What fraction of the molecules in a gas at 300 K collide with an energy equal to or greater than Ea when Ea equals 50 kJ/mol? What is the value of this fraction when Ea is 100 kJ/mol?

Verified step by step guidance

1. insert step 1> Convert the activation energy (Ea) from kJ/mol to J/mol by multiplying by 1000, since 1 kJ = 1000 J.
2. insert step 2> Use the Arrhenius equation to find the fraction of molecules with energy equal to or greater than Ea: \( f = e^{-\frac{E_a}{RT}} \), where R is the gas constant (8.314 J/mol·K) and T is the temperature in Kelvin.
3. insert step 3> Substitute the values for Ea (in J/mol), R, and T (300 K) into the equation for both cases: Ea = 50,000 J/mol and Ea = 100,000 J/mol.
4. insert step 4> Calculate the exponent \( -\frac{E_a}{RT} \) for each case to find the fraction of molecules.
5. insert step 5> Interpret the results to understand how the fraction of molecules changes with different activation energies.

Key Concepts

Kinetic Molecular Theory

Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It posits that gas molecules move randomly and collide with each other and the walls of their container. The average kinetic energy of these molecules is directly proportional to the temperature of the gas, which is crucial for understanding how temperature affects collision energy.

Arrhenius Equation

The Arrhenius Equation relates the rate of a chemical reaction to temperature and activation energy (Ea). It shows that the fraction of molecules with energy equal to or greater than Ea increases with temperature. This equation is essential for calculating the fraction of gas molecules that can overcome the activation energy barrier during collisions.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution describes the distribution of speeds (and thus kinetic energies) of particles in a gas. It indicates that at a given temperature, only a fraction of molecules have sufficient energy to overcome the activation energy barrier. This concept is key to determining the fraction of molecules that can successfully collide with energy equal to or greater than Ea.