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

Rate constants were measured at various temperatures for the reaction, $\mathrm{HI}(\mathrm{g})+\mathrm{CH}_{3} \mathrm{I}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{I}_{2}(g)$. The following data were obtained: $$\begin{array}{cc}\hline \text { Rate Constant }\left(\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}\right) & \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) \\\hline 1.91 \times 10^{-2} & 2.05 \times 10^{2} \\2.74 \times 10^{-2} & 2.10 \times 10^{2} \\3.90 \times 10^{-2} & 2.15 \times 10^{2} \\5.51 \times 10^{-2} & 2.20 \times 10^{2} \\7.73 \times 10^{-2} & 2.25 \times 10^{2} \\1.08 \times 10^{-1} & 2.30 \times 10^{2}\end{array}$$ Determine the activation energy in $\mathrm{kJ} / \mathrm{mol}$ both graphically and by calculation using Equation $14.15$. For the calculation of $E_{n}$, use the first and last sets of data in the table.

Rate constants were measured at various temperatures for the reaction, $\mathrm{HI}(\mathrm{g})+\mathrm{CH}_{3} \mathrm{I}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{I}_{2}(g)$.
The following data were obtained:
$$\begin{array}{cc}\hline \text { Rate Constant }\left(\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}\right) & \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) \\\hline 1.91 \times 10^{-2} & 2.05 \times 10^{2} \\2.74 \times 10^{-2} & 2.10 \times 10^{2} \\3.90 \times 10^{-2} & 2.15 \times 10^{2} \\5.51 \times 10^{-2} & 2.20 \times 10^{2} \\7.73 \times 10^{-2} & 2.25 \times 10^{2} \\1.08 \times 10^{-1} & 2.30 \times 10^{2}\end{array}$$
Determine the activation energy in $\mathrm{kJ} / \mathrm{mol}$ both graphically and by calculation using Equation $14.15$. For the calculation of $E_{n}$, use the first and last sets of data in the table.

Key Concepts

Arrhenius Equation

The Arrhenius equation relates the rate constant of a chemical reaction to temperature and activation energy. It is expressed in the form k = A exp(–Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation underpins the analysis of temperature effects on reaction rates and is essential for quantitatively understanding how and why reaction rates change with temperature.

Activation Energy

Activation energy is the minimum amount of energy required for reactants to successfully collide and form products. It represents the energy barrier that must be overcome for a reaction to proceed. In kinetics, a higher activation energy usually corresponds to a slower reaction rate at a given temperature, making it a key parameter in characterizing the temperature sensitivity of chemical processes.

Graphical Determination Using the Arrhenius Plot

A common method for determining activation energy graphically is to plot the natural logarithm of the rate constant (ln k) against the reciprocal of the absolute temperature (1/T). According to the linearized form of the Arrhenius equation (ln k = ln A – Ea/RT), the slope of the resulting straight line is –Ea/R. Analyzing this plot allows one to extract the activation energy directly from the slope, offering a visual and intuitive method for data analysis.

Calculation Using Two Data Points

An alternative approach to determine activation energy involves using the Arrhenius equation with two sets of kinetic data. By taking the ratio or difference of the natural logarithms of the rate constants at two different temperatures, one can isolate the activation energy without requiring a full graphical analysis. This method is particularly useful when only a few data points are available, allowing for a straightforward calculation of the activation energy based on the change in ln k with 1/T.

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