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## Q. 13.8

using the Two-Point Form of the arrhenius Equation
The reaction between nitrogen dioxide and carbon monoxide is:
$NO_{2}(g) + CO(g) → NO(g) + CO_{2}$(g)
The rate constant at 701 K is measured as 2.57 M$^{-1} . s^{-1}$, and that at 895 K is measured as 567 M$^{-1} . s^{-1}$. Find the activation energy for the reaction in kJ/mol.

 SORT You are given the rate constant of a reaction at two different tem-peratures and asked to find the activation energy. GIVEN T$_{1} = 701 K, k_{1} = 2.57 M^{-1} . s^{-1}$ T$_{2} = 895 K, k_{2} = 567 M^{-1} . s^{-1}$ FIND E$_{a}$ STRATEGIZE Use the two-point form of the Arrhenius equation, which relates the activation energy to the given information and R (a constant). EQUATION ln $\frac{k_{2}}{k_{1}}= \frac{E_{a}}{R} \left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)$ SOLVE Substitute the two rate constants and the two temperatures into the equation. Solve the equation for E$_{a}$, the activation energy, and convert to kJ/mol.

## Verified Solution

ln$\frac{ 567\cancel{ M^{-1} . s^{-1}}}{2.57\cancel{ M^{-1} . s^{-1}}} = \frac{E_{a}}{R} \left(\frac{1}{701K}-\frac{1}{895K}\right)$
5.40 = $\frac{E_{a}}{R} \left(\frac{3.\underline{0}9\times10^{-4}}{K}\right)$
E$_{a} = 5.40 \left(\frac{K}{3.\underline{0}9\times10^{-4}}\right)$R
= 5.40$\left(\frac{\cancel{K}}{3.\underline{0}9\times10^{-4}}\right) 8.314 \frac{J}{mol . \cancel{K}}$
= 1.5 × 10$^{5}$ J/mol
= 1.5 × 10² kJ/mol

CHECK The magnitude of the answer is reasonable. Activation energies for most reactions range from tens to hundreds of kilojoules per mole.