Rate constants were measured at various temperatures for the...

Rate constants were measured at various temperatures for the reaction, $\mathrm{HI}(g)+\mathrm{CH}_{3} \mathrm{I}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{I}_{2}(g)$. The following data were obtained: $$ \begin{array}{cc}\text { Rate Constant }\left(\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}\right) & \text {Temperature }\left({ }^{\circ} \mathrm{C}\right) \\1.91 \times 10^{-2} & 2.05 \times 10^{2} \\2.74 \times 10^{-2} & 2.10 \times 10^{2} \\3.90 \times 10^{-2} & 2.15 \times 10^{2} \\5.51 \times 10^{-2} & 2.20 \times 10^{2} \\7.73 \times 10^{-2} & 2.25 \times 10^{2} \\1.08 \times 10^{-1} & 2.30 \times 10^{2} \\\end{array}$$ Determine the activation energy in $\mathrm{kJ} / \mathrm{mol}$ both graphically and by calculation using Equation 13.16. For the calculation of $E_{\mathrm{n}}$, use the first and last sets of data in the table.

Key Concepts

Rate Constant and Temperature Dependence

Rate constants are specific to individual reactions and vary with temperature according to the Arrhenius Equation. Their measurement at different temperatures allows for quantitative analysis of the reaction’s energy profile and kinetic parameters, including the activation energy, through both direct calculation and graphical methods.

Graphical Analysis Using an Arrhenius Plot

A common method for determining activation energy is to plot the natural logarithm of the rate constant (ln k) against the inverse of the absolute temperature (1/T). The resultant straight line has a slope equal to –Ea/R, from which the activation energy can be calculated. This graphical approach provides a visual and analytical way to interpret how temperature affects reaction rates.

Arrhenius Equation

The Arrhenius Equation expresses the relationship between the rate constant (k) and temperature (T) through an exponential term involving the activation energy (Ea). It is typically written as k = A exp(–Ea/RT), where A is the pre-exponential factor and R is the gas constant. This equation is fundamental to understanding how reaction rates increase with temperature.

Activation Energy

Activation energy is the minimum energy required for reactants to transform into products. It represents the energy barrier that must be overcome for a reaction to occur. Experimentally, activation energy can be determined either graphically via the Arrhenius plot or by using experimental rate constants in an appropriate form of the Arrhenius Equation.