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Collision theory is a principle of
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
used to predict the rates of 
chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
s. It states that when suitable particles of the
reactant In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
hit each other with correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as
activation energy In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently by Max Trautz in 1916 and William Lewis in 1918. When a catalyst is involved in the collision between the reactant molecules, less energy is required for the chemical change to take place, and hence more collisions have sufficient energy for reaction to occur. The reaction rate therefore increases. Collision theory is closely related to
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
.


Rate constant

The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is : r(T) = kn_\textn_\text= Z \rho \exp \left( \frac \right) where: *''k'' is the rate constant in units of (number of molecules)−1⋅s−1⋅m3. * ''n''A is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
of A in the gas in units of m−3. * ''n''B is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
of B in the gas in units of m−3. E.g. for a gas mixture with gas A concentration 0.1 mol⋅L−1 and B concentration 0.2 mol⋅L−1, the number of density of A is 0.1×6.02×1023÷10−3 = 6.02×1025 m−3, the number of density of B is 0.2×6.02×1023÷10−3 = 1.2×1026 m−3. * ''Z'' is the
collision frequency Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of specie ...
in units of m−3⋅s−1. * \rho is the
steric factor The steric factor, usually denoted ''ρ'', is a quantity used in collision theory. Also called the ''probability factor'', the steric factor is defined as the ratio between the experimental value of the rate constant and the one predicted by co ...
. * ''E''a is the
activation energy In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
of the reaction, in units of J⋅mol−1. * ''T'' is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
in units of K. * ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
in units of J mol−1K−1. The unit of ''r''(''T'') can be converted to mol⋅L−1⋅s−1, after divided by (1000×''N''A), where ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
. For a reaction between A and B, the
collision frequency Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of specie ...
calculated with the hard-sphere model is: : Z = n_\text n_\text \sigma_\text \sqrt\frac = 10^6N_A^2\text \sigma_\text \sqrt\frac where: *''σ''AB is the reaction cross section (unit m2), the area when two molecules collide with each other, simplified to \sigma_\text = \pi(r_\text+r_\text)^2 , where ''r''A the radius of A and ''r''B the radius of B in unit m. * ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
unit J⋅K−1. * ''T'' is the absolute temperature (unit K). * ''μAB'' is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
of the reactants A and B, \mu_\text = \frac (unit kg). * ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
. * is molar concentration of A in unit mol⋅L−1. * is molar concentration of B in unit mol⋅L−1. If all the units that are related to dimension are converted to dm, i.e. mol⋅dm−3 for and dm2 for ''σ''AB, dm2⋅kg⋅s−2⋅K−1 for the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, then : Z = N_\text \sigma_\text \sqrt\frac
text Text may refer to: Written word * Text (literary theory), any object that can be read, including: **Religious text, a writing that a religious tradition considers to be sacred **Text, a verse or passage from scripture used in expository preachin ...
\text] unit mol⋅dm−3⋅s−1.


Collision in diluted solution

Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from
Fick's laws of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
. For a diluted solution in the gas or the liquid phase, the above equation is not suitable when
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. For any given molecule A, it has to collide with a lot of solvent molecules, let's say molecule C, before finding the B molecule to react with.


Quantitative insights


Derivation

Consider the bimolecular elementary reaction: :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 sectionAB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius (r_) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume \pi r^_ c_A per second as it moves, where c_A is the average velocity of the particle. (This solely represents the classical notion of a collision of solid balls. As molecules are quantum-mechanical many-particle systems of electrons and nuclei based upon the Coulomb and exchange interactions, generally they neither obey rotational symmetry nor do they have a box potential. Therefore, more generally the cross section is defined as the reaction probability of a ray of A particles per areal density of B targets, which makes the definition independent from the nature of the interaction between A and B. Consequently, the radius r_ is related to the length scale of their interaction potential.) From kinetic theory it is known that a molecule of A has an
average velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
(different from
root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
velocity) of c_A = \sqrt \frac, where k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, and m_A is the mass of the molecule. The solution of the
two-body problem In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
states that two different moving bodies can be treated as one body which has the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
of both and moves with the velocity of the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, so, in this system \mu_ must be used instead of m_A. Thus, for a given molecule A, it travels t=l/c_A=1/(n_B\sigma_c_A) before hitting a molecule B if all B is fixed with no movement, where l is the average traveling distance. Since B also moves, the relative velocity can be calculated using the reduced mass of A and B. Therefore, the total collision frequency, of all A molecules, with all B molecules, is : Z = n_\text n_\text \sigma_ \sqrt\frac = 10^6N_A^2 \sigma_ \sqrt\frac = z From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is e^. Therefore, the rate of a bimolecular reaction for ideal gases will be :r = z \rho \exp\left( \frac \right), in unit number of molecular reactions s−1⋅m−3, where: * ''Z'' is the collision frequency with unit s−1⋅m−3. The ''z'' is ''Z'' without B]. * \rho is the
steric factor The steric factor, usually denoted ''ρ'', is a quantity used in collision theory. Also called the ''probability factor'', the steric factor is defined as the ratio between the experimental value of the rate constant and the one predicted by co ...
, which will be discussed in detail in the next section, * ''Ea'' is the
activation energy In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
(per mole) of the reaction in unit J/mol, *''T'' is the absolute temperature in unit K, * ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
in unit J/mol/K. * is molar concentration of A in unit mol/L, * is molar concentration of B in unit mol/L. The product ''zρ'' is equivalent to the preexponential factor of the
Arrhenius equation In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in ...
.


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 In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reac ...
for an elementary bimolecular reaction, r = k(T) B], it is noticed that : k(T) = N_A \sigma_\rho \sqrt \frac \exp \left( \frac \right) unit M−1⋅s−1 (= dm3⋅mol−1⋅s−1), with all dimension unit dm including ''k''B. This expression is similar to the
Arrhenius equation In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in ...
and gives the first theoretical explanation for the Arrhenius equation on a molecular basis. 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".Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers.


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, which is not true, as the orientation of the collisions is not always proper for the reaction. For example, in the
hydrogenation Hydrogenation is a chemical reaction between molecular hydrogen (H2) and another compound or element, usually in the presence of a catalyst such as nickel, palladium or platinum. The process is commonly employed to reduce or saturate organic ...
reaction of
ethylene Ethylene (IUPAC name: ethene) is a hydrocarbon which has the formula or . It is a colourless, flammable gas with a faint "sweet and musky" odour when pure. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds). Ethylene ...
the H2 molecule must approach the bonding zone between the atoms, and only a few of all the possible collisions fulfill this requirement. To alleviate this problem, 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): : \rho = \frac, and it is most often less than unity. 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
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s, producing ions. 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. : 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. :


See also

* Two-dimensional gas


References


External links


Introduction to Collision Theory
{{DEFAULTSORT:Collision Theory Chemical kinetics