Most chemical reactions proceed more rapidly as the temperature increases. The increased molecular motion leads to more frequent collisions, and a larger fraction of those collisions have sufficient energy to produce a chemical reaction.
In 1889, Svante Arrhenius proposed an empirical relationship describing how the rate constant depends on temperature:
\[ k=Ae^{-E_a/RT} \]where
Although the Arrhenius equation was developed empirically, it successfully describes the temperature dependence of many chemical reactions.
The activation energy, \(E_a\), is the minimum energy barrier that reacting molecules must overcome before products can form. Increasing the temperature does not change the activation energy itself; instead, it increases the fraction of molecules with enough kinetic energy to overcome the barrier.
::contentReference[oaicite:0]{index=0}A reaction with a small activation energy has a larger rate constant and proceeds more rapidly than one with a larger activation energy under the same conditions.
If the rate constant is measured at two different temperatures, the Arrhenius equation can be written as
\[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]This form is especially useful because it allows the activation energy to be determined from only two experimental measurements of the rate constant.
A more reliable method is to measure the rate constant at several temperatures and rearrange the Arrhenius equation into linear form:
\[ \ln k = -\frac{E_a}{R} \left(\frac{1}{T}\right) + \ln A \]This equation has the form
\[ y=mx+b \]where
\[ \underbrace{\ln k}_{y} = \underbrace{-\frac{E_a}{R}}_{m} \underbrace{\left(\frac{1}{T}\right)}_{x} + \underbrace{\ln A}_{b} \]Therefore, a plot of \(\ln k\) versus \(1/T\) should produce a straight line. The slope is
\[ -\frac{E_a}{R} \]and the intercept is
\[ \ln A. \]| If... | Then... |
|---|---|
| Temperature increases | The rate constant usually increases. |
| Activation energy increases | The rate constant decreases. |
| The Arrhenius plot has a steeper negative slope | The activation energy is larger. |
| The Arrhenius plot has a larger intercept | The Arrhenius factor \(A\) is larger. |
It is important to remember that the activation energy and the Arrhenius factor are properties of the reaction mechanism. Changing the temperature changes the value of the rate constant, but it does not change either \(E_a\) or \(A\).
Big picture: The Arrhenius equation explains why reaction rates generally increase with temperature by relating the rate constant to the activation energy. Measuring how the rate constant changes with temperature provides a way to determine the activation energy and offers insight into the energy barrier that must be overcome for a reaction to occur.
A reaction follows the second-order rate law
\[ \text{Rate}=k[A]^2 \]The rate constant was measured at two temperatures:
| Temperature | Rate constant |
|---|---|
| \(25.0\ ^\circ\mathrm{C}\) | \(1.25\times10^{-2}\ \mathrm{M^{-1}\,s^{-1}}\) |
| \(45.0\ ^\circ\mathrm{C}\) | \(5.80\times10^{-2}\ \mathrm{M^{-1}\,s^{-1}}\) |
Calculate the activation energy for the reaction.
Temperatures used in the Arrhenius equation must be expressed in kelvin:
\[ T_1 = 25.0+273.15 = 298.15\ \mathrm{K} \] \[ T_2 = 45.0+273.15 = 318.15\ \mathrm{K} \]Rearranging to solve for the activation energy gives
\[ E_a = -R \frac{ \ln(k_2/k_1) }{ \left(1/T_2-1/T_1\right) } \]The ratio of the rate constants is
\[ \frac{k_2}{k_1} = \frac{5.80\times10^{-2}} {1.25\times10^{-2}} = 4.64 \]Therefore,
\[ \ln\left(\frac{k_2}{k_1}\right) = \ln(4.64) = 1.535 \]The reciprocal-temperature difference is
\[ \frac{1}{T_2}-\frac{1}{T_1} = \frac{1}{318.15} - \frac{1}{298.15} = -2.108\times10^{-4}\ \mathrm{K^{-1}} \]Substitution gives
\[ E_a = -(8.314) \frac{1.535} {-2.108\times10^{-4}} \] \[ E_a = 6.05\times10^{4}\ \mathrm{J\,mol^{-1}} \]Converting to kilojoules per mole,
\[ \boxed{E_a=60.5\ \mathrm{kJ\,mol^{-1}}} \]Because the reaction is second order, both rate constants have units of \(\mathrm{M^{-1}\,s^{-1}}\). Their ratio is therefore dimensionless:
\[ \frac{k_2}{k_1} = \frac{\mathrm{M^{-1}\,s^{-1}}} {\mathrm{M^{-1}\,s^{-1}}} \] \[ \frac{k_2}{k_1} = \text{dimensionless} \]This is necessary because the argument of a logarithm must be dimensionless. The Arrhenius calculation is otherwise performed in exactly the same way for zeroth-, first-, and second-order reactions.
Physical interpretation: Increasing the temperature from \(25.0\ ^\circ\mathrm{C}\) to \(45.0\ ^\circ\mathrm{C}\) increases the rate constant by a factor of 4.64. The calculated activation energy of \(60.5\ \mathrm{kJ\,mol^{-1}}\) describes the sensitivity of the rate constant to temperature; it does not depend on the concentrations used in the rate law.
Use the two-temperature form of the Arrhenius equation to calculate the missing quantity:
\[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
Big picture: The Arrhenius equation provides the quantitative link between temperature and reaction rate. By measuring how the rate constant changes with temperature, chemists can determine the activation energy of a reaction and gain insight into the energy barrier that must be overcome before reactants can be converted into products.