Chapter 3: Rate Laws
Examples of Rate Laws
First Order Reactions
(1)
\[ \ce{C2H6 -> C2H4 + H2} \]
\[ \boxed{-r_A = k C_{\ce{C2H6}}} \]
\[ \text{with } k = 0.072 \, \text{s}^{-1} \, e^{\frac{82 \, \text{kcal/mol} \left(\frac{1}{1000} - \frac{1}{T}\right)}{1}} \]
(2)
\[ \phi\text{N} = \text{NC}\ell \longrightarrow \phi\text{C}\ell + \text{N}_2 \]
\[ \boxed{-r_A = k C_{\phi\text{N}=\text{NC}\ell}} \]
\[ \text{with } k = 0.00717 \, \text{s}^{-1} \, e^{\frac{-28.7 \, \text{kcal/mol} \left(\frac{1}{333} - \frac{1}{T}\right)}{R}} \]
(3)
\(\ce{CH2OCH2 + H2O ->[H2SO4] HOCH2CH2OH}\)
\[ \boxed{-r_A = k C_{\ce{CH2OCH2}}} \]
(4)
\[ \ce{C2H4 + 1/2 O2 ->[Ag] CH2OCH2} \]
\[ \boxed{-r_A = k P_{\ce{C2H4}}^{1/3} P_{\ce{O2}}^{2/3}} \]
with \(k = 0.014 \, \frac{\text{lbmol}}{\text{atm} \cdot \text{lb cat} \cdot \text{hr}}\), while overall this reaction is first order, it is \(1/3\) order in ethylene and \(2/3\) order in oxygen.
(5)
\[ \underset{A}{\ce{(CH3)3COO(CH3)3}} \rightarrow \ce{C2H6 + 2CH3COCH3} \]
\[ \boxed{-r_A = k C_A} \]
\[ \text{with } k_A = 10^{-4} \, \text{min}^{-1} \, @ \, 50^\circ \text{C}, \, \text{and } E = 85 \, \text{kJ/mol} \]
(6)
\(\ce{CH3COCH3 -> CH2CO + CH4}\)
\(\boxed{-r_A = k \, C_{\ce{CH3COCH3}}}\)
\(\text{with} \, \ln k = 34.34 - \frac{34,322}{T}\)
\(\text{where } k \text{ is in } \text{s}^{-1} \text{ and } T \text{ is in Kelvin}\)
(7)
\(\text{n-}C_4H_{10} \leftrightharpoons \text{i-}C_4H_{10}\)
\((\text{n}) \hspace{4.5em} (\text{i})\)
\[ \boxed{-r_n = k \left[ C_{\text{n}C_4} - \frac{C_{\text{i}C_4}}{K_C} \right]} \]
\(\text{with} \, \ln k = 34.34 - \frac{34,322}{T}\)
where \[ \boxed{k = 31.1 \, \exp \left[ \frac{7906 \, (T - 360)}{360T} \right]} \, \text{and} \, \boxed{K_C = 3.03 \, \exp \left[ \frac{-830.3 \, (T - 333)}{333T} \right]} \]
Second Order Reactions
(1)
\[ \ce{C₆H₆(NO₃)(Cl)(ONCB) + 2NH3 -> NO2C₆H₆NH2 + NH4Cl} \]
\[ \boxed{-r_A = k C_{\text{ONCB}} C_{\text{NH}_3}} \]
with \(k = 0.0017 \, \frac{\text{m}^3}{\text{kmol} \cdot \text{min}} \, \text{at} \, 188^\circ \text{C}\)
and \(E = 11,273 \, \frac{\text{cal}}{\text{mol}}\)
This reaction is first order in ONCB, first order in ammonia and overall second order.
(2)
\(\ce{CNBr + CH3NH2 -> CH3Br + NCNH2}\)
\[ \boxed{-r_A = k C_{\text{CNBr}} C_{\text{CH}_3\text{NH}_2}} \]
with \(k = \frac{2.2 \, \text{dm}^3}{\text{s} \cdot \text{mol}}\)
This reaction is first order in CNBr, first order in CH3NH2 and overall second order.
(3)
\(\ce{CH3COOC2H5 + C4H9OH \xrightleftharpoons{} CH3COOC4H9 + C2H5OH}\)
\(A + B \xrightleftharpoons{} C + D\)
\[ \boxed{-r_A = k \left[ C_A C_B - \frac{C_C C_D}{K_C} \right]} \]
with \(K_C = 1.08\) and \(k = 9 \times 10^{-5} \, \frac{\text{dm}^3}{\text{mol} \cdot \text{s}} \, \text{at} \, 300\,\text{K}\)
Nonelementary Reactions
(1)
\(\ce{CH(CH3)C6H6 -> C6H6 + C3H6}\)
\(\ce{Cumene(C) -> Benzene(B) + Propylene(P)}\)
\[ \boxed{-r_C = \frac{k \left[P_C - P_B P_P / K_P \right]}{1 + K_B P_B + K_C P_C}} \]
(2)
\[ \ce{C2H6 -> C4H10 + H2 + 2CH4 + C2H4} \]
\[ -r_{\ce{C2H6}} = k_1 C_{\ce{C2H6}} + k_2 C_{\ce{C2H6}}^{1/2} \]