Maxwell-Boltzmann Distribution
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Core Concept
Maxwell-Boltzmann Distribution describes the range of speeds that gas particles can have at a given temperature. It provides insight into how molecular speed and energy vary among particles in a gas.
Practice Tips
Most Particles Are Near the Most Probable Speed: While some particles are moving very slowly and others very fast, most are near the most probable speed.
Temperature’s Impact: Increasing temperature shifts the distribution curve to the right and flattens it, increasing the number of fast-moving particles.
Mass Matters: At the same temperature, lighter gas particles have a broader distribution and move faster on average than heavier gas particles.
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Key Concepts
Distribution of Molecular Speeds:
Not all particles in a gas move at the same speed. Some move faster, and some slower, depending on factors like temperature and the mass of the gas particles.
The Maxwell-Boltzmann Distribution curve shows the range of speeds, with most particles clustering around a most probable speed.
Temperature and Kinetic Energy:
Temperature directly affects the distribution of molecular speeds.
As temperature increases, the average speed and kinetic energy of particles increase, and the distribution curve flattens and shifts to the right (indicating higher speeds).
Effect of Particle Mass:
Heavier gas particles move more slowly on average than lighter gas particles at the same temperature.
Lighter gases, like helium, have broader distributions and higher average speeds compared to heavier gases like xenon at the same temperature.
Maxwell-Boltzmann Distribution Curve
Shape of the Curve:
The curve is asymmetrical, starting at the origin (0 speed, where no particles exist) and rising to a peak that represents the most probable speed.
The curve then tails off to the right, indicating a small number of particles with very high speeds.
Key Points on the Curve:
Most Probable Speed ($u_{\text{mp}}$): The speed at which the largest number of particles are moving. This is the peak of the distribution.
Average Speed ($u_{\text{avg}}$): The mean speed of all particles, slightly to the right of the most probable speed.
Root Mean Square Speed ($u_{\text{rms}}$): A statistical measure of the speed of particles, even farther to the right. It takes into account the square of particle speeds.
Typically, these speeds relate as follows:
$u_{\text{mp}} < u_{\text{avg}} < u_{\text{rms}}$
Effect of Temperature on the Curve:
At higher temperatures, the curve becomes broader and flatter, shifting to the right. This shows that more particles have higher speeds.
At lower temperatures, the curve is narrower and steeper, with more particles near the most probable speed.
Maxwell-Boltzmann Equation (Advanced Concept)
The Maxwell-Boltzmann distribution of speeds in a gas can be described mathematically (optional for AP Chemistry):
$f(u) = 4 \pi \left( \frac{m}{2 \pi k T} \right)^{3/2} u^2 e^{-\frac{mu^2}{2kT}}$f
Where:
f(u) is the fraction of particles with speed u.
m is the mass of a gas particle.
k is the Boltzmann constant (1.38×10−23J/K).
T is the temperature in Kelvin.
This equation is not necessary to memorize but helps to understand that speed distribution depends on particle mass and temperature.
Visualizing the Maxwell-Boltzmann Distribution Curve
Comparing Temperatures:
For the same gas, a distribution curve at a higher temperature is broader and flatter, indicating a greater range of particle speeds.
Comparing Gases at the Same Temperature:
Lighter gases have a wider distribution (more particles with high speeds) than heavier gases, which have a narrower distribution.
