Why the Equilibrium of Molecular Systems needs Three Definitions
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 thermodynamic equilibrium,
kinetically controlled equilibrium and non equilibrium steady states. A simple model based on
driving forces and mechanisms explains why three types of equilibrium occur and that they are a
result of the statistical behaviour of molecular Systems.
1
Section 1 Introduction
In this article we use the word System to denote a molecular system with one or more moles of
molecules and use capitalised letters for the thermodynamic properties of the System. 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 normally 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 the
interconversion of energy and work. The Gibbs Function, G, is the thermodynamic property
that describes this behaviour of this behaviour 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
2
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.
Thermodynamic equilibrium alone cannot explain all types of equilibrium. Many Systems have a
range of conditions in which they cannot reach thermodynamic equilibrium. In section 3 we
introduce one definition of equilibrium for such Systems when we define kinetic controlled
equilibrium and relate it to the statistical behaviour of the System using the MBD of energies for
some of these systems. This leads to a discussion of a simple model based on driving forces and
mechanisms 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.
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 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
enthalpy or energy. To describe equilibrium at constant pressure it is essential to have a function
that takes account of the 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
(1)
On rearranging this can be written as:
Enthalpy convertible to Work = Enthalpy – Enthalpy not convertible to Work
(2)
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.
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.
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
4
minimum, and mechanisms are available for all energies changes involved as the System
changes from its initial condition to its final equilibrium state.
Section 3. Kinetically controlled 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.
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 not 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 equilibrium.
Introduce a platinum catalyst and the reaction to form water produces an equilibrium mixture of
hydrogen oxygen and water as steam. These considerations lead to the definition of kinetically
5
controlled equilibrium where a reaction does not occur at the conditions of the System because
there is no mechanism available to the System at these conditions.
A System is at a kinetically 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 at conditions different to those of the
System for all energies changes involved as the System changes from its initial condition to its
final equilibrium state.
Kinetically controlled equilibrium arises because the MBD restricts the energy levels available to
a System at constant conditions. Textbook treatments of the equilibrium state that a System at a
kinetically controlled equilibrium has a local minimum in the Free Energy but this author knows
of no analytical demonstration of that. It is clear that a System at a kinetically controlled
equilibrium will be at 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, though this is never stated in standard treatments. Such Systems at constant
temperature will have a MBD of energies. The thermodynamic properties of these Systems can
be calculated from the properties of the individual molecules using the usual formulae to
calculate the energy, U, and the pressure from the energies of the molecules in exactly the same
way as for thermodynamic equilibrium.
The MBD determines what energy levels are available to a System at the conditions of the
System and whether there are enough molecules able to react and continue the reaction at those
conditions. We take up the discussion of the roles of mechanism and driving forces in the enxt
section.
6
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 on the Systems.
Experimental scientists are well aware that a System will not reach equilibrium left to itself but
requires constant effort by experimenters if they are 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
7
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 average of the result 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 average 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. A consequence of this is that kinetically controlled equilibrium with
differences of pressure do not occur.
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 kinetic controlled equilibrium
frequently occurs in Systems where 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
8
currents that have remained separated for millions of years are just one example of this type of
behaviour.
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
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 the other two types of 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
Most Systems display all three types of 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
9
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.
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. At
temperatures below the ignition temperature the mixture will not ignite, shown in area B. At low
10
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.
Section 7 Conclusion
It clear from the discussion above that the concept of thermodynamic equilibrium cannot explain
all the types of equilibrium that are observed either in the laboratory or in nature. All Systems
are capable of displaying each of the three 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 three different
definitions of equilibrium to account for what is observed.
List of Figures
Figure 1 Equilibrium Diagram for Mixtures of Combustible Gases and Air
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Equilibrium for the Chemical Reactions of Ideal Gases
By
R M Gibbons
BSc PhD DIC
Proofs to:
R M Gibbons
4 Little Acre
Beckenham
Kent BR3 3ST
Abstract
The thermodynamic properties of ideal gases and their mixtures behave in a very simple way.
Expressions are available for the energy, enthalpy and the reversible work done by an ideal gas
system.
A simple introduction to the Gibbs Function, G, leads to the definition of thermodynamic
equilibrium at which the G must be a minimum. Combining this definition of G with the
expressions for the properties of ideal gases gives an expression for the equilibrium constant, K,
provides an explanation of why chemical reactions reach an equilibrium in which all the
reactants are not converted to products, and explains how exothermic and endothermic reactions
can occur.
12
Section 1 Introduction
The purpose of this article is to provide simple explanations of some key topics in A Level
syllabuses which relate to equilibrium. In particular the aim is to provide explanations of
chemical reactions that do not go to completion, equilibrium constants, and how exothermic and
endothermic reactions arise. It is too difficult to provide explanations of these topics for all types
of Systems and we restrict the discussion on these topics to the reactions of ideal gases.
It is not possible to discuss equilibrium at constant pressure without introducing the Gibbs
Function, G. In Section 2 we give a direct derivation of G from the second law of
thermodynamics and show it must be a minimum at equilibrium using the third law of
thermodynamics. To obtain expressions for the equilibrium of ideal gases using the minimum
value of G we require some properties of ideal gases including reversible work of an ideal gas
and discuss these are in Section 3. We go on in Section 4 to use as an example the reaction of
oxygen and hydrogen to form water at 1 atm and 100 C to show the minimum in G leads to the
conditions for equilibrium of this System. This leads to an expression for the equilibrium
constant, K and to an explanation for exothermic and endothermic reaction. We discuss briefly
in Section 5 how this treatment must be extended to apply to a wider range of Systems.
Section 2. Thermodynamic Equilibrium and the Gibbs Function
In this article we use the word System to denote a molecular system with one or more moles of
molecules and use capitalised letters for the thermodynamic properties of the System. 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.
13
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 normally 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
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
enthalpy, H, or energy, U.
H, is closely related to the U by the definition
H = U +PV
(1)
where P is the pressure and V the volume. For a change in temperature at constant pressure H
automatically includes the work done arising from the change in volume. H then is the heat
supplied to a System. This makes H the natural property to use in describing Systems at constant
Pressure
To describe equilibrium it is essential to have a function that takes account of the 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.
14
Enthalpy = Enthalpy convertible to Work + Enthalpy not convertible to Work
(2)
On rearranging this can be written as:
Enthalpy convertible to Work = Enthalpy – Enthalpy not convertible to Work
(3)
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 any change between two states
is the reversible work. This is the amount of work a System can do when its surroundings have a
pressure infinitesimally less than the Pressure at each point on the path between the initial and
final states of the System.
We can show the Gibbs Function is a minimum at equilibrium by
combining reversible work 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.
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.
This defines the Gibbs Function but leaves a problem because it cannot be measured directly; nor
can the enthalpy not convertible to work. Values of both these quantities 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
15
measurements of the pressure, temperature and volume of the System. To set conditions for
equilibrium it is sufficient that the 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.
There are no general expressions applicable for either the reversible work a System can do or for
the enthalpy not convertible to work. In fact the only Systems for which we have closed
expressions for the reversible work between pressures are ideal gases and for that reason these
are the only Systems discussed further in this article.
To use this condition of G having a minimum to determine equilibrium for reactions of ideal
gases we first need to obtain expressions for reversible work an ideal gas System can do. We
discuss this in Section 3.
Section 3 The Properties of Ideal Gases
Ideal Gases have a special place in the development of theories of Physical Chemistry. One of
the first laws of Chemistry to be discovered, the Ideal Gas Law continues to have an important
role to play in theories of Chemistry. It describes exactly the behaviour of a System of
molecules with negligible molecular volumes which move freely and exchange energy and
momentum in collisions but do not interact with one another in any way. It importance comes
in part because it describes the behaviour of all substances at low enough Pressures at all
16
Temperatures and is a good approximation to the behaviour of many gases at atmospheric
pressure and room temperature. The other main reason is that it is the only model for which
explicit expressions are available for the energy, enthalpy and the equation relating the pressure
and volume, usually referred to as the equation of state:
PV = RT
(4)
Now the work done by the System for a change in volume, d V, is
∆W = -P d V
(5)
Replacing P by RT/V, using equation 4 and carrying out the integration leads to the expression
for reversible work done between P1 and P0 at constant Temperature
∆W = - RT log (P1/P0)
(6)
The actual work done by an ideal gas at constant temperature depends on the pressure of the
surroundings as the System Pressure changes between the initial and final states of the System.
The System does as much work as it can against the surroundings but when the System gets to its
final state of equilibrium the enthalpy convertible to work at that state will be the reversible work
the System could do between the state of the System and 0 K.
Two other simplifications for ideal gases are that the enthalpy and energy depend solely on
temperature and do not vary with pressure. As a consequence the only changes in the values of
G at constant Temperature arise from the work done as the System changes between P1 and P0.
For reversible changes therefore we can write, using equation 6,
G (T, P1) = G (T, P0) + RT log (P1/P0)
(7)
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The value of P0 is usually taken as 1 atm and values of G (T, P 0) are available for a large number
of gases in the data books used with A Level courses (e g 1). How values of G (T, P0) can be
obtained will be discussed in a later article. In the next Section we show how equation 7 leads to
the conditions for equilibrium for chemical reactions of ideal gases and for that purpose we have
no need of values for G (T, P 0).
This is the expression for the Gibbs Function of each ideal gas at pressure P on its own. In a
mixture the total pressure is the sum of the pressures of the components and the Gibbs Function
of the mixture is the sum of the Gibbs Functions of the components. For a mixture of nO2 moles
of oxygen and nH2 moles of hydrogen we can write for Gm, the Gibbs Function of the mixture,
G m = nO2 GO2 + nH2 G H2
(8)
Section 4 Equilibrium for Ideal Gas Reactions
In this Section we show how the minimum in the Gibbs Function leads to the condition for
equilibrium for the chemical reactions of ideal gases. We use the reaction of hydrogen and
oxygen at 100 C and 1atm to form water, as a vapour, to illustrate this using the usual equation:
2 H2 + O 2 = 2 H2 O
(9)
We can use this equation to write expressions for G for each of hydrogen, oxygen and water.
From equations 3, 5 and 8 we can obtain the value of GR for the reactants from adding G for
two moles of hydrogen to one of oxygen. Similarly the value of GP for the product is that for two
moles of water. The value of GS for the total System is
GS = GR + GP
(10)
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Now at equilibrium GS must be a minimum. For that to happen GR and G P must be equal. If one
is lower than the other, GS can always be reduced by producing more of the lower one. So for GS
to be a minimum GR and GP must be equal. When we equate GR and GP and collect all the log
terms on one side of the equation and the non-log terms on the other we find:
∆G = RT log (P H2O 2 / (P H2 2 P 2 ) )
(11)
and ∆G is the differences of the values of the terms Gn (T,P0 ) in GR and GP .
The argument of the log term is the Equilibrium Constant, K
K = P H2O 2 / (P H22 P O 2 )
(12)
Equation 10 also allows us to calculate the heat of reaction from the difference of the enthalpies
of the products and reactants. In this reaction heat is given out and the reaction is exothermic.
Because the equality in equation 10 is based on the Gibbs Function and not the enthalpies it is
perfectly possible for heat to be absorbed in the reaction and this does occur in endothermic
reactions, such as that between oxygen and nitrogen.
Section 5 Conclusion
The discussion in Section 4 shows how the minimum in the Gibbs Function is essential to
explaining why and how equilibrium occurs in reactions of ideal gases. Without the Gibbs
Function there can be no explanation for why reactions should occur and, if they do occur, do not
go to completion.
The account of equilibrium for chemical reactions of ideal gases given above explains how
balanced reactions reach equilibrium. It is incomplete because it does not explain how values of
19
G (T, P0) can be obtained for each of the substances in the equation. As pointed out in the
discussion there is no need to use these quantities in determining the condition for equilibrium.
These quantities can be obtained fairly simply using A level mathematical methods but the
derivations involve the Maxwell Boltzmann Distribution, which included in most A Level
syllabuses and the entropy which is not included in those syllabuses. These topics will be taken
up in a further article.
Currently the Gibbs Function is thought to be too advanced for A Level chemistry courses.
Using standard treatments for the Gibbs Function this is true; the Gibbs Function is normally
taught in advanced physical chemistry courses on thermodynamics.
We have been able to apply this treatment to mixtures of to ideal gases because, apart from
reactions and collisions, these Systems have no interactions between molecules and we have an
expression for the reversible work of an ideal gas. Any general treatment must allow for mutual
interactions of molecules and this introduces extra mathematical complications. The same
approach, based on the minimum in G at equilibrium, applies but a number of extra terms and
properties must be introduced. These topics are best left to a more advance course
I hope that the simple treatment of the Gibbs Function introduced here will find acceptance at A
level. The concept of the Gibbs Function is central to most of physical chemistry and its early
introduction provides a framework for discussion of theories of all types of equilibrium. The
major benefits to A Level courses of doing so, would be that explanations can be given of why
chemical reactions can occur, how equilibrium constants arise, and exothermic and endothermic
reactions occur. Current syllabuses offer no explanations for these topics which is not helpful for
students. This will enable physical chemistry to be taught in the same way as physics with
20
explanations of why things happen. Current approaches emphasise how things happen with
students expected to remember what occurs but without explanations as to why they happen. I
believe that early explanations of these topics will make physical chemistry more understandable
and less of a test of memory
References
G Attwood and G Skipwood, “Chemistry Data Book, John Murray, London, 1992
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