Effect of the strength of the gravitational field on the rate of chemical reactions
Mirza Wasif Baig*
E-mail: [email protected]
Abstract
The magnitude of the rate of chemical reactions also depends on the position in the gravitational field, where a chemical reaction is being carried out. At higher gravitational field rate of reaction is greater than the rate of reaction at the lower gravitational field provided temperature and pressure are kept constant at two positions in the gravitational field. Effect of gravity on the rates of reactions has been shown by formulating the rate constants from basic theories of chemical kinetics i.e. transition state theory, collision theory, RRKM and Marcus theory in the language of the general theory of relativity. Gravitational transformation of Boltzmann constant and energy quantum levels of molecules has been developed quantum mechanically. Gravitational transformation of thermodynamic state functions has been formulated that successfully explains quasi-equilibrium existing between reactants and activated complex; at different gravitational fields. Gravitational mass dilation has been developed that explains at higher gravitational fields the transition states possess more kinetic energy to sweep translation on the reaction coordinate, resulting in the faster conversion of reactants into products. Gravitational transformation of the half-life equation shows gravitational time dilation for the half-life period of chemical reactions and thus renders the general theory of relativity and present theory are in accord with each other.
Key Words gravitational time dilation; energy spacing, chemical kinetics; gravitational Boltzmann constant; general theory of relativity.
Introduction The special theory of relativity proposed by Einstein appeared in 19051,2 that discarded the absolute notion of space and time.3 General relativity was born as a result of efforts to extend the special theory of relativity to non-inertial frames.4 It describes gravity as a effect rather than a force. It completely interwove space and time to one entity space-time. This space-time behaves as a flexible fabric. Warps and curves in this fabric of space-time is origin of gravity. Geometry of this four-dimensional space-time fabric completely defines the behavior of gravity. General relativity explains that clocks located in closer proximity to a planet i.e. lower gravitational potential run slower than clocks located at a certain height from the surface of a planet i.e. higher gravitational potential. This effect has also already been experimentally verified.5 In past, there had been few attempts to study the effect of gravity on chemical reactions6,7 but a theory which can truly explain how gravity will really affect the rate of chemical reactions is missing so far. The present study is the first theoretical attempt to invoke gravitational time dilation in chemical kinetics that can explain the effect of gravity on rates of chemical reactions in the gravitational potential of an arbitrary massive planet with mass M.
Theory
In order to compare the rates of chemical reactions at higher and lower positions in the gravitational field temperature and pressure at two positions should be same as rates of chemical reactions are functions of temperature and pressure.8 Since temperature remains same in entire gravitational field; even at extreme and strange places likes black holes.9 This is quite similar to Lorentz invariance of temperature which is recently reported in relativistic theory of chemical kinetics that completely explains relativistic time dilation in chemical and nuclear reactions.10 To explain gravitational time dilation we first need to consider relativistic increase in mass for observers moving at fractions of speed of light. For particle moving at fraction of speed of light increase in its mass will be given is defined as, 22 mucmu =−11 0 (1) Now it is well known that mass possessed by particles is due to their interaction with Higgs field through Higgs Mechanism.11-13 When particles are moving at sufficient higher speed there is increase in their mass. So in moving frame particles interact more intensely with Higgs field via Higgs Mechanism resulting in increase of particles mass. This relativistic mass for particles moving at higher speed is a consequence of special theory of relativity.1-3 To have compatibility between special theory of relativity and general relativity, mass dilation for particles in a gravitational field should exist like gravitational time dilation. 3 It can be developed considering a simple thought experiment. It is known when a beam of photon is moved straight from region of lower gravitational potential to region of higher gravitational potential they go under red-shift 20 i.e. their wavelength decreases at region of higher gravitational potential.14 Now consider a thought experiment in which either beam of electrons or atoms are projected against gravitational potential (from region of lower gravitational potential to region of higher gravitational potential) due to gravitational red shift de Broglie wavelength of either atoms or electrons will be greater at higher gravitational region than at lower gravitational potential this will should mass of electrons and atoms to be less at higher gravitational region than at lower gravitational potential due to de Broglie relation.15 This lead to mathematical formulation of gravitational mass dilation as,
2 mhl=− 1 2 GM rc m (2)
So in presence of strong gravitational force particles interact more effectively with Higgs field as compared to particles displaced away from surface of massive planets feeling less gravitational force, this will result in decrease in mass of subatomic particles in higher gravitational fields. So by increasing either velocity of particles or strength of gravitational field interaction of subatomic particles with Higgs field is more pronounced. This clearly indicates that there exists some hidden relation between gravitational field and Higgs field. Mathematical theory providing deep insight of correlation between Higgs field and gravitational field is out of scope of the current manuscript. In this paper all physical quantities that change with strength of gravitational field are defined by subscript h at higher gravitational field while at lower gravitational field they are denoted by subscript l; mathematically related to one another by factor =−12GM rc2 where M is the mass of heavy planet and r is the radial coordinate of observer.3
Gravitational transformation of Statistical Mechanics In nature it is found for molecules that electronic transitions are fastest followed by vibrational transition which are at last followed by rotational transitions.16-18 This is a direct consequence of Heisenberg time energy uncertainty principle which unravels that greater the spacing between two quantum states of system, longer the system can survive in the excited stat. At higher gravitational potential Heisenberg time energy uncertainty relation gives the uncertainty of the energy of a state as,19 t (3) j h h At lower gravitational potential Heisenberg time energy uncertainty relation can be written as,19 t (4) j l l Heisenberg time energy uncertainty shows that the energy of a state would be truly j h stationary only if the life time t is infinite. Since practically life time t is not infinite h h and thus state is smeared over a range . To be consistent with gravitational time dilation, j h life time of an energy state should decrease in higher gravitational potential i.e. =tt . ( )hl( ) For Heisenberg time energy uncertainty to remain valid in all gravitational fields, energy spacing between the states should increase i.e. = −1 (5) jjhl From spectroscopic signatures of molecules at lower gravitational potential it is observed that following inequalities are found in nature for electronic, vibrational and rotational transitions of molecules at room temperature i.e.19
kT (6) ( elec)ll( B ) kT (7) ( vib)ll( B ) ( ) (kT) (8) rotB ll
Inequalities of Eq. s. (6), (7) and (8) for electronic, vibrational and rotational transitions of molecules at room temperature should symmetrically also exist at higher gravitational field i.e.
kT (9) ( elecB)hh( ) kT (10) ( vibB)hh( ) kT (11) ( rotB)hh( )
Gravitational time dilation dictates spacing between allowed energy levels to be greater at higher gravitational potential than at lower gravitational potential i.e. = −1 . So from jjhl Eq.s. (9)-(11) it can be deduced that Boltzmann constant will increase at higher gravitational potential i.e. kk= −1 . ( BB)hl( ) Maxwell Boltzmann Distribution Law
Maxwell Boltzmann Distribution law defines molecules in jth energy level j at higher gravitational potential as,20
( j ) n =−exph (12) j h (kT) B h Gravitational transformation of energy levels and Boltzmann constant straight forward makes Maxwell Boltzmann distribution of molecules remain same at higher and lower gravitational potentials i.e. n== n n (13) jhl j j
Molecular Partition Function Maxwell Boltzmann statistics is most successful theoretical explanation of distribution of atoms and molecules among various energy states accessible to them in thermal equilibrium. Higher temperature and low density switches off the quantum effects.20-21 Product of translational, rotational, vibrational and electronic partition functions gives total partition function for Maxwell Boltzmann Statistics. At higher gravitational potential total partition function can be mathematically expressed in terms of individual translational, rotational, vibrational and electronic partition functions as,
1 TRVENNNN (Qtotal ) = ( q) ( q) ( q) ( q ) (14) h N ! h h h h Translational Partition Function Translational partition function for molecules of mass mh interacting in volume V at higher gravitational field translational partition function can be defined as, 20-21
32 2 mkT T hB( )h (qV) = 2 (15) h h
Gravitational transformations of Boltzmann constant and mass proves that translational partition functions remain same in the gravitational field irrespective of its position in it i.e.
qqqTTT == (16) ( )hl( )
Rotational temperature and Rotational Partition Function Rotational temperature for rotating diatomic molecules at higher gravitational potential can be formulated as,20-21
hcB (=) h (17) R h k ( B )h While rotational constant at higher gravitational potential can be formulated as,
h Bh = 2 (18) 8 cIh According to Eq. (2) at higher gravitational field mass of electron will decrease so according to Heisenberg’s Uncertainty principle i.e. m v x /2 velocity of electron revolving should increase with same factor 휉−1 rendering the total momentum of electron unchanged that keeps bond length same at all gravitational potentials. Gravitational transformation of mass at higher 20,21 potential will also make moment of inertia to decrease i.e. 퐼ℎ = 휉퐼푙. This increases the rotational constant at higher gravitational potential i.e. −1 BBhl= (19) Gravitational transformation of Boltzmann constant and rotational constant straight forward gives, == (20) ( RRR)hl( )
Rotational partition function in terms of rotational temperature at higher gravitational potential can be defined as, T (qR ) = (21) h ( R )h
As from Eq. (21) it follows that rotational will remains same in all gravitational potential irrespective of its position in it i.e.
qRRR== q q (22) ( )hl( )
Vibrational temperature and Vibrational Rotational Partition Function
Vibrational partition function for vibrating molecules at higher gravitational potential can be formulated as,20-21 1 (qV ) = (23) h ( ) 1− vib h T
Vibrational temperature in terms of gravitational frequency of harmonic oscillator will be defined as, h (=) h (24) vib h k ( B )h
Since frequency is inversely related to time so frequency of harmonic oscillator at higher gravitational potential will increase because mass of atoms will decrease at higher gravitational −1 potential and they will oscillate much faster than at higher gravitational potential i.e. hl = . Thus gravitational Boltzmann constant and gravitational frequency will render vibrational temperature to remain same at higher and lower gravitational potentials i.e. == (25) ( vibvibvib)hl( ) As it follows from Eq. (25) that vibrational temperature remains unchanged at different gravitational potentials so this renders vibrational partition function remain unchanged,
qqqVVV == (26) ( )hl( )
Electronic Partition Function
At higher gravitational field electronic partition function for atoms and molecules can be formulated in terms of Electronic temperature as, 20,21 N ( ) qgE =−exp E h (27) ( )h i i T
Electronic temperature at higher gravitational potential will be,
E h(i ) (=) h (28) E h k ( B )h
Gravitational transformations of Boltzmann constant and energy levels straight forward give,
== (29) ( EEE)hl( )
Thus Eq. (29) shows that electronic partition functions remain same at higher and lower gravitational potentials i.e.
qqqEEE == (30) ( )u ( )0
Thus from all respective individual partition functions gives total molecular partition function will remain same at higher and lower gravitational potentials i.e.
QQQQ=== (31) ( totaltotaltotal)hl( )
Gravitational transformation of Statistical Thermodynamics
Statistical mechanics gives the molecular-level view of all the macroscopic thermodynamic quantities such as work, free energy and entropy. In statistical thermodynamics all properties of system in thermodynamics equilibrium are encoded in terms of partition function Q. All thermodynamic state variables and equilibrium constant is are mathematically expressed in terms of partition function.20,21 At higher gravitational potential they all can be mathematically expressed as,
Q UNk=− (32) hB( )h T V
HTkTQVkQ=+lnln (33) hBB ( )hh ( ) TVVT
ln Q SkTkQhBB=+ ln( ) ( ) (34) hhT
AkTQ= ln (35) uB( )h
GkTQVQ=−lnln (36) uB( )h ( ) V T
Gravitational Boltzmann constant straight forward gives the following gravitational transformations for all thermodynamic state functions.
−1 UUhl= (37) −1 HHhl= (38) −1 SShl= (39) −1 AAhl= (40) −1 GGhl= (41)
Thus all thermodynamic state functions increases at higher gravitational field. Gravitational transformations of thermodynamic state functions will be used to explain quasi equilibrium existing between reactants and activated complexes during the chemical reaction.
Gravitational transformation of chemical kinetics
To theoretically explain the effect of the strength of the gravitational field on chemical reactions the necessary mathematical forms of the rate laws in four basic theories of chemical kinetics, meeting the requirements of general relativity, are derived in the following.
Gravitational transformation of rate constant from Transition state theory Transition state theory which separates reactants and products on potential energy surface while taking in account Born-Oppenheimer approximation formulates an expression for thermal rate constant. Maxwell Boltzmann Distribution is used to distribute reactant, products and transition states among their different quantum states even in the absence of equilibrium between reactant and product molecules.22-23There exist special type of equilibrium between reactants and activated complexes named as quasi-equilibrium. In the transition state motion along the reaction coordinate is separated from the other motions and explicitly treated as translational motion in a classical regime. RRKM theory also exclusively treats the motion of transition state along the reaction coordinate as a simple translational motion.24 According to time energy Heisenberg uncertainty principle’s at higher gravitational field life time of transition state t must be ( )h greater than kTto execute translation along the reaction coordinate. Mathematically it ( B )h can be written as kTt which according to gravitational time dilation naturally leads ( B )hh( ) to gravitational transformation of Boltzmann constant i.e. kk= -1 . Eyring equation ( BB)hl( ) for nth order reaction in thermodynamic terms, at higher gravitational field will be defined as; 25,26 (kT) (−SH††) ( ) B hhh (kn ) = (n− 1) expexp (42) h o RRhh T hc( ) Placing gravitational transformations of Boltzmann constant, enthalpy and entropy gives, kk= −1 (43) ( nn)hl( ) Arrhenius factor gives quantitative expression for number of reactant molecules crossing energy barrier and transforming into products. At higher gravitational potential it transforms as; 20,22 (kT) (S † ) B hh Anh =−exp1exp −( ) (n− 1) (43) o Rh hc( ) Placing gravitational transformations of the Boltzmann constant, the universal gas constant and entropy gives.
-1 AAhl= (44) Gravitational transformation of rate constant from Collision Theory of Bimolecular Reactions The collision theory for reaction rates treats the molecules of reactants as hard spheres colliding with each other expressing the rate of chemical reaction with number of collisions. The theory mathematically expresses the rate of reaction in terms of three important parameters (i) collision frequency, (ii) collision cross section and (iii) relative velocity. Collision theory is mostly successful for bimolecular reactions like A + B P. At higher gravitational field collision theory definition of the rate constant in terms of collision cross section and relative velocity of colliding molecules can be stated as,25,26 kN= () (45) ( 2 )h AABrh At higher gravitational field relative velocity between the colliding atoms and molecules can be mathematically expressed as, 1 8(kT) 2 v = B h (46) r h h
Placing gravitational transformations of the Boltzmann constant and mass in Eq. (46) gives gravitational transformation of relative velocity as. vv= −1 rrhl (47)
Relative velocity between the colliding atoms and molecules increases at higher gravitational potential due to gravitational transformation of Boltzmann constant and mass thus molecules to rapidly execute translational motion. Placing value of vv= −1 from Eq. (47) in Eq. (45) rrhl again gives the same gravitational transformation of rate constant,
kk= −1 (48) 22hl At higher gravitational field collision frequency for bimolecular is mathematically defined in 25,26 terms of mole densities AB, collision cross section and relative velocity as, ZN = () (49) ( ABAABrhAB)h Substituting gravitational transformations of relative velocity in Eq. (50) reveals that collision frequency increases at higher gravitational potential, ZZ = −1 (50) ( ABAB)hl( ) Gravitational transformation of collision frequency again goes in the favor of gravitational time dilation.
Gravitational transformation of rate constant from Marcus Theory of Electron Transfer Marcus theory is most successful theoretical model for electron transfer reactions in general and most importantly for outer sphere electron transfer reactions.27-31 This model is based on the rearrangement of solvent molecules around the reactant ions to configure it for a favorable electron transfer. There is a solvent arrangement around each reactant ion having Gibb’s free energy G as a minimum, change in this solvent structure shoots up its free energy. For successful electron transfer the transition state is attained by reduction in the separation between the two reactant ions and reorganization of the solvent structure around each of them. A reaction coordinate for electron transfer can be regarded as combination of these ion-ion separations and solvent reorganization coordinates. Gibb’s free energy of reactants and products versus reaction coordinate behave as a parabolic function. Point of intersection of two parabolic curves corresponding to free energies of reactants and products locates the Transition state. Marcus theory mathematically encodes the activation energy based in terms of reorganization energy. At higher gravitational field the Marcus expression for rate constant of electron transfer reaction AZA + BZB AZA + ∆Z + BZB - ∆Z can be expressed as, 27-31
o 2 −+(())GABhh ()()expkZABhABh= (51) 4hhRT
h is the reorganization energy, it quantitatively explains the energy required to reorganize the system structure from initial to final coordinates, avoiding hopping between any two different electronic states. Like other thermodynamic state functions, reorganization energy should have same gravitational transformation. Since reorganization energy is composed of solvational and vibrational and components respectively. At higher gravitational field vibrational ( o )h ( i )h reorganization energy can be formulated in terms of reduced force constant k of the jth ( j )h r p normal mode coordinates of reactants q j and products q j as, 1 2 =−kqq rp (52) ( ijjj)h ( )h ( ) 2 j At higher gravitational field reduced force constant k of the jth normal mode can be defined ( j )h as k = 4 22 . Gravitational transform of reduced mass = −1 and oscillation ( jhh)h hl frequency = −1 will straight forward gives kk= −1 . Using this gravitational hl ( jj)hl( ) transformation for force constant in Eq. (52) formulates following gravitational transformation of vibrational reorganization energy, = −1 (53) ( ii)u ( )0
At higher gravitational field solvational reorganization energy for ∆e charge transferred between the reactants can be mathematically defined in terms of reactants ionic radii a1 and a2, their centre to centre separation distance W, refractive index and dielectric constants of the solvent which are nh and εh respectively i.e. 2 11111 ( ) =( +−−e) (54) o h 2 22aaW12 (n ) ( s ) s h h Since decrease in mass at higher gravitational potential will certainly give gravitational transformation for dielectric constant and refractive index of solvent as = and ( ss)hl( ) nn= −1/2 respectively. Gravitational transformation of refractive index and dielectric ( ss)hl( ) constant in Eq. (54) gives,
= −1 (55) ( oo)hl( )
Therefore, from Eq. (53) and (54) gravitational transformation for total reorganization energy −1 can be formulated as hl= . Gravitational transformations for free energy, reorganization energy, collision frequency and ideal gas constant in Eq. (51) proves yields, −1 ()()kkAB h= AB l (56) Now consider an electron transfer reaction in different perspective. Let A, B are reactants and X*, X are hypothetical initial and final thermodynamic states of the system defined as intermediates. A + B X* X P (57) When reactants approach each other to undergo electron-transfer reaction suitable solvent fluctuation leads to formation of state X*, which have the same atomic configuration of the reacting pair and of the solvent as that of the activated complex, while it owns electronic configuration of the reactant. X* have two choices either to form the reactant following disorganization of some of the oriented solvent molecules, or it transform to state X by undergoing an electronic transition, X possess the same atomic configuration as that of X* but it owns the electronic configuration of the products. The pair of states X* and X together constitutes activated complex. At higher gravitational field rate constant for electron transfer reaction expressed in terms of electronic coupling (HAB)h between the reaction intermediates of the electron transfer reaction (i.e., the overlap of the electronic wave functions of the two states) can be formulated as,31 o 2 21 2 −+(()G hh ) ()expkH= ( ) (58) et hAB h 4 RT 4 RT hh hh oo−1 Gravitational transformations for free energy i.e. ()()GGhl= , electronic coupling −1 −1 −1 ()()HHABhABl= , universal gas constant RRhl= and reorganization energy hl= results in following gravitational transformation for electron transfer rate constant,
−1 ()()kkethetl = (59)
Gravitational transformation of RRKM rate constant RRKM theory whcih unifies statistical RRK theory with transition state theory have following expression for the rate constant at high pressure limit,24
† † ∞ (푘퐵)ℎ푇 푄푟 푄푣 exp(−(퐸0)ℎ⁄(푘퐵)ℎ푇) (푘푢푛𝑖)ℎ = (60) ℎ 푄푟푄푣
Gravitational transformations of Boltzmann constant and activation energy simply give, ∞ −1 ∞ (푘푢푛𝑖)ℎ = 휉 (푘푢푛𝑖)푙 (61)
Gravitational transformation of rate of reaction From basic knowledge of chemical kinetics, it is well known that the rate of a chemical reaction is defined as the rate of change of concentration “C” with respect to time t.25,26 In case of gas phase reaction “C” is replaced by pressure “P” and number of molecules or atoms “N” in solid phase reactions (nuclear reactions). At higher gravitational filed the rate law can be stated as, dC n r== k C ( nn)hh( ) (62) dth
Substituting gravitational transformations of rate constant in Eq. (62) shows that rate of reaction at higher gravitational field is faster than at low gravitational field.
rr= −1 (63) ( nn)hl( )
Gravitational transformation of Half Life Half-life period is the time period during which one half of the initial concentration Co of a reactant converts in to products. Half life period of reaction at higher gravitational field can be defined as, J (t1/2 ) = (64) h kC(1)n− ( no)h J is coefficient for nth order reaction. Placing gravitational transformations of rate constant kk= −1 simply gives gravitational half- life equation, ( nn)hl( ) tt= (65) ( 1/21/2)hl( )
Gravitational transformation for half life is completely the mirror image of Einstein’s gravitational time dilation equation. This is capable of explaining time dilation at molecular level as Einstein’s equation does in the physical world.
Relativistic Equilibrium Constant
Considering a chemical reaction at chemical equilibrium,20 aA + bB cC + dD (66) Equilibrium constant for this reaction can be expressed in terms of partition function at higher gravitational field as, cd QQCD KR T =− exp ( eqh)h ab( h ) QQAB (67)
Gravitational transformations of difference in zero point energies of reactants and products and ideal gas constant simply shows that chemical equilibrium constant remain same in the entire gravitational field, KKK == (68) ( eqeqeq)hl( ) System at chemical equilibrium should appear the same irrespective of its position in the gravitational field. Rate constant of forward and backward reaction will definitely increases at higher gravitational potential but since Equilibrium constant is their ratio so it remains unchanged. Thus amount of reactant and product in equilibrium with one another remain the same independent of position of chemical system in the gravitational field.
Discussions
Discussion on gravitational statistical thermodynamics
The present theory shows that the gravitational transformation of Boltzmann constant is consistent with the gravitational transformation of energy spacing between permitted quantum levels of molecules. Since molecular transition between two states is dictated by the spacing among them which results in electronic transitions to be quicker than vibronic transitions and vibronic transitions to be quicker than rotational transitions. As general theory of relativity works on the principle that time flows quicker at higher the gravitational field, this should result in faster de-excitation of an excited state. This can only be afforded at the expense of an increase in energy spacing between different quantum states i.e. = −1 . Experimentally it is ( )hl( ) known that in molecules possessing lighter isotopes of elements constituting them have elevated quantum levels than those with heavier isotopes.36 Thus based on this observation, the gravitational mass dilation which has been formulated in current work fully supports the gravitational transformation of energy spacing between different quantum levels of the system under study. Fermi-Dirac statistics, Bose-Einstein statistics and Maxwell Boltzmann statistics all describing distribution of the total number of particles among different quantum states under wkT=−exp 20-21 different scenarios contains a common exponential factor i.e. hB( ( )hh( ) ) . The gravitational transformations of energy levels and Boltzmann constant together keep this exponential factor constant with change in the gravitational field. So partition function of all three kinds of statistics and their distribution in the respected permitted energy states do not change with change in the gravitational field due to this exponential factor. This will result in the number of molecules in particular energy levels to remain the same irrespective of the position of the system in the gravitational field. Thus the ratio of the distribution of the total energy among the total number of molecules comprising the system under study and their all internal degrees of freedom (translational, vibrational, rotational and electronic) will remain the same irrespective of the position of the system in the gravitational field. At higher gravitational potential mass of molecules decreases that will result in a decrease of their moment of inertia i.e. IIhl= . To conserve the law of conservation of angular momentum angular velocity increases at higher the −1 gravitational potential i.e. hl= . This will result in an increase of rotational speed of molecules at higher the gravitational fields which supports the gravitational time dilation. The gravitational transformation of rotational inertia of molecules increases rotational constant at −1 higher the gravitational potential i.e. BBhl= . This again supports the gravitational time dilation. The gravitational transformation of rotational constant and Boltzmann constant keeps the rotational temperature same at all positions in the gravitational field. This makes rotational partition function to remain the same at all positions in the gravitational field. Since molecules execute vibrations, the gravitational transformation of the mass of atoms at higher gravitational field will increase the vibrational frequencies of all vibrational modes present in the molecules −1 resulting in the gravitational vibrational transformation i.e. hl= , this again goes in agreement with the gravitational time dilation. As at higher the gravitational potential time flows at a faster rate, so atoms in molecules will undergo rapid compressions and extensions. The gravitational transformation of vibrational frequency and Boltzmann constant at higher the gravitational field keeps the vibrational temperature the same at all positions in the gravitational field. Similarly, this makes vibrational partition function to remain the same at all positions in the gravitational field. Electronic temperature and electronic partition function will not change with change in the strength of the gravitational field due to the gravitational transformations of the mass and velocities of electrons and their permitted quantum states. Thus all kinds of molecular partition functions i.e. translational, rotational, vibrational and electronic will thus remain the same throughout the entire gravitational potential. Thus decrease in mass of molecules and an increase in spacing between quantum levels at higher the gravitational potential keeps the translational, vibrational, rotational and electronic partition functions constant at different the gravitational potentials. Since translational, rotational, vibrational and electronic temperature already had been mathematically proved invariant under Lorentz boosts.10 Similarly there also exists no gravitational transformation for translational, rotational, vibrational and electronic temperature. Since all thermodynamic state variables are mathematically expressed in terms of partition functions together with Boltzmann constant. The gravitational transformation of Boltzmann constant will make all thermodynamic state functions to increase at higher the gravitational potential. At higher the gravitational potential mass of particles increase that result in an increase of all forms of their internal motions and thus all molecular transformations defining thermodynamic state functions increase at higher the gravitational potential.
Discussion on gravitational chemical kinetics
Transition state theory is the most successful and universal theory of chemical kinetics for evaluating reaction rates. This theory first time coins the concept of an activated complex called transition state which is responsible for conversion of reactants into products by making a translational sweep over the reaction coordinate. According to Eq. (2) mass of transition state †† decreases at higher gravitational field i.e. mmhl= while according to the gravitational transformation of velocity transition state velocity will increase at higher gravitational field i.e. †1† − vvhl= so this will make the momentum of the transition state to remain same at all positions in the gravitational field i.e. =ppphl = . Since de Broglie relation shows that mass and de Broglie wavelength associated with a transition state are inversely related to one another i.e. 21 h =hph . As the momentum of transition state remain same at all positions in the gravitational field, so de Broglie associated with transition state remains same in the entire gravitational potential i.e. hl==. This de Broglie wave associated with transition state should not be confused with de Broglie wave explained earlier, while formulating gravitational mass dilation. In that thought experiment stream of atoms or electrons are pumped out straight in a beam against gravitational potential leading to gravitational red-shift in their de Broglie wavelength, while here transition states for two same reactions are discussed which are carried out at different positions in a gravitational field, whose de Broglie wave remains same at lower and higher gravitational fields. At the higher gravitational field transition state produced will be less massive than transition state produced at lower gravitational potential, so to have same de Broglie wavelength transition state at different gravitational fields; it should possess greater velocity at the higher gravitational field. This fully supports the gravitational time dilation. According to Heisenberg’s Uncertainty principle reaction coordinate sufficient to accommodate transition state should be equal to its size of de Broglie wavelength i.e. q /2 .20 Since ( )h h de Broglie wavelength of transition state is same at all positions in the gravitational field i.e. this keeps reaction coordinate to be same at all positions in the gravitational field i.e. q = q = q . Accommodation of activated complexes along the reaction coordinate is ( )hl( ) mathematically defined as ††= h2 m k T it can be shown that it remains the same at h( h( B )h ) all positions in the gravitational field by gravitational transformations of Boltzmann constant and ††† mass i.e. hl==. At the higher gravitational field, the average rate of passage of activated complexes over the barrier along the coordinate can be formulated as rkTm††= 2 hBh ( )h which on substituting gravitational transformations for Boltzmann constant and mass gives † 1 † − rrhl= . Therefore, at higher gravitational field transition state will have more kinetic energy and thus it will undergo fast translations over the reaction coordinate. Gravitational transformation of the Boltzmann constant facilitates the transition state with more kinetic energy and speeds up its motion along the reaction coordinate. The rate constant of fastest reaction possible at the higher gravitational field will be an order of k T h .34 At the higher ( B )h gravitational field increase in Boltzmann constant will increase the rate constant by increasing the frequency of the passage of reactants through the transition state. Gravitational transformation of rate constant holds for all kinds of reactions regardless of what type of kinetics they follow i.e. either zero order, first order, second order or third order etc. Transitions state theory defines the Arrhenius factor in terms of thermodynamic state functions. Since for all thermodynamic state functions and universal gas constant there exists gravitational transformations thus Arrhenius factor also does have it. Arrhenius factor appearing as a pre- exponential factor in the rate equation controls the frequency for the passage through the transition state. Gravitational transformation of a mass of reacting atoms and molecules increases the Arrhenius which is fully in agreement with gravitational time dilation. Rate constant gives a quantitative picture of the speed of reaction; as the rate constant increases, the rates of reaction also increases, at the higher gravitational potential. This is for the first time mathematically shown in current work that the rate of reaction has a gravitational transformation equivalent to gravitational time dilation in Eq. (63). Gravitational transformation of the half-life period equation is also developed first time here as shown in Eq. (65) and it is same as Einstein’s gravitational time dilation equation. This shows a deep connection between the present theory of rates of reactions with the general theory of relativity. Boosting up the rate of reaction at higher gravitational potential can be compared to kinetic isotopic effect in chemistry.33 When lighter isotopes are present in molecules rate of reaction is increased. Collision theory majorly focusing on the kinetics of bimolecular reactions expresses the rate of reactions as the frequency of bimolecular collisions. Rate of reaction is dictated by the number of fruitful collisions occurring per second. Frequency of collision is responsible for the rate of reaction. Rate constant in collision theory is a product of the relative velocity of colliding molecules and collision cross-section area undergoing in the bimolecular collision. Gravitational transformation of a mass of molecules increases the relative velocities between the colliding molecules. Gravitational transformation of Boltzmann constant also elevates the energy available per molecule, thus shooting up the relative velocity between the molecules. Collision frequency defines the rate of reaction as a product of the area of collision cross-section, mole densities and relative velocity of colliding molecules. Since collision cross-section area and mole densities (concentration) remains the same at all positions in the gravitational field, while relative velocity increases that give gravitational transformation of collision frequency as shown in Eq. (50). Increase in collision frequency of molecules at the higher gravitational field increases the reaction rate. Gravitational collision frequency transformation fully agrees with the general theory of relativity. Since time flows faster at higher gravitational potential, so does frequency to increases at the higher gravitational field. The Marcus theory is a statistical mechanical approach employing potential energy surfaces to describe several important redox processes in chemistry and biology.30,31 Marcus theory majorly dealing with electron transfer reactions that occur from donor to acceptors. These reactions occur on a scale much faster than the nuclear vibrations. Therefore, the nuclei do not appreciably move during electron transfer phenomenon. During the transfer energies of the donor and acceptor, orbitals must match. The energy levels of the donor and acceptor orbitals in the reactants and products come in continual flux due to internal nuclear and the solvent motions. For a successful transfer, the donor and acceptor molecules must attain definite geometries and suitable solvation arrangements that result in the matching of energy levels between the donor and acceptor orbitals. The nuclei of donor and acceptor molecules relax to their optimum positions once electron transfer has occurred. The energy needed to modify the solvation sphere and internal structures making the donor and acceptor orbital of the same energy is defined as reorganization energy. This energy is a barrier to electron transfer and appears in two forms. One of them is vibrational reorganization energy ()ih that is the energy difference between changes in bond length, angles etc. which occur upon electron transfer. Gravitational transformation of a mass of nuclei of reacting species increases their characteristic oscillation frequency and renders thus reduces the force constant at a higher gravitational field. So energy needed to bring changes in bond length, angles etc. for a successful transfer of electron increases at the higher gravitational field will increase as shown in Eq. (53). While solvational reorganization energy ()oh that counts the energy required in the reorganization of the solvent shell is shown in Eq. (54). When electron transfer reaction is carried out in higher gravitational potential gravitational transformation of mass at higher gravitational field makes density to decrease. Therefore, at higher gravitational field solvent becomes less dense than at lower gravitational potential resulting in the gravitational transformation of refractive index and dielectric constant making solvational reorganization energy to increase the successful transfer of electron which is shown in Eq. (55). Eq.s (54) and (55) together makes the gravitational transformation of total reorganization energy possible. This is similar to gravitational transformations of other thermodynamic state functions expressed earlier in this work. Electron transfer reaction occurs via a very slight spatial overlap of the electronic orbitals of the two reacting molecules in the activated complex. The assumption of slight overlap is defined in terms of electronic coupling between intermediates X* and X. The intermediate state X* has two choices either to disappear and reform reactants, or undergo electronic jump to form a state X in which the ions resembles characteristics of products. Rate constant equation for such electron transfer reaction have electronic coupling ()H ABh as a pre-exponential factor that defines the electronic overlap between the intermediates X* and X. According to Heisenberg Uncertainty principle, the greater the overlap shorter will be the lifetimes of states X* and X.29 At higher gravitational field gravitational mass transformation will increase the velocity of electrons keeping the net momentum same while space occupied by these electrons remains unchanged due to momentum- space Uncertainty principle i.e. bond length remains same at all positions in the gravitational field. This increase in electronic oscillations confined in the same space (orbitals) will be made the spatial overlap of orbitals to increase at the higher gravitational field, and thus electronic −1 coupling transform as ()()HHAB h= AB l . Thus rate constant for electron transfer reaction increases at high gravitational field agreeing with gravitational time dilation effect.
Application Gas phase reaction of hydroxyl radical with 2-methyl-1-propanol To demonstrate the effectiveness of the present theory, we consider example of gas-phase reactions of hydroxyl radical with 2-methyl-1-propanol36 at two different positions in earth’s gravitational field provided temperature and pressure to be same at two positions. Effect of gravity on rate constant of this gas phase reaction is elaborated in Table 1.
Table 1. Comparison of rate constants at two different positions in earth’s gravitational field provided temperature and pressure to be same i.e. 298 K and 1 atm at both positions Rate constant at surface of earth41 Rate constant at height of 20200 km (cm3molecule-1s-1) (cm3molecule-1s-1) 0.92×10-11 0.9200000001535×10-11a [a] This value has been developed from Eq. (43)
Conclusions All gravitational transformations developed in current theory have opposite trend to all corresponding Lorentz transformations developed previously in the relativistic theory of chemical kinetics. Since relativistic time dilation has opposite effect to gravitational time dilation.5 All thermodynamic state functions increase at higher gravitational field. Thus their respective gravitational transformations formulated in this paper are completely compatible with gravitational time dilation. Rate constant of reaction increases at higher gravitational potential and thus successfully explains faster rate of all molecular processes at higher gravitational field. Mass dilation developed in current theory indicates hidden influence of gravity on Higgs mechanism. Thus faster time flow in molecular rate processes at higher gravitational potential is consequence of decrease in mass of subatomic particles constituting atoms and molecules and increase in energy spacing of all quantum levels associated with them.
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