# How does the rate constant change with the change of the temperature?

In class, we discussed the following multiple-choice question.

The rate expression for a reaction is: $$rate = k[X][Y]$$ Which statement is correct?

A. As the temperature increases, the rate constant decreases.
B. The rate constant increases with increased temperature, but eventually reaches a constant value.
C. As the temperature increases, the rate constant increases.
D. The rate constant is not affected by a change in temperature.

Considering the Arrhenius equation given below, the rate constant is dependent on the absolute temperature of the reactants and activation energy of the reaction.

$$k = Ae^{-\frac{E_\mathrm{A}}{RT}}$$

Taking the following limit as the absolute temperature approaches infinity reveals that the rate constant converges to the Arrhenius constant at very high temperatures. This is also evident when graphing the Arrhenius equation. Therefore, we conclude that option B is correct.

$$\lim_{T \to \infty} Ae^{-\frac{E_\mathrm{A}}{RT}} = A\lim_{T \to \infty} e^{-\frac{E_\mathrm{A}}{RT}} = A$$

However, when checking the mark-scheme, option C is listed as the correct answer. Is there a mistake in my reasoning?

• The Arrhenius "constant" is actually only a pre-exponential factor which is in itself dependent on the temperature. The activation energy is also dependent on the temperature. – Martin - マーチン Feb 6 '20 at 16:42
• @Martin-マーチン"The activation energy is also dependent on the temperature."How? – Adnan AL-Amleh Feb 7 '20 at 3:02
• If the rate constant is not affected by a change of temperature, it means that the activation energy is zero. – Maurice Mar 7 '20 at 20:29

If this is a test question on Arrhenius' Law, at more elementary levels, in my opinion, it maybe an acceptable question. Otherwise per this source, for example, a possible issue from this reference, to quote::

Defying Arrhenius’ law

In a theoretical paper, Ian Smith and colleagues at the University of Birmingham, UK, proposed several mechanisms by which neutral species reacting under the near-vacuum pressure and extremely low temperature conditions of interstellar clouds could defy Arrhenius’ law and actually react more rapidly at lower temperatures [2].

Also:

Arrhenius’ law doesn’t apply here, said Smith, because ’there is no well-defined barrier on the minimum energy path leading from reactants to products in these reactions.’

And:

Chemical reactions at extremely low temperatures, for instance in interstellar clouds, can run at surprisingly fast rates, report astrochemists who have used a combination of theoretical and experimental methods to work out why.

Answer B is technically correct although C could also be accepted. Assuming $$E_A$$ to be constant, at low temperatures because of the inverse $$T$$ in the exponential $$\exp(-E_A/(R\cdot \text{small number}) \equiv \exp(-\text{big})$$ which is a small number. At high temperatures $$\exp(-E_A/(R\cdot \text{big number})\equiv \exp(-\text{small})$$ is a big number, so the rate constant increases with temperature.

At temperature such that the exponential $$\to 1$$ the rate constant is $$A$$ for all practical purposes. Usually this limit is not reached unless $$E_A$$ is very small and then as $$A$$ is also a function of temperature this ($$A$$) term becomes now important. Generally, however, for the vast majority of reactions the exponential term is the most important.