The **rate constant** is a proportionality factor in the rate law of chemical kinetics that relates the molar concentration of reactants to reaction rate. It is also known as the **reaction rate constant** or **reaction rate coefficient** and is indicated in an equation by the letter *k*.

### Key Takeaways: Rate Constant

- The rate constant, k, is a proportionality constant that indicates the relationship between the molar concentration of reactants and the rate of a chemical reaction.
- The rate constant may be found experimentally, using the molar concentrations of the reactants and the order of reaction. Alternatively, it may be calculated using the Arrhenius equation.
- The units of the rate constant depend on the order of reaction.
- The rate constant isn't a true constant, since its value depends on temperature and other factors.

## Rate Constant Equation

There are a few different ways to write the rate constant equation. There is a form for a general reaction, a first order reaction, and a second order reaction. Also, you can find the rate constant using the Arrhenius equation.

For a general chemical reaction:

aA + bB → cC + dD

the rate of the chemical reaction may be calculated as:

Rate = k[A]^{a}[B]^{b}

Rearranging the terms, the rate constant is:

**rate constant (k) = Rate / ([A] ^{a}[B]^{a})**

Here, k is the rate constant and [A] and [B] are the molar concentrations of the reactants A and B.

The letters a and b represent the order of the reaction with respect to A and the order of the reaction with respect to b. Their values are determined experimentally. Together, they give the order of the reaction, n:

a + b = n

For example, if doubling the concentration of A doubles the reaction rate or quadrupling the concentration of A quadruples the reaction rate, then the reaction is first order with respect to A. The rate constant is:

k = Rate / [A]

If you double the concentration of A and the reaction rate increases four times, the rate of the reaction is proportional to the square of the concentration of A. The reaction is second order with respect to A.

k = Rate / [A]^{2}

## Rate Constant From the Arrhenius Equation

The rate constant may also be expressed using the Arrhenius equation:

k = Ae^{-Ea/RT}

Here, A is a constant for the frequency of particle collisions, Ea is the activation energy of the reaction, R is the universal gas constant, and T is the absolute temperature. From the Arrhenius equation, it is apparent that temperature is the main factor that affects the rate of a chemical reaction. Ideally, the rate constant accounts for all of the variables impacting reaction rate.

## Rate Constant Units

The units of the rate constant depend on the order of reaction. In general, for a reaction with order a + b, the units of the rate constant are mol^{1−(m+n)}·L^{(m+n)−1}·s^{−1}

- For a zero order reaction, the rate constant has units molar per second (M/s) or mole per liter per second (mol·L
^{−1}·s^{−1}) - For a first order reaction, the rate constant has units of per second of s
^{-1} - For a second order reaction, the rate constant has units of liter per mole per second (L·mol
^{−1}·s^{−1}) or (M^{−1}·s^{−1}) - For a third order reaction, the rate constant has units of liter squared per mole squares per second (L
^{2}·mol^{−2}·s^{−1}) or (M^{−2}·s^{−1})

## Other Calculations and Simulations

For higher order reactions or for dynamic chemical reactions, chemists apply a variety of molecular dynamics simulations using computer software. These methods include Divided Saddle Theory, the Bennett Chandler procedure, and Milestoning.

## Not a True Constant

Despite its name, the rate constant isn't actually a constant. It **only holds true at a constant temperature**. It's affected by adding or changing a catalyst, changing the pressure, or even by stirring the chemicals. It doesn't apply if anything changes in a reaction besides the concentration of the reactants. Also, it doesn't work very well if a reaction contains large molecules at a high concentration because the Arrhenius equation assumes reactants are perfect spheres that perform ideal collisions.

## Sources

- Connors, Kenneth (1990).
*Chemical Kinetics: The Study of Reaction Rates in Solution*. John Wiley & Sons. ISBN 978-0-471-72020-1. - Daru, János; Stirling, András (2014). "Divided Saddle Theory: A New Idea for Rate Constant Calculation".
*J. Chem. Theory Comput*. 10 (3): 1121–1127. doi:10.1021/ct400970y - Isaacs, Neil S. (1995). "Section 2.8.3".
*Physical Organic Chemistry*(2nd ed.). Harlow: Addison Wesley Longman. ISBN 9780582218635. - IUPAC (1997).
*(Compendium of Chemical Terminology*2nd ed.) (the "Gold Book"). - Laidler, K. J., Meiser, J.H. (1982).
*Physical Chemistry*. Benjamin/Cummings. ISBN 0-8053-5682-7.