Collision theory

Collision theory, proposed by Max Trautz^[1] and William Lewis in 1916 and 1918, qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. Max Trautz ( March 19 1880 &ndash August 19, 1960) was a German Chemist. William Lewis may refer to William Lewis (athlete (1876–1962 American hurdler in the 1900 Olympic Games William Lewis (chemist A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called The reaction rate or rate of reaction for a Reactant or product in a particular reaction is intuitively defined as how fast a reaction takes ^[2] This theory is based on the idea that reactant particles must collide for a reaction to occur, but only a certain fraction of the total collisions have the energy to connect effectively and cause the reactants to transform into products. This is because only a portion of the molecules have enough energy and the right orientation (or "angle") at the moment of impact to break any existing bonds and form new ones. The minimal amount of energy needed for this to occur is known as activation energy. In Chemistry, activation energy, also called midnight energy, is a term introduced in 1889 by the Swedish scientist Svante Arrhenius, that is defined Collision theory is closely related to chemical kinetics. Chemical kinetics, also known as reaction kinetics is the study of rates of chemical processes

Reaction rate tends to increase with concentration - a phenomenon explained by collision theory

1 Rate constant
2 Qualitative overview
3 Quantitative insights
- 3.1 Derivation
- 3.2 Validity of the theory and steric factor
  - 3.2.1 Steric factor
4 References
5 External links

Rate constant

The rate constant for a bimolecular gas phase reaction, as predicted by collision theory is:

$k(T) = Z \rho \exp \left( \frac{-E_{a}}{RT} \right)$ . The reaction rate or rate of reaction for a Reactant or product in a particular reaction is intuitively defined as how fast a reaction takes In Chemistry, concentration is the measure of how much of a given substance there is mixed with another substance In Chemical kinetics a reaction rate constant k or \lambda quantifies the speed of a Chemical reaction.

Z is the collision frequency. See also Collision theory Collision frequency is defined in Chemical kinetics, in the background of theoretical kinetics, as the average number ^[3]
$\scriptstyle \rho$ is the steric factor. See also Collision theory Steric factor P is a term used in collision theory ^[4]
E_a is the activation energy of the reaction. In Chemistry, activation energy, also called midnight energy, is a term introduced in 1889 by the Swedish scientist Svante Arrhenius, that is defined
T is the temperature. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature
R is gas constant. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working

And the collision frequency is:

$Z = N_A \sigma_{AB} \sqrt \frac{8 k_B T}{\pi \mu_{AB}}$

N_A is Avogadro's number
σ_AB is the reaction cross section
k_B is Boltzmann's constant
μ_AB is the reduced mass of the reactants

Qualitative overview

Fundamentally collision theory is based on kinetic theory and therefore it can only be applied strictly to ideal gases, otherwise approximations are used. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics Reduced mass is the "effective" Inertial mass appearing in the Two-body problem of Newtonian mechanics. Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Qualitatively, it assumes that the molecules of the reactants are rigid, uncharged spheres that physically collide prior to reacting. Moreover, it postulates that the majority of collisions do not lead to a reaction, but only those in which the colliding species have:

A kinetic energy greater than a certain minimum, called the activation energy, E_a
The correct spatial orientation (steric factor) with respect to each other. In Chemistry, activation energy, also called midnight energy, is a term introduced in 1889 by the Swedish scientist Svante Arrhenius, that is defined See also Collision theory Steric factor P is a term used in collision theory

These collisions which lead to reaction are called effective collisions. The reaction rate, may be defined as the number of effective collisions per unit time. The reaction rate or rate of reaction for a Reactant or product in a particular reaction is intuitively defined as how fast a reaction takes

According to collision theory, two significant factors determine reaction rates:

Concentration: Increase in concentration of reactants increases the collision frequency between the reactants. Thus the effective collision frequency also increases.
Temperature: The kinetic energy of particles follows the Maxwell-Boltzmann distribution. The Maxwell–Boltzmann distribution is a Probability distribution with applications in Physics and Chemistry. An increase in temperature not only increases the average speed of the reactant particles and the number of collisions, but also the fraction of particles having kinetic energy higher than the activation energy. Thus, the effective collision frequency increases.

If a heterogeneous reaction takes place, then the surface area of the solid is also important: the more reactive centers exposed on the surface (due to the porosity of the solid and how finely divided it is), the more collisions with reacting molecules.

Quantitative insights

Derivation

Collision theory can only be applied quantitatively to bimolecular reactions, of the kind:^[5]

A + B → C

In collision theory it is considered that two particles A and B will collide if their nuclei get closer than a certain distance. The area around a molecule A in which it can collide with an approaching B molecule is called the cross section (σ_AB) of the reaction and is, in principle, the area corresponding to a circle whose radius (r_AB) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume $\scriptstyle \pi r^{2}_{AB} c_A$ per second as it moves, where $\scriptstyle c_A$ is the average velocity of the particle.

From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of $c_A = \sqrt \frac{8 k_B T}{\pi m_A}$ . Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering The Maxwell–Boltzmann distribution is a Probability distribution with applications in Physics and Chemistry. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the Where $\scriptstyle k_B$ is Boltzmann constant and $\scriptstyle m_A$ is the mass of the molecule. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics

The solution of the two body problem states that two different moving bodies can be treated as one body which has the reduced mass of both and moves with the velocity of the center of mass, so, in this system $μ A B$ must be used instead of $m A$ . In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other Reduced mass is the "effective" Inertial mass appearing in the Two-body problem of Newtonian mechanics.

Therefore, the total collision frequency,^[3] of all A molecules, with all B molecules, is:

$N_A^{2} \sigma_{AB} \sqrt \frac{8 k_B T}{\pi \mu_{AB}}[A][B] =N_A^{2} r^{2}_{AB} \sqrt \frac{8 \pi k_B T}{ \mu_{AB}}[A][B] = Z [A][B]$

From Maxwell Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is $e^{\frac{-E_a}{k_BT}}$ . Therefore the rate of a bimolecular reaction for ideal gases will be:

$r = Z \rho [A][B] \exp \left( \frac{-E_{a}}{RT} \right)$

Where:

Z is the collision frequency.
$\scriptstyle \rho$ is the steric factor, which will be discussed in detail in the next section. See also Collision theory Steric factor P is a term used in collision theory
E_a is the activation energy of the reaction. In Chemistry, activation energy, also called midnight energy, is a term introduced in 1889 by the Swedish scientist Svante Arrhenius, that is defined
T is the absolute temperature.
R is gas constant. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working

The product Zρ is equivalent to the preexponential factor of the Arrhenius equation. See also Arrhenius equation In Chemical kinetics, the preexponential factor or A factor is the pre-exponential constant in the Arrhenius equation The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the Rate constant, and therefore rate of a chemical reaction

Validity of the theory and steric factor

Once a theory is formulated, its validity must be tested, that is, compare its predictions with the results of the experiments.

When the expression form of the rate constant is compared with the rate equation for an elementary bimolecular reaction, $\scriptstyle r =k(T) [A][B]$ , it is noticed that $k(T) = N_A \sigma_{AB} \sqrt \frac{8 k_B T}{\pi m_A} \exp \left( \frac{-E_{a}}{RT} \right)$ . The rate law or rate equation for a Chemical reaction is an equation which links the Reaction rate with concentrations or pressures of reactants and constant

That expression is similar to the Arrhenius equation, and gives the first theoretical explanation for the Arrhenius equation on a molecular basis. The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the Rate constant, and therefore rate of a chemical reaction The weak temperature dependence of the preexponential factor is so small compared to the exponential factor that it cannot be measured experimentally, that is, "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T^½ dependence of the preexponential factor is observed experimentally"^[5]

Steric factor

If the values of the predicted rate constants are compared with the values of known rate constants it is noticed that collision theory fails to estimate the constants correctly and the more complex the molecules are, the more it fails. The reason for this is that particles have been supposed to be spherical and able to react in all directions; that is not true, as the orientation of the collisions is not always the right one. For example in the hydrogenation reaction of ethylene the H₂ molecule must approach the bonding zone between the atoms, and only a few of all the possible collisions fulfill this requirement. Hydrogenation is the Chemical reaction that results in addition of Hydrogen (H2 Structure This Hydrocarbon has four Hydrogen Atoms bound to a pair of Carbon atoms that are connected by a Double bond.

A new concept must be introduced: the steric factor, $ρ$ . It is defined as the ratio between the experimental value and the predicted one (or the ratio between the frequency factor and the collision frequency, and it is most often less than unity(one). See also Arrhenius equation In Chemical kinetics, the preexponential factor or A factor is the pre-exponential constant in the Arrhenius equation ^[4] $\rho = \frac{A_{observed}}{Z_{calculated}}$

Usually, the more complex the reactant molecules, the lower the steric factor. Nevertheless, some reactions exhibit steric factors greater than unity: the harpoon reactions, which involve atoms that exchange electrons, producing ions. Harpoon reactions are a type of Chemical reaction between two substances one of them prone to form a Cation, generally a Metal, and the other one prone The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J An ion is an Atom or Molecule which has lost or gained one or more Valence electrons giving it a positive or negative electrical charge The deviation from unity can have different causes: the molecules are not spherical, so different geometries are possible; not all the kinetic energy is delivered into the right spot; the presence of a solvent (when applied to solutions), etc.

Experimental rate constants compared to the ones predicted by collision theory for gas phase reactions
Reaction	A (Arrhenius frequency factor)	Z (collision frequency)	Steric factor
2ClNO → 2Cl + 2NO	9. In Chemical kinetics a reaction rate constant k or \lambda quantifies the speed of a Chemical reaction. See also Arrhenius equation In Chemical kinetics, the preexponential factor or A factor is the pre-exponential constant in the Arrhenius equation See also Collision theory Collision frequency is defined in Chemical kinetics, in the background of theoretical kinetics, as the average number 4 10⁹	5. 9 10¹⁰	0. 16
2ClO → Cl₂ + O₂	6. 3 10⁷	2. 5 10¹⁰	2. 3 10^-3
H₂ + C₂H₄ → C₂H₆	1. 24 10⁶	7. 3 10¹¹	1. 7 10^-6
Br₂ + K → KBr + Br	10¹²	2. 1 10¹¹	4. 3

Collision theory can be applied to reactions in solution; in that case, the solvent cage has an effect on the reactant molecules and several collisions can take place in a single encounter, which leads to predicted preexponential factors being too large. ρ values greater than unity can be attributed to favorable entropic contributions. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy

Experimental rate constants compared to the ones predicted by collision theory for reactions in solution^[6]
Reaction	Solvent	A 10^-11	Z 10^-11	Steric factor
C₂H₅Br + OH^-	C₂H₅OH	4. See also Arrhenius equation In Chemical kinetics, the preexponential factor or A factor is the pre-exponential constant in the Arrhenius equation See also Collision theory Collision frequency is defined in Chemical kinetics, in the background of theoretical kinetics, as the average number Bromoethane, also known as ethyl bromide is a Chemical compound of the Haloalkanes group 30	3. 86	1. 11
C₂H₅O^- + CH₃I	C₂H₅OH	2. Iodomethane, commonly called Methyl iodide and commonly abbreviated "MeI" is the Chemical compound with the formula CH3I 42	1. 93	1. 25
ClCH₂CO₂^- + OH^-	water	4. Water is a common Chemical substance that is essential for the survival of all known forms of Life. 55	2. 86	1. 59
C₃H₆Br₂ + I^-	CH₃OH	1. Methanol, also known as methyl alcohol, carbinol, wood alcohol, wood naphtha or wood spirits, is a Chemical compound 07	1. 39	0. 77
HOCH₂CH₂Cl + OH^-	water	25. 5	2. 78	9. 17
4-CH₃C₆H₄O^- + CH₃I	ethanol	8. Cresols are Organic compounds which are methyl[[phenol]]s They are a widely occurring natural and manufactured group of Aromatic Organic compounds 49	1. 99	4. 27
CH₃(CH₂)₂Cl + I^-	(CH₃)₂CO	0. 3-Pentanone (also known as diethyl ketone but with systematic name pentan-3-one) is a colorless liquid Ketone with an odor like that of Acetone 085	1. 57	0. 054
C₅H₅N + CH₃I	C₂H₂Cl₄	-	-	2. Pyridine is a Chemical compound with the formula C5[[Hydrogen H5]] N. 0 10^-6

References

^ Trautz, Max. Das Gesetz der Reaktionsgeschwindigkeit und der Gleichgewichte in Gasen. Bestätigung der Additivität von Cv-3/2R. Neue Bestimmung der Integrationskonstanten und der Moleküldurchmesser, Zeitschrift für anorganische und allgemeine Chemie, Volume 96, Issue 1, Pages 1 - 28, 1916, [1]
^ International Union of Pure and Applied Chemistry. The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "collision theory". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ ^a ^b International Union of Pure and Applied Chemistry. The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "collision frequency". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ ^a ^b International Union of Pure and Applied Chemistry. The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "steric factor". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ ^a ^b Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers
^ Moelwyn-Hughes

External links

Introduction to Collision Theory

Dictionary

-noun

A theory that relates collisions among particles to reaction rate; reaction rate depends on such factors as concentration, surface area, temperature, stirring, and the presence of either a catalyst or an inhibitor.

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