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 : ${\displaystyle k}$ ${\displaystyle r_{i}}$ ${\displaystyle n=\sum _{i}r_{i}}$

${\displaystyle [k]=\mathrm {\left({\frac {Liter}{mol}}\right)^{n-1}\cdot {\frac {1}{s}}} }$

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. ${\displaystyle v={\frac {1}{\gamma }}{\frac {\mathrm {d} c_{\mathrm {C} }}{dt}}=k\cdot a_{\mathrm {A} }^{\alpha }\cdot a_{\mathrm {B} }^{\beta },}$ ${\displaystyle \mathrm {\frac {mol}{Liter\cdot s}} }$

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.

See also

• Damköhler number

Web links

• Directory of reference works and databases with speed constants