KEY2CHEM

Relationship between Temperature and Rate Constant

The relationship between temperature and rate constant can be defined by the Arrhenius equation ($$k = A e^\frac{-E_a}{RT}$$), where $$k$$ is rate constant, $$A$$ is frequency factor (related to the reaction requirements for orientation), $$e$$ is base of natural log (ln), $$E_a$$ is activation energy, $$R$$ is universal gas constant in energy units ($$8.314\tfrac{J}{mol \cdot K}$$), and $$T$$ is temperature in Kelvin. Increasing the temperature increases the rate constant (which increases the reaction rate), since it increases the fraction of collisions with energy sufficient to overcome the activation barrier. Similarly, lowering the activation energy increases the rate constant (which increases the reaction rate), since the threshold for energy the particles must have is now lower. These ideas can be summarized in a reaction energy diagram, where the potential energy of the reaction is tracked as reactants are converted into products through a transition state (instant of highest potential energy where bonds are partially broken and partially formed). The activation energy is the energy needed to achieve the activated (transition) state.

Example 1.

Increasing the temperature _______ the rate constant and increasing the activation energy ______ the rate constant.

A. increases, increases

B. increases, decreases

C. decreases, increases

Solution

B. increases, decreases

As described by the Arrhenius equation ($$k = A e^\frac{-E_a}{RT}$$), increasing temperature increases the rate constant (which increases reaction rate). Increasing the activation energy decreases the rate constant.

Example 2.

In the diagram below, the same reaction is performed at two temperatures: $$T_1$$ and $$T_2$$. Which statement about the two temperatures is correct?

A. $$T_1 = T_2$$

B. $$T_1 > T_2$$

C. $$T_1 < T_2$$

Solution

C. $$T_1 <T_2$$

Since a larger fraction of collisions have sufficient energy to surpass the activation energy barrier at $$T_2$$ compared for $$T_1$$, $$T_1 < T_2$$.

Example 3.

In the reaction energy diagram below, which point describes the transition state?

A. A

B. B

C. C

Solution

B. B

The transition state is the instant of highest potential energy and represents a species with bonds that are partially broken and partially formed. The activation energy is the energy required to achieve the transition state.