1.7.3 Maxwell–Boltzmann Distributions

What is the Maxwell-Boltzmann Distribution?

The Maxwell-Boltzmann distribution describes the distribution of molecular energies in a gas at a given temperature. It provides a graphical representation of how many gas molecules have various amounts of kinetic energy.

• The distribution is based on the fact that at any temperature, gas molecules will have a range of energies, from very low to very high.
• A Maxwell-Boltzmann curve shows this energy distribution and helps explain how temperature and other factors affect reaction rates.

Key Features of the Maxwell-Boltzmann Distribution

• The graph is asymptotic, meaning the curve approaches the horizontal axis but never touches it.
• The curve shows a peak where most molecules have a moderate amount of energy. This is the most probable energy.
• There are fewer molecules with very low or very high energies, as shown by the tails of the curve.
• The area under the curve represents the total number of molecules.

Maxwell-Boltzmann Distribution at Different Temperatures

Temperature has a significant effect on the distribution of molecular energies:

At Higher Temperatures

• The curve shifts to the right and flattens. This is because the average kinetic energy of the molecules increases.
• More molecules have energies above the activation energy, which leads to a higher rate of successful collisions and thus a faster reaction rate.
• The peak of the curve is lower and occurs at a higher energy.

At Lower Temperatures

• The curve shifts to the left and becomes taller. This means more molecules have lower kinetic energy.
• Fewer molecules have enough energy to surpass the activation energy, so the reaction rate is slower.
• The peak of the curve is higher and occurs at a lower energy.

Interpreting the Maxwell-Boltzmann Distribution

• Most probable energy: The peak of the curve represents the energy that the largest number of molecules possess.
• Average energy: Located to the right of the peak, the average energy is higher than the most probable energy because the distribution is skewed.
• Activation energy (Ea): The energy threshold that molecules must exceed for a reaction to occur. Only the molecules with energy greater than (Ea) will successfully react.
• As the temperature increases, more molecules will have energies exceeding the activation energy, increasing the rate of reaction.

The Effect of Catalysts on the Maxwell-Boltzmann Distribution

A catalyst does not change the shape of the Maxwell-Boltzmann distribution curve but it lowers the activation energy (Ea).

• This means that a greater proportion of molecules now have enough energy to surpass the activation energy, even without increasing the temperature.
• As a result, the rate of reaction increases because more molecules can successfully collide and react.