# Rate constant

The rate constant is used in reaction kinetics to show the proportionality of the reaction rate to the concentrations of the starting materials. It is indirectly a measure of the speed of a chemical reaction. ${\ displaystyle k}$ ${\ displaystyle v}$ For an example reaction is the reaction rate${\ displaystyle \ mathrm {\ alpha \, A + \ beta \, B \ \ rightarrow \ \ gamma \, C}}$ ${\ displaystyle v = {\ frac {1} {\ gamma}} {\ frac {\ mathrm {d} c _ {\ mathrm {C}}} {dt}} = k \ cdot c _ {\ mathrm {A}} ^ {r_ {a}} \ cdot c _ {\ mathrm {B}} ^ {r_ {b}},}$ where is the rate constant of the reaction and the respective partial reaction order. The unit of the rate constant depends on the overall reaction order : ${\ displaystyle k}$ ${\ displaystyle r_ {i}}$ ${\ displaystyle n = \ sum _ {i} r_ {i}}$ ${\ displaystyle [k] = \ mathrm {\ left ({\ frac {liters} {mol}} \ right) ^ {n-1} \ cdot {\ frac {1} {s}}}}$ Occasionally, the rate equation is also with activities at: . Then the occurring rate constant has the unit and is not to be confused with the rate constant above. ${\ displaystyle v = {\ frac {1} {\ gamma}} {\ frac {\ mathrm {d} c _ {\ mathrm {C}}} {dt}} = k \ cdot a _ {\ mathrm {A}} ^ {\ alpha} \ cdot a _ {\ mathrm {B}} ^ {\ beta},}$ ${\ displaystyle \ mathrm {\ frac {mol} {liters \ cdot s}}}$ ## calculation

The rate constant can be calculated using the empirical Arrhenius equation (or using transition state theory ):

${\ displaystyle k = A \ cdot e ^ {- {\ frac {E _ {\ mathrm {A}}} {R \ cdot T}}}}$ With

• Frequency factor  or pre-exponential factor A ( assumed not to be temperature dependent: which is a sufficiently accurate approximation for most purposes)${\ displaystyle A \ neq f (T),}$ • Activation energy in  J / mol${\ displaystyle E _ {\ mathrm {A}}}$ • universal gas constant  R  = 8.314 J / (mol K )
• absolute temperature  T in K.

The temperature dependence of the frequency factor is derived from the impact theory :

${\ displaystyle A = \ sigma \ cdot {\ sqrt {\ frac {8 \ cdot {\ mathit {k _ {\ mathrm {B}}}} \ cdot T} {\ pi \ cdot \ mu}}} \ cdot N_ {\ mathrm {A}}}$ With

• Impact cross-section ${\ displaystyle \ sigma}$ • Boltzmann's constant ${\ displaystyle {\ mathit {k _ {\ mathrm {B}}}}}$ • reduced mass ${\ displaystyle \ mu}$ • Avogadro's constant ${\ displaystyle N _ {\ mathrm {A}}.}$ The frequency factor corresponds to the product of the collision number  Z and the orientation factor  P . The frequency factor thus indicates the maximum number of collisions in the gas phase, taking into account the orientation of the molecules necessary for the reaction.