Given Information:
Overall rate constant: \( K = \frac{k_1 k_2}{k_3} \)
Overall activation energy: \( E_a = 400 \, \text{kJ/mol} \)
Activation energies for each step:
\( E_{a1} = 300 \, \text{kJ/mol}, \, E_{a2} = 200 \, \text{kJ/mol}, \, E_{a3} = ? \)
Using the Arrhenius Equation:
The overall rate constant \( K \) and overall activation energy \( E_a \) can be determined by combining the individual rate constants and activation energies as follows:
\[ K = \frac{k_1 k_2}{k_3} \]
According to the Arrhenius equation, we can write: \[ \ln K = \ln \left(\frac{k_1 k_2}{k_3}\right) = \ln k_1 + \ln k_2 - \ln k_3 \] The corresponding activation energy \( E_a \) for \( K \) is: \[ E_a = E_{a1} + E_{a2} - E_{a3} \]
Substituting the Given Values:
\[ 400 = 300 + 200 - E_{a3} \]
Solving for \( E_{a3} \):
\[ E_{a3} = 500 - 400 = 100 \, \text{kJ/mol} \]
Conclusion:
The value of \( E_{a3} \) is \( 100 \, \text{kJ/mol} \).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32