aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b The logic of behavior of atoms
Madjid Eshaghi Gordji , Gholamreza Askari and AlirezaTavakoli Targhi Abstract
Understanding of the atomic structures and ways in which the atoms interacting iscritical to the understanding of chemistry. In this paper applied the concepts of gametheory in explaining reactions between elements of the periodic table. The findings inthis study suggest that the coordination and anti coordination, also cooperation andnon-cooperation between atoms and molecules lead to bonding formation according tothe game theory. Therefore, the chemical behavior and physical treats of any chemicalreaction can be predicted. Multiple examples of each batch of the chemical reactionsis expressed to support the useability of our claims.
Keywords
Game theory; Nash equilibria; Chemical bond .
Game theory was for first time recognized by Von Neumann and Morgenstern(1953) in reference to human economic behavior. It is widely used in social andbehavioral sciences, ranging from politics and market economics to ecosystemsand biological phenomena. In game theory, a player is said to be rational if heseeks to play in a manner which maximizes his own payoff. The first explicit useof game theory terminology in the field of evolution was by Hamilton (1967),who sought for an “ unbeatable strategy ” for the sex ratio when there is localcompetition for mates. Hamilton’s “ unbeatable strategy ” is essentially the sameas an ESS as defined by Maynard Smith and Price ( 1973). Evolutionary gametheory is a way of thinking about evolution at the phenotypic level when thefitness of particular phenotypes depend on their frequencies in the population [1].Chemistry is the science that is concerned with the characterization,composition and transformations of matter [2]. In 1869, the Russian chemist Department of Mathematics, Semnan University P.O. Box 35195-363, Semnan, Iran ISPR Lab., Department of Mathematics, Shahid Beheshti University, Tehran, Iran.Email: [email protected]; [email protected]; a [email protected] () Dmitri Mendeleev produced the first orderly arrangement, or periodic table, ofall 63 elements known at the time. Moseley discovered that appropriate structureof the periodic table correlated to the atomic number. The periodic table is avery useful for correlating the properties of the elements. Molecules and ions aretwo important types of particles derived form atoms. Atoms are held together inmolecules by chemical bonds [2].A central assumption of classical game theory is that the players will behaverationally, and according to some criterion of self-interest. Such an assumptionwould clearly be out of place in a chemical context. Instead, the criterion ofrationality is replaced by that of population dynamics and stability, and thecriterion of self-interest bonding formation. By these explanations, now followingquestions raised as fundamental aims of the paper. • Why are doing atoms bond together? • Why more materials in combination can be found in nature? • Why the noble gases are to be atomic in nature?Response to these questions, help us to explain the relation of the stability andthe game theory. The answers must be found in the properties of atoms. Noblegases have eight electrons in their valence shell (except helium), that is the reasonfor their stability. To answer the first and second mentioned questions, it shouldpoint out that the atoms bond together to bring stability. Thus, the elements mayacquire stability through the bonding formation.Reactions of atoms depends upon many factors. The nuclear charge, the electronconfiguration and effective size of the atom, are the most important factors amongthem. Behavior of elements is determined by the distribution of electrons in aroundthe nucleus. In other words, the electrons in the valence shell and energy levelslead to the formation of a pattern of behavior in elements. These behavior iscalled “ type ” or “ behavior kind ” which are related to the model configuration. Thediversity or differences in their properties leads to heterogeneous in periodic table.Therefore, various properties observe in a period or group. The kind of behaviorby any member of the population is not selective, because of specific combinationsof electrons, protons and neutrons are selected naturally. Atom portion has twoparts that are electrically charged, the electron has a negative charge and thenucleus has a positive charge. Atoms of each element or molecules in the reactioncalled a player . The players do not make conscious choices. Interaction betweenplayers leading to bonding formation. This interaction between atoms leading toattraction force and repulsion force which these forces cause approaching atomsand bonding formation. In other words, each player achieve stable state. Noblegases has little tendency to bonding formation. It follows that they are usuallystable.The chemical reactions are evaluated based on collision theory and transition state theory. A necessary condition for both mentioned theories is collisions betweenparticles. This work is established based on the collision theory in which thecollisions between particles is formulate as a game between the elements.The remainder of this paper the game of collisions is described. Chemical bond isformed when atoms combine by changes in electron distribution. There are threefundamental types of bonding: • Ionic bonding • Covalent bonding • Metallic bonding.All reactions involving two reactants required collisions between particles toproceed [2]. Elements can acquire stability through bonding formation. Ourframework allows for implement different games for these type of reactions. Ionic bonding
An ion is a particle that is made up of an atom or a group of atoms and thatbears an electric charge. There are two types: • A cation has a positive charge (because one or more electrons have beenlost). • An anion has a negative charge (because one or more electrons have beengained).Ionic bonding results when electrons are transferred from type of atom toanother. The atoms of one of the reacting elements lose electron and becomepositively charged ions. The atoms of other reactant gain electrons and becomenegatively charged ions. The electrostatic (plus-minus) attraction between theoppositely charged ions holds them together in a crystal structure. The netattraction holds the crystal together and may be considered to be ionic bond [2].Consider the reaction of a sodium atom and a chlorine atom. Sodium is in group IA . Chlorine is a member of group V IIA . The sodium atom loses one electron;the chlorine atom gains an electron.Consider a population of two atoms. Suppose that, this population is made upof two types, which can gain electron (G) or lose electron (L). The members ofthis population can not shared electrons. Is desirable to loses electrons per atomor electron gain, to achieve stable condition.The important point of this game for particles is stability through bondingformation which has benefit and gain for both particles. As stated earlier, webelieve that each atom as a player in the position of the collisions. Each atomplay one role of several kind in the population. Two players with same role andsinge each earns a reward S and for non-kind pair receive payment P . We canrepresent the game in a Table 1, each row of the matrix represents a kind ofpossible types in the population, and each column represents a possible kind of () types in the population. It should be noted that player cannot choose from “ loseelectron ” or “ gain electron ”, however, is from kind lose electron or kind gainelectron. Two players randomly chosen to play the following table:Table 1. A symmetric game between sodium and chlorineatoms with no symmetric Nash equilibriumG LG (S, S) (P, P)L (P, P) (S, S)This game has two Nash equilibria ( G, L ), (
L, G ) and ( 0 S < P ) and twoNash equilibriums are polymorphism homogeneous of stable strategy. This gamelooks like playing anti coordination. Thus, the number of sodium ions producedis the same as the number of chlorine ions produced, and the formula, NaCl, givethe simplest ratio of ions present in the compound (1 to 1) [2].
N a + + Cl − −→ N aCl.
In above example, the population is changed due to the new types of mutations.New population consists of two types of ions, means cation (C) and anion (A)that they called mutated. In collision any member of the new population, congenerrepulse each other and Non-kind attract each other. In case of congener each ofthem earns a reward S and for the case of non-kind, then both receive a payment P . To map this collision to the Nash equilibria, we consider cation and anion asplayers in the game theory. Two players randomly chosen to play the followingtable:Table 2. The payoff matrix game between cations and anions.If S < P the game has asymmetric Nash equilibrium.If
S > P the game has symmetric Nash equilibrium.C AC (S, S) (P, P)A (P, P) (S, S)To formulate the game of the players, we consider the repulse energy ofthe species of congener is equal. We have two Nash equilibriums (
C, A ) and(
A, C ), if
S < P which are polymorphism homogeneous of stable strategy.Consequently, the attraction force between non-kind players is more than therepulsion force of between congener players. This causes the crystal body toformed. Accordingly, if
S > P , then we have two Nash equilibriums (
C, C ),(
A, A ) and two Nash equilibriums are polymorphism homogeneous (polymorphismheterogeneous) stable strategy. Thus, the attractive force towards each the non-player type is less than the force of repulsion fellow players and consequentlyrupture crystal body. Covalent bonding
In covalent bonding electrons are shared, not transferred. A single covalentbond consists of a pair of electrons shared by two atoms. Molecules are made upof atoms covalently bonded to each other. A molecule is a particle that is formedfrom two or more atoms that are bound tightly together. The collision theorydescribes reactions in terms of collisions between reacting molecules (players).Assume that the hypothetical gas-phase reaction: A ( g ) + B ( g ) → AB ( g )takes place through collisions between A and B molecule (players). An A and B molecule strike one another. The old A − A and B − B bonds break whilesimultaneously two new A − B bonds form, and two AB molecules leave the sceneof the collision. When two slow-moving molecules approach one another closely,they rebound because of the repulsion due to the charges of their electron clouds[2].Consider the bond formed by two fluorine atoms. The fluorine molecule can berepresented by the symbol F − F . Assume that, there exist population of twoatoms. Each atom alone can not achieve a stable situation. Atoms shared electronsto reach a stable state. Bonding pair of electrons from the nuclei of two atomsabsorbed and from electrons excreted. So, two kinds of atoms compete for absorbbonding pair of electrons. Each player has two strategies that can attract andrepulse the bonding pair of electrons. One F atom attracts electrons to the sameextent as any atoms, such as F . In other words, the electron cloud density of thebond is symmetrically distributed around two nuclei. If both attract(A), then eachearns a reward S . If one repulse (R) and the other attract, then the repulse gainsa payoff P and attract gets S ( S > P ). If both repulse, then both have payment P . The table below show the game:Table 3. A purely covalent bond in moleculeformed form two identical atoms, such as F A RA (S, S) (S, P)R (P, S) (P, P)The strictly dominates strategies for each player attract the bonding pair ofelectrons (
S > P ). We see that (A,A) is the unique Nash equilibrium. Therefore,the attract is a monomorphism homogeneous stable strategy.Accordingly again we consider the bond formed by two hydrogen and chlorineatoms. The hydrogen chloride molecule can be represented by the symbol H-Cl. We consider a population of two atoms which shared electrons to reach astable state. Each atom alone can not achieve a stable situation. Bonding pair ofelectrons from the nuclei of two atoms absorbed and from electrons excreted. So, () two kinds of atoms compete for absorb bonding pair of electrons. Each player hastwo strategies, that can attract(A) and repulse(R) the bonding pair of electrons. Inthe HCl molecule the electrons of the covalent bond are more strongly attractedby the chlorine atom than by the hydrogen atom. Hence, the electrons clouddensity of the bond is not symmetrically distributed around two nuclei. If player1(hydrogen) and player 2(chlorine) attract, then each earns a reward S and P respectively. If player 1 and player 2 repulse, then each earns a reward K and T respectively. The payoffs rank for this game is given by K < T < S < P . Thenthe payoff matrix of our game is thus given by Table 4.Table 4. The outcomes of play betweenhydrogen and chlorine atomsA RA (S, P) (S, T)R (K, P) (K, T)Since for each player the action A is strictly dominates the action R, thestrategy pair (A,A) is a Nash equilibrium. Thus, the attract is a polymorphismhomogeneous stable strategy.Pair of members of a population engage in the following game. Atoms in amolecules can shared electrons with congener or non-kind atom in other moleculeand to reach stable state. Each player can shared electrons with non-kind, wesay has chosen cooperation or shared electrons with congener, has chosen non-cooperation (defect). If both cooperate(C), then each earns a reward S . If onedefect(D) and the other cooperate, the defector, gains a payoff P, and cooperatorgets zero. If both defect, then both a payment P , where payoffs satisfies theinequality 0 < P < S .Table 5. A symmetric game betweenatoms two molecule differentC DC (S, S) (0, P)D (P, 0) (P, P)As mentioned earlier in this case, the game has two Nash equilibriums (C,C)and (D,D). Accordingly, the cooperation is a polymorphism homogeneous stablestrategy. Also, the defect is a polymorphism heterogeneous (polymorphismhomogeneous) stable strategy. Our game looks like the stag hunt game.Under normal conditions the group of noble gases and other elements do notbe reactive. The properties of the noble gases reflect their very stable electronicconfigurations. Game between the molecules or atoms(except fluorine and oxygenatoms) and noble gases: If player 1 and player 2 cooperate (C), then each earns a reward S. If one defects(D) and the other cooperate, the defector gets an even larger payment P, and thecooperator gets zero. However, if both defect, then both get a payment T. Thepayoff rank for the game Table 6 is given by 0 < S < T < P .Table 6. The payoff matrix game betweena molecule and noble gasesC DC (S, S) (0, P)D (P, 0) (T, T)The dominates strategy for each player is a defect. The actions pair (D,D) isa Nash equilibrium. The defect is a polymorphism homogeneous (polymorphismheterogeneous) stable strategy. Our game looks like the deadlock game.Games between the fluorine and the group of noble gases:Table 7. Game between thefluorine and the noble gasesC DC (S, S) (0, P)D (P, 0) (T, T)The payoff rank for the game Table 7 satisfies the inequality 0 < T < P < S .We conclude that the game has two Nash equilibriums in pure strategies, namely(C,C) and (D,D), that two Nash equilibriums are polymorphism homogeneousstable strategy. Our game looks like the security dilemma game.Suppose that, population is made up of two types, which can be H moleculeor I molecule. The collision H and I , can lead to a reaction. Collisions thatproduce reactions, called “ effective collisions ”. We assume that collisions areeffective collision. If two H molecules collision, then each earns a reward zero. Ifan H and I molecule strike one another, then both a payment P ( P > I molecules collision, gains a payoff zero.Table 8. Collision H and I moleculesfor iodine chloride formed I H I (0, 0) (P, P) H (P, P) (0, 0)Observe that this game has two Nash equilibriums ( I , H ) and ( H , I ).Consequently, two Nash equilibriums are polymorphism homogeneous of stablestrategy. () Chemical equilibria
All reversible processes tend to attain a state of equilibrium. For a reversiblechemical reaction, an equilibrium state is attained when the rate at which achemical reaction is proceeding equals the rate at which the reverse reaction isproceeding. Consider a hypothetical reversible reaction A + B ⇋ AB.