Thermodynamic Equilibrium of Molecular Systems
By
R M Gibbons
BSc PhD DIC
Proofs to:
R M Gibbons
4 Little Acre
Beckenham
Kent BR3 3ST
Abstract
All Systems at constant conditions are in some sort of equilibrium. We show that three types of
equilibrium routinely occur in molecular Systems leading to two types of thermodynamic
equilibrium and one of non equilibrium steady states. A simple model based on driving forces
and mechanisms explains why these types of equilibrium occur and shows that they are a result
of the statistical behaviour of molecular Systems.
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Section 1 Introduction
In this article we use the word System to denote a molecular system with one or more moles of
molecule. Any System of molecules with constant values of its principal properties, such as
temperature pressure, density, energy and enthalpy or composition is in some sort of equilibrium.
A simple model based on driving forces and mechanisms explains all the different types of
equilibrium. Because of the molecular nature of chemical systems this single model produces
three different types of equilibrium. The type of equilibrium produced in a System at any set of
conditions depends on the mechanisms available to the System at those conditions. The statistical
behaviour of the System, in the form of the Maxwell Boltzmann Distribution (MBD), determines
what mechanisms are available at those conditions. In addition outside interference can provide
additional mechanisms and this is the basis of all experimental methods for making accurate
measurements of the properties of Systems in the laboratory.
Changes in any System must obey the laws of thermodynamics and so must take account of
interconversion of energy and work. The Gibbs Function, G, is the thermodynamic property that
describes this behaviour at constant pressure and we introduce this is section 2 along with the
concept of reversible work. We show how to obtain an expression for the Gibbs Function
directly from the second law of thermodynamic in terms of enthalpy convertible to work and
enthalpy not convertible to work. We demonstrate that G must have a minimum at equilibrium
using the third law of thermodynamics and arguments based on the reversible work the System
could do if it were returned to a crystalline state at 0 K. This leads to the definition of
thermodynamic equilibrium in terms of G and the minimum in G at equilibrium.
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Thermodynamic equilibrium alone explains all types of equilibrium but we must recognise that
the Gibbs Function is a minimum for the System at the conditions of the System. In section 2
we introduce one definition of thermodynamic equilibrium for such Systems and relate it to the
statistical behaviour of the System using the MBD of energies for some of these systems. If all
MBD energy levels were always available to a System this would d be all that we had to
consider. In fact many Systems have ranges of conditions, for which the MBD energy levels
needed for a reaction to occur are not available, and this leads to the need for a second definition
of thermodynamic equilibrium, kinetically controlled (thermodynamic) equilibrium, for Systems
at conditions where reactions cannot occur. We go on to discuss thermodynamic equilibrium and
the MBD in section 3. This leads to a discussion of a simple model based on driving forces and
mechanisms in section 4 which shows how both these types of equilibrium are related and
introduces the need for a third type of equilibrium, non equilibrium steady states, which is
discussed in section 5. Obviously, from the definition, this is not a true equilibrium but we need a
term to describe Systems that in some cases have been unchanged over millions of years.
We go on to show in section 6 that these ideas have practical applications as they are the basis of
safety standards for the use of natural gas and all other combustible gases and conclude with a
brief discussion in section 7.
Section 2 Thermodynamic Equilibrium and the Gibbs Function
The driving forces producing equilibrium arise from differences in the properties of the System.
In elementary introductions to equilibrium it is usual to introduce differences of temperature and
pressure as independent driving forces which reduce to zero at equilibrium. In fact we know
from the laws of thermodynamics that in any change there is some interconversion of work and
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enthalpy or energy The enthalpy, H, is simply related to the energy , U, by adding the term PV,
where P is the Pressure and V the volume,
H =U + PV
(1)
Now the work done in a change at constant pressure is given by P∆V so ∆H is the function that
automatically includes the work done in any change at constant pressure
∆H =∆U + P∆V
(2)
Note that when the volume of the System increases this definition means work done by the
System is negative.
To describe equilibrium at constant pressure it is essential to have a function that takes account
of interconversion of work and enthalpy. The Gibbs Function for changes at constant pressure is
the function required and can be obtained directly from a statement of the second law of
thermodynamics when it is written as an enthalpy balance.
Enthalpy = Enthalpy convertible to Work + Enthalpy not convertible to Work
(3)
On rearranging this can be written as:
Enthalpy convertible to Work = Enthalpy – Enthalpy not convertible to Work
(4)
The enthalpy convertible to work is the Gibbs Function, G. To prove G is a minimum at
equilibrium we note that the maximum work a System can do in a change is the reversible work
involved in that change and the following statement of the third law of thermodynamics:
At 0 K all the molecules of a System are at their rest positions in a crystalline state which
maximise their interactions with other molecules and the System can do no work.
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It follows that the amount of enthalpy convertible to work at any equilibrium will be equal to the
amount of work that could be obtained from the System if it were returned reversibly to 0 K. As
enthalpy is leaving the System it follows that G has a minimum (most negative) value at
equilibrium. Note it is assumed in this statement that the System will not undergo a chemical
reaction in the course of the change in the state of the System between the current conditions and
its state at 0 K.
This defines the Gibbs Function but leaves a problem because values of G cannot be measured
directly. Values of them must be obtained from measurements that allow us to determine the
enthalpy of the System and the amount of work it can do. Enthalpy values can be obtained from
heat capacity data, Cp, combined with data for the changes of enthalpy in phase changes. The
work capacity of a System can be determined from measurements of the pressure, temperature
and volume of the System. To set conditions for equilibrium it is sufficient that the value of G is
a minimum and there is no need for actual values of G.
This leads to the definition of thermodynamic equilibrium.
A System is at thermodynamic equilibrium when the values of its temperature pressure density
and composition have values that are uniform and independent of time, the Gibbs Function is a
minimum, and mechanisms are available for all energies changes involved as the System
changes from its initial condition to its final equilibrium state.
All Systems have ranges of conditions where they can and do reach thermodynamic equilibrium.
Many of them have ranges of conditions where that thermodynamic equilibrium does not include
a reaction which would occur at other conditions. It is convenient to introduce a definition of
thermodynamic equilibrium where a reaction cannot occur, even though the reaction does occur
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at other conditions. Mixtures of combustible gases and air are examples of where this sort of
distinction is useful as it enables us to designate safe ranges of compositions where combustion
will not occur. For that purpose we introduce the definition of kinetic controlled equilibrium:
A System is at kinetic controlled equilibrium when the values of its temperature pressure density
and composition have values that are uniform and independent of time, the Gibbs Function is a
minimum, and mechanisms are not available involving the energy levels needed for chemical
reactions at the conditions of the System.
Section 3 Thermodynamic Equilibrium and the MBD
The definition given above of thermodynamic equilibrium is the first to ensure a System will
reach the equilibrium specified. The energy levels available to a System at any set of conditions
are determined by the statistical behaviour of the molecules. For all chemical systems this means
the MBD of energies. The key consideration that determines what energy levels are available to
a System is:
At given conditions for a System only those energy levels that are occupied by enough molecules
can contribute to the properties of a system.
In practice for the MBD this means that only those energy levels within the range nought to four
times the average energy per molecule can contribute to the properties. Energy values outside
this range occur too infrequently to contribute in a measurable way. This observation is the key
to developing greatly simplified methods, given the MBD, using just GCSE Mathematics to
obtain the average values of the energy, U, and other properties of a System from the properties
of the molecules.
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We can make a similar statement about the energy levels that can contribute to the mechanisms
by which the System changes from one equilibrium state to another without any outside
interference.
At given conditions for a System only those energy levels occupied by enough molecules can
contribute to the mechanisms by which the System changes from one state to another.
The MBD energy levels contributing to the mechanisms include all those that occur at each
condition of the System as its changes from the initial to the final state. For changes involving
reactions this includes high temperatures that may be produced as a result of exothermic
reactions as well as MBD energy states at the conditions of the System.
The consequence of this is that merely observing constant values of properties at the conditions
of the System, which is how in practice we determine equilibrium, is enough to ensure that a
System is at thermodynamic equilibrium. A simple example will demonstrate this. At 100 C
and 1 atm a mixture of hydrogen and oxygen will not react and are in a stable thermodynamic
equilibrium. Introduce a platinum catalyst and the reaction to form water produces an
equilibrium mixture of hydrogen oxygen and water as water vapour. The presence of the catalyst
does not alter the energy states available to the System but it does alter the energy states involved
in the mechanism for the reaction to form water.
Thermodynamic equilibrium for the System of hydrogen and oxygen arises because the MBD
restricts the energy levels available to a System at constant conditions. At thermodynamic
equilibrium the System will have a minimum of the value of G, provided no reaction takes place,
so the condition for equilibrium must include some unspecified energy barrier to ensure the
reaction does not occur. The MBD determines what energy levels are available to a System at
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the conditions of the System and whether there are enough molecules able to react, and continue
the reaction, at those conditions.
This example is just one of many where Systems at given conditions are at a stable
thermodynamic equilibrium with a minimum value of the Gibbs Function but could produce a
much lower value of the Gibbs Function for the System if other energy states were available to
provide a mechanism for changes to occur. Examples of such Systems are:
Nitro-glycerine – stable at room temperature but explodes when detonated,
Ethyl alcohol and oxidising agents which produce methanal, ethanoic acid or carbon dioxide and
water depending on the strength of the oxidising agent used.
We take up the discussion of the roles of mechanism and driving forces in the next section and
defer a full discuss the implications for the definition of equilibrium in molecular Systems to
section 7.
Section 4 Mechanisms Driving Forces and the MBD
Mechanisms at the molecular level are about transfers of energy and momentum between
molecules via collisions. The MBD of energies controls how many molecules can contribute to
these transfers and hence the rate at which they occur. In this article we use the idea of
mechanisms in a much wider sense. Mechanisms are all processes that contribute to producing
equilibrium. Processes that occur without outside interference will all ultimately involve
transfers of energies between molecules. Many mechanisms involve external interference to
promote equilibrium. This includes all experimental techniques which scientists use to produce
well controlled uniform Systems so that they can make accurate measurements. Experimental
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scientists are well aware that a System will not reach equilibrium left to itself but requires
constant effort by experimenters to obtain accurate measurements of the properties of a System.
To ensure a rapid approach to equilibrium all experimental techniques must produce large
numbers of molecules that can contribute to the transfers of energies and momentum.
The principal driving forces producing equilibrium are the difference of temperature, pressure
composition and the Gibbs Function. Differences of temperature in a single phase produce
transfers of energy between molecules in which higher energy molecules transfer energy through
collisions until a common average energy is reached and the molecules have a MBD. The
pressure similarly reaches a common value throughout the System by transfers of momentum
between molecules. That the Gibbs Function of the System is a minimum at equilibrium arises
as a result of the changes in the System that are needed for the transfers of energy momentum
and composition to conform with the laws of thermodynamics as they must.
The properties of the System are just the sums of the corresponding properties of the molecules.
The pressure arises from the momentum of each individual molecule which exchange
momentum via random collisions. It is the sum of the results of the changes in momentum that
the molecules experience at the boundary of the System. It appears to be a fixed value because
the standard deviation of the pressure calculated from the sum of the momentum changes is so
small it cannot be measured. The momentum of each of the molecules contributes to the pressure
and is determined by the energy levels associated with the kinetic energies of the molecules. As
a result the response of the pressure of a System to change is always quick. And as the energy
levels involved are closely spaced the change in pressure is always determined by the MBD as
these energy levels are always available, in the terms of this article, as mechanisms to produce
the pressure. Consequently equilibrium with differences of pressure do not occur.
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The energy, U, is similarly the average of the sum of the energies of the molecules. It too has
such a small standard deviation that it appears as a fixed value at a constant temperature. While
changes in pressures always involve all the molecules, transfers of energy only occur between
higher energy molecules and those with a lower energy. Consequently transfers of energy
between molecules occurs more slowly than pressure changes and in many Systems the higher
energy molecules physically separate from lower energy molecule of the System. This happens
frequently in liquid Systems where warmer less dense layers remain separate from colder denser
layers; warm surface and cold deep ocean currents that have remained separated for millions of
years are just one example of this type of behaviour.
In principle, at thermodynamic equilibrium, a System, has a uniform temperature and pressure;
in practice there will be some fluctuations about these averages. It is not feasible to expect, for
example, a container of unstirred liquid or gas to be totally uniform in temperature at the end of a
change of state. That is why all experiments involving containers of liquids and gases provide
additional mixing, stirring and temperature control to ensure the System is as uniform as
possible. For physical changes such fluctuations are usually minor but for changes involving
chemical reactions fluctuations can be much more important, for reasons which we will discuss
later.
Section 5 Non Equilibrium Steady States
The other type Equilibrium that can occur is when energy barriers occur at the conditions of the
System that prevent changes occurring that would bring the System to thermodynamic
equilibrium. Most of these energy barriers relate to diffusion processes at the condition of the
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System. These types of equilibrium are non equilibrium steady states and are frequently found in
nature. Their definition is:
A System is at a non equilibrium steady state when the values of its temperature pressure density
and composition have values that are independent of time and mechanisms are not available at
the conditions of the System, for all energies changes involved, as the System changes from its
initial condition to its final equilibrium state.
The System has constant values for each of its properties but there is nothing to ensure there is
uniformity in the System and average values of the System properties cannot be calculated
without further information. Unlike thermodynamic equilibrium, we cannot hope to predict the
non equilibrium steady states and can only use this definition to explain what has been observed.
Section 6 Equilibrium Diagram for Mixtures of combustible Gases and Air
We have introduced in section 2 a definition of thermodynamic equilibrium, kinetic controlled
equilibrium, where a reaction cannot occur, even though the reaction does occur at other
conditions. Mixtures of combustible gases and air are examples of where this sort of distinction
is useful as it enables us to designate safe ranges of compositions and temperatures where
combustion will not occur.
Most Systems display both thermodynamic and kinetic controlled equilibrium depending on the
conditions. To illustrate their importance let us show how they provide the basis of safety
standards for mixtures of combustible gases and air. For these Systems we show schematically in
Figure 1 the conditions that are safe for the use of combustible gases, based on the types of
equilibrium discussed above. We show the areas where the gases react to obtain thermodynamic
equilibrium in red while the kinetically controlled equilibrium areas are shown in blue.
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For all mixtures of combustible gases and air there is an ignition temperature which is the lowest
temperature at which the mixture can ignite. For temperature greater than the ignition
temperature the mixture will react and reach thermodynamic equilibrium, as shown in area A. In
modern gas appliances a piezo electric spark is used to produce ignition. The heat released by the
molecules, which react as a result of the spark, go on to heat up sufficient molecules, to
temperature greater than the ignition temperature, for the reaction to continue. At temperatures
below the ignition temperature the mixture will not ignite; we show this in area B of figure 1. At
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low concentrations of the gas the mixture will not ignite even for temperatures greater than the
ignition temperature, shown in area C. This is the basis of safety standards for natural gas used
in homes which is odorised so that the presence of gas can be detected at concentrations much
lower than that at which the gas can to be ignited. In both these states the mixtures will be at a
kinetically controlled equilibrium. Finally at high enough concentrations of gas, area D, there is
insufficient oxygen for the reaction to occur and this too leads to kinetically controlled
equilibrium as indicated in Figure 1.
The mechanism for the reaction to occur depends on the heat released by molecules that have
interacted successfully. This heat is used to heat the whole System and is not concentrated on
those molecules which must be heated to the ignition temperature for the reaction to be
sustained. As soon as there is sufficient dilution of the reactants there are an insufficient numbers
of effective collisions, in which a reaction occurs, for the reaction to be sustained even above the
ignition temperature.
Section 7 Conclusion
It clear from the discussion above that the concept of thermodynamic equilibrium explains all the
types of genuine equilibrium that are observed either in the laboratory or in nature. All Systems
are capable of displaying each of the types of equilibrium identified above. Simple statistics
based on the MBD make clear the connection between different types of equilibrium and explain
why a simple model based on driving forces and mechanisms requires two different definitions
of thermodynamic equilibrium to account for what is observed. For steady state non equilibrium
systems the definition we introduced describes what occurs in such Systems but does not enable
us to calculate or predict this type of behaviour.
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Our example of the equilibrium of mixtures of hydrogen and oxygen at 100 C and 1 atm shows
that the normal method of defining equilibrium conditions in terms of temperature pressure and
composition is incomplete. As it does not address the question of whether the gases can react at
these conditions it is implicitly assumed that they cannot, so in effect a condition of a “high
enough” activation energy is introduced to ensure a reaction does not occur.
There are two ways of interpreting this behaviour. If we say that thermodynamic equilibrium is
determined solely by the energy states available to this System of hydrogen and oxygen, then the
equilibrium in the absence of a catalyst is not a thermodynamic equilibrium and the System is in
a meta-stable state which, with an appropriate catalyst, will react to produce a thermodynamic
equilibrium mixture. However introduce a better catalyst and a new equilibrium will be
produced, so that all equilibriums, except that produced by a perfect catalyst, will be meta-stable
states. This is unsatisfactory.
The alternative explanation also introduces a new idea namely that the energy levels involved in
the mechanism also form part of the definition of thermodynamic equilibrium. In this
explanation thermodynamic equilibrium is determined by the energy states available to the
System and by the energy states available to the System that are required by the mechanism for
the change to occur. The introduction of a catalyst does not change the energy levels of the
System but does change the energy levels involved in the mechanism and consequently produces
a new thermodynamic equilibrium. It is usual to give the conditions of equilibrium in terms of
the temperature pressure and composition without any mention of whether the System can react
or not.
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It is clear from our example of mixtures of hydrogen and oxygen there is an assumption that the
gases will not react at 100 C and 1atm. This implies that the gases cannot react at these
conditions and is equivalent to saying there is an activation energy barrier which ensures the
reaction will not occur at these conditions. So a full definition of thermodynamic equilibrium
should include a statement about the activation energy barrier for a reaction as well as the
temperature pressure and composition. The introduction of a catalyst then by definition alters
the thermodynamic equilibrium by altering the activation energy for the reaction. As the values
of the energy barriers involved are not known, in practice a statement of whether a reaction can
or cannot occur at the conditions of the System needs to be included in the definition of
thermodynamic equilibrium.
In conclusion the arguments presented here show that for a System at given conditions, the only
changes that can occur, are those for which the energy levels involved in the mechanism are
available for that change. When a System reaches its thermodynamic equilibrium after a change
the Gibbs Function will be a minimum at the final condition of the System.
List of Figures
Figure 1 Equilibrium Diagram for Mixtures of Combustible Gases and Air
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