Understanding Activation Energy from Effective Collisions

Given that only 0.0001% of collisions are effective at 400 K, we can use the Arrhenius equation to find the activation energy. The fraction of effective collisions is given by the exponential term in Arrhenius equation:

$f=e^{-\frac{E_a}{RT}}$

Where $f$ is the fraction of effective collisions, $E_a$ is activation energy, $R$ is gas constant, and $T$ is temperature.

Step-by-Step Solution:

Step 1: Convert the percentage to fraction

0.0001% = $\frac{0.0001}{100}=0.000001={10}^{-6}$

Step 2: Write the equation with known values

${10}^{-6}=e^{-\frac{E_a}{RT}}$

Step 3: Take natural logarithm on both sides

$\ln \left({10}^{-6}\right)=-\frac{E_a}{RT}$

$-13.8155=-\frac{E_a}{RT}$

Step 4: Solve for activation energy

$E_a=13.8155\times R\times T$

Using R = 1.987 cal/mol·K and T = 400 K:

$E_a=13.8155\times 1.987\times 400$

$E_a=10978.5\text{cal/mol}$

Step 5: Convert to kcal/mol

$E_a=10.9785\text{kcal/mol}\approx 11.05\text{kcal/mol}$

Final Answer: The activation energy is approximately 11.05 kcal/mol

Related Topics:

• Arrhenius Equation
• Collision Theory
• Activation Energy
• Reaction Kinetics
• Temperature Dependence of Reaction Rates

Key Formulae:

Arrhenius Equation: $k=A{e}^{-\frac{E_a}{RT}}$

Fraction of effective collisions: $f=e^{-\frac{E_a}{RT}}$

Gas constant: R = 1.987 cal/mol·K = 8.314 J/mol·K

Theory:

The Arrhenius equation describes how the rate constant of a chemical reaction depends on temperature. The exponential term represents the fraction of molecules that have sufficient energy to overcome the activation energy barrier. A higher activation energy means fewer molecules have the required energy, resulting in fewer effective collisions and a slower reaction rate.