In any gas or liquid sample, individual particles are constantly colliding and exchanging energy. Because of these millions of collisions, the particles do not all possess the same kinetic energy. Some particles move very slowly, while others move extremely fast. The distribution of these energies is described by the Maxwell-Boltzmann distribution.
š Key Principle
The total area under a Maxwell-Boltzmann curve represents the total number of particles in the sample. Therefore, if the size of the sample does not change, the area under the curve must remain constant, even if temperature changes alter the shape of the curve.
A mathematical curve that shows the distribution of kinetic energies among the particles in a gas or liquid at a specific temperature.
The kinetic energy value possessed by the largest number of particles in a sample, represented by the peak of the Maxwell-Boltzmann curve.
The average kinetic energy of all particles in the sample. Due to the asymmetry of the curve, \( E_{mean} \) lies slightly to the right of \( E_{mp} \).
Key Characteristics of the Distribution Curve
Every Maxwell-Boltzmann distribution curve exhibits several fundamental features that you must know and be able to explain:
- Starts at the origin (0,0): This represents the fact that no particles in a sample have zero kinetic energy, as all particles possess some motion.
- Asymptote at high energy: The curve approaches the x-axis but never touches it. This is because there is no theoretical maximum energy for a particle, meaning there is always a tiny probability of finding a particle with extremely high energy.
- Peak and skewness: The curve is not symmetrical; it has a long tail extending to the right (positively skewed). This asymmetry causes the mean energy (\( E_{mean} \)) to be positioned to the right of the most probable energy (\( E_{mp} \)).
Effect of Temperature on the Curve
When the temperature of a sample is increased, the average kinetic energy of the particles increases. This changes the distribution curve in very specific ways:
- The peak of the curve shifts to the right (indicating that the most probable energy increases).
- The peak of the curve shifts down (because the area under the curve must remain constant; since the curve is wider, it must be lower).
- The tail of the curve on the right-hand side is higher than the original curve, showing that a much larger fraction of particles now have high energies.
Effect of a Catalyst on the Distribution
A catalyst increases the rate of reaction by providing an alternative pathway with a lower activation energy (\( E_a \)). It is vital to understand that:
- The addition of a catalyst does not change the kinetic energy of the particles.
- Therefore, the Maxwell-Boltzmann distribution curve remains completely unchanged.
- Instead, the line representing the activation energy (\( E_a \)) shifts to the left (to a lower value, \( E_{cat} \)).
- This shifts the threshold for a successful collision, meaning a significantly larger fraction of particles now possess energy equal to or greater than the new activation energy (\( E \ge E_{cat} \)).
A very common exam error is to state that a catalyst "gives particles more energy" or "increases particle speeds". A catalyst does neither of these. The particle speeds and energies are determined entirely by the temperature. A catalyst simply lowers the energy barrier required, so that more of the existing particles have enough energy to react.
Worked Examples
Solution:
1. An increase in temperature shifts the Maxwell-Boltzmann distribution curve to the right and flattens it.
2. While the total number of particles (the area under the curve) remains constant, the peak moves to a higher energy and the tail at high energies rises.
3. This shift causes a much larger fraction of the particles to possess kinetic energies equal to or greater than the activation energy (\( E \ge E_a \)).
4. Consequently, the frequency of successful collisions increases significantly, leading to a large increase in the reaction rate.
a) The area under the curve.
b) The position of the most probable energy peak.
c) The proportion of particles with energy greater than the activation energy.
Solution:
a) The area under the curve: Unchanged, as the total number of particles in the sample is constant.
b) The position of the peak: Moves to a lower value on the vertical axis (shifts down) and a higher value on the horizontal axis (shifts to the right).
c) The proportion of particles with \( E \ge E_a \): Increases, as shown by the larger shaded area under the curve to the right of the \( E_a \) line.
Get flashcards and quizzes in ChemEasy, or plan your revision with ChemPlan IB.