# 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.