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. 1 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. 2 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 3 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. 4 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 5 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. 6 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 7 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 8 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. 9 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 10 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. 11 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 12 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. 13 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. 14 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 15 16