Δ𝐻𝑖𝑛𝑡
𝑇
for reversible process
at constant temperature and pressure
Δ𝑆 𝑒𝑥𝑡 =
Δ𝑄 𝑟𝑒𝑣,𝑒𝑥𝑡
𝑇
=
−Δ𝑈𝑖𝑛𝑡
𝑇
for reversible process
at constant temperature and volume
C8863 Free Energy Calculations -15Spontaneity
of Process
int
ext In an isolated system, the entropy increases
until equilibrium is reached.
𝑑𝑆 𝑎𝑙𝑙 > 0
irreversible action
(spontaneous process)all
Δ𝑆int +
−Δ𝐻𝑖𝑛𝑡
𝑇
= Δ𝑆all
Q
Δ𝐺int = Δ𝐻𝑖𝑛𝑡 − 𝑇Δ𝑆int = −𝑇Δ𝑆all
Reorganization:
for reversible process
at constant temperature and
pressure
Gibbs energy (free energy)
C8863 Free Energy Calculations -16Spontaneity
of Process
int
ext In an isolated system, the entropy increases
until equilibrium is reached.
𝑑𝑆 𝑎𝑙𝑙 > 0
irreversible action
(spontaneous process)all
Δ𝑆int +
−Δ𝐻𝑖𝑛𝑡
𝑇
= Δ𝑆all
Q
Δ𝐺int = Δ𝐻𝑖𝑛𝑡 − 𝑇Δ𝑆int = −𝑇Δ𝑆all
Reorganization:
for reversible process
at constant temperature and
pressure
Δ𝐴int = Δ𝑈𝑖𝑛𝑡 − 𝑇Δ𝑆int = −𝑇Δ𝑆all
for reversible process
at constant temperature and
volume
Helmholtz energy (free energy)
C8863 Free Energy Calculations -17Free
energy and spontaneity
0−= STHG
0=−= STHG
0−= STHG
spontaneous process
non-spontaneous process
the system is in equilibrium
The change in Gibbs free energy indicates whether the process can occur spontaneously.
However, it does not determine in what time the actual transformation will take place.
for process at constant temperature and pressure
Similar relations are valid for Helmholtz energy.
C8863 Free Energy Calculations -18Ideal
Gas
Or what you should already know….
C8863 Free Energy Calculations -19Ideal
Gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that
are not subject to interparticle interactions.
The ideal gas obeys the following equation of state:
𝑝𝑉 = 𝑛𝑅𝑇 = 𝑛𝑁𝐴 𝑘 𝐵 𝑇
pressure volume molar amount
thermodynamic temperature
the ideal gas constant
Boltzmann constant
Avogadro constant
The empirical form of the equation of state was derived by combining four laws (Benoît
Paul Émile Clapeyron, 1834) :
• Boyle's law (1662)
• Charles's law (1801)
• Avogadro's law (1812)
• Gay-Lussac's law (1809)
Other derivation are possible employing, for
example, statistical thermodynamics.
https://en.wikipedia.org/wiki/Ideal_gas
C8863 Free Energy Calculations -20Ideal
Gas - Internal Energy
Internal energy of the ideal gas is (see later statistical thermodynamics for derivation):
𝑈 =
3
2
𝑛𝑅𝑇
Then the following statement is true:
𝜕𝑈
𝜕𝑉 𝑇
= 0
This is consequence of the fact that there is no interaction
between particles.
C8863 Free Energy Calculations -21Recommended
Literature
• Atkins, P. W. Physical Chemistry, 5. ed., repr. (with correct.).; Oxford Univ. Press: Oxford,
1994.
• Bokshteĭn, B. S.; Mendelev, M. I.; Srolovitz, D. J. Thermodynamics and Kinetics in
Materials Science: A Short Course; Oxford University Press: New York, 2005.
• Dill, K. A.; Bromberg, S. Molecular Driving Forces: Statistical Thermodynamics in Biology,
Chemistry, Physics, and Nanoscience, 2nd ed.; Garland Science: London ; New York,
2011.