# Is the Arrhenius Equation only valid for reactants in the gas state?

I couldn't find a source that explicitly say this, but given the gas constant R is used, is the Arrhenius equation only valid when all reactants are gases? Do they have to be ideal gases?

If the above is true, is there a similar equation for reactions where at least one reactant is in liquid/solid form?

• No, it has no such limits, obviously R being named "gas constant" is misleading. – Mithoron Jun 14 '15 at 22:56
• $R$ is called the gas constant because it was first determined in relation to gases. However, $R$ is related to two other constants that have wide usage outside the gas world: Boltzmann's constant $k_B$ and Avogadro's number $N_A$: $$R=N_A \cdot k_B$$ – Ben Norris Jun 15 '15 at 10:59
• @BenNorris From what I understand the exponential is a probability function, is that the same regardless of phase? If the equation has no phase restriction then I am curious about the theoretical reasoning behind using the gas constant R as opposed to some other constant (i.e. Boltzmann constant)? – Yandle Jun 15 '15 at 15:54
• @Yandle - calling the 'gas' constant such is a misnomer. The constant has applications all over thermodynamics at the molar scale. It just happened to be discovered during the study of gases. We choose not to use $k_B$ very often because it is more convenient for small numbers of particles. – Ben Norris Jun 15 '15 at 22:32
• @BenNorris My textbook describes the $e^{\frac{-E_a}{RT}}$ as an energy factor that expresses the frequency of collisions that occur with an energy above threshold for reaction ($E_a$). Am I correct to interpret this expression as valid regardless of what phase or combination of phases the reactants are composed of? – Yandle Jun 16 '15 at 1:49

## 1 Answer

As Ben Norris explained, $R$ can be just more convenient so that one can deal with quantities per mole.

I.e. one could write either $e^{(-E_a/RT)}$ or $e^{(-E_a/K_BT)}$, where in the first instance one would use $E_a$ in $\ce{J~mol^{-1}}$, in the second instance just in $\ce{J}$. The term in the exponent should of course be dimensionless.

Since the numbers in $\ce{kJ~mol^{-1}}$ are much more familiar for chemists, one would often use the first one to avoid having to use very small numbers.

The crux is that the Boltzmann factor in the Arrhenius expression is related to the chance for particles having enough energy at a given temperature, $T$, to surmount the activation barrier.

• So is the exponential (to my understanding is expresses the probability of a collision resulting in successful reaction) in the Arrhenius equation valid for all phases and not just gas? Is there an explanation/proof as to why this is the case? – Yandle Jun 19 '15 at 5:05
• Indeed, but not exactly. The exponential relates to the probability of having sufficient energy to surmount the barrier upon collision. Whether or not, given sufficient energy, the collision leads to reaction depends on other factors. E.g. in gas phase theory, the reaction probability depends on the collision cross section and a 'geometrical' factor depending on whether the reaction partners are oriented in the right way to react. (For reactions involving hydrogen atoms and/or reactions at low T, also collisions with insufficient E can lead to reactions: tunneling). – Fedor Jun 28 '15 at 18:49