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Q3.22:

The rate constant for the decomposition of (N_{2}O_{5}) at various temperatures is given below:

Draw a graph between (\ln k) and (1/T) and calculate the values of (A) and (E_{a}). Predict the rate constant at 30° and 50°C.

Solution :

Step 1: Data Tabulation and Unit Conversion

To perform the Arrhenius analysis, convert temperatures ($T$) from Celsius to Kelvin ($T(K) = T(^\circ C) + 273.15$) and compute $1/T$ and $\ln k$.

$T (K)$ $1/T (K^{-1})$ $k (s^{-1})$ $\ln k$
293 $3.413 \times 10^{-3}$ $1.7 \times 10^{-5}$ $-10.98$
303 $3.300 \times 10^{-3}$ $6.6 \times 10^{-5}$ $-9.62$
313 $3.195 \times 10^{-3}$ $2.5 \times 10^{-4}$ $-8.29$
323 $3.096 \times 10^{-3}$ $8.4 \times 10^{-4}$ $-7.08$
333 $3.003 \times 10^{-3}$ $2.4 \times 10^{-3}$ $-6.03$

Step 2: Linear Regression Analysis

Per the Arrhenius Equation: $\ln k = \ln A - \frac{E_a}{R} \left( \frac{1}{T} \right)$. This follows the form $y = mx + c$, where $m = -\frac{E_a}{R}$ and $c = \ln A$.

Using the slope formula $m = \frac{\Delta y}{\Delta x}$ between extreme points $(3.413 \times 10^{-3}, -10.98)$ and $(3.003 \times 10^{-3}, -6.03)$:

$m = \frac{-6.03 - (-10.98)}{3.003 \times 10^{-3} - 3.413 \times 10^{-3}} = \frac{4.95}{-0.41 \times 10^{-3}} \approx -12073 K$

Step 3: Calculating $E_a$ and $A$

Since $m = -E_a / R$ and $R = 8.314 \, J \cdot K^{-1} \cdot mol^{-1}$:

$E_a = -m \times R = 12073 \times 8.314 = 100375 \, J/mol \approx 100.38 \, kJ/mol$ [Arrhenius Law]

To find $A$, use point $(303 K, -9.62)$:

$-9.62 = \ln A - (12073 \times 3.300 \times 10^{-3}) \implies \ln A = -9.62 + 39.84 = 30.22$

$A = e^{30.22} \approx 1.34 \times 10^{13} \, s^{-1}$

Step 4: Graphical Representation

1/T (K^-1) ln k

Step 5: Rate Constant Prediction

Using $\ln k = \ln A - \frac{E_a}{R} (\frac{1}{T})$:

  • For $30^\circ C$ ($303.15 K$): $\ln k = 30.22 - \frac{100375}{8.314 \times 303.15} = -9.62 \implies k = 6.6 \times 10^{-5} \, s^{-1}$.
  • For $50^\circ C$ ($323.15 K$): $\ln k = 30.22 - \frac{100375}{8.314 \times 323.15} = -7.08 \implies k = 8.4 \times 10^{-4} \, s^{-1}$.

Final Solution: The activation energy $E_a$ is $100.38 \, kJ/mol$, the pre-exponential factor $A$ is $1.34 \times 10^{13} \, s^{-1}$. The predicted rate constants are $6.6 \times 10^{-5} \, s^{-1}$ at $30^\circ C$ and $8.4 \times 10^{-4} \, s^{-1}$ at $50^\circ C$.


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