Given Data:
Rate constant (k1) = 8.25 x 10^5 M^-1 s^-1 at T1 = 302.0°C
Activation energy (Ea) = 23 kJ/mol
Temperature T2 = 323.0°C
Conversion of temperatures:
T1(K) = 302.0 + 273.15 = 575.15 K
T2(K) = 323.0 + 273.15 = 596.15 K
Formula to find k2:
[ \ln k_2 - \ln k_1 = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) ]
Constants Required:
R = 8.314 J/mol*K
Convert Ea:
23 kJ/mol = 23000 J/mol
Calculation:
[ \ln k_1 = \ln (8.25 \times 10^5) = 14.01 ]
Plugging Values into the Formula:
[ \ln k_2 - 14.01 = \frac{23000}{8.314} \left( \frac{1}{575.15} - \frac{1}{596.15} \right) ]
Calculate ( \frac{1}{T_1} - \frac{1}{T_2} ):
Result: 0.1695 K^-1
[ \ln k_2 - 14.01 = 0.1695 ]
[ \ln k_2 = 14.01 + 0.1695 \approx 14.1795 ]
[ k_2 = e^{14.1795} \approx 9.8 \times 10^5 M^{-1} s^{-1} ]
Given Steps of Mechanism:
Br2 (g) → 2Br (g)
Br (g) + OCl2 (g) → BrOCl (g) + Cl (g)
Br (g) + Cl (g) → BrCl (g)
Overall Reaction:
[ \text{Overall: } Br_2(g) + OCl_2(g) → BrOCl(g) + BrCl(g) ]
Intermediates present:
Br (g), Cl (g)
Given Mechanism:
Step 1: 2NO (g) → N2O2 (g) (rate constant = k1)
Step 2: N2O2 (g) + O2 (g) → 2NO2 (g) (rate constant = k2)
Observation:
k1 << k2 indicates that Step 1 is slower and thus rate-determining.
Observable Rate Law:
Rate = k [NO]^2
This reflects the dependence on the concentration of NO due to the slow initial step.