The Maxwell-Boltzmann distribution describes how molecular speeds are spread out in an ideal gas. Even when a gas has one temperature, its molecules do not all move at the same speed. Some move slowly, many move near a typical speed, and a few move very fast.
This distribution helps connect microscopic particle motion to macroscopic quantities like temperature and pressure.
As temperature increases, the speed distribution curve becomes lower, wider, and shifts to the right because more molecules have higher speeds. The area under the curve represents the total fraction of molecules, so it stays equal to 1. Three important speeds are the most probable speed, average speed, and root-mean-square speed.
These speeds depend on temperature and molecular mass, which is why lighter gases move faster than heavier gases at the same temperature.
Understanding Physics: The Maxwell-Boltzmann Distribution
A gas reaches this pattern through countless collisions. Molecules constantly exchange energy when they hit each other. A fast molecule can lose some kinetic energy in one collision, while a slower one can gain it.
After enough collisions, the gas settles into a stable statistical pattern called thermal equilibrium. Individual molecules still change speed all the time.
The overall spread stays steady as long as the temperature stays steady. This is important because temperature describes the average kinetic energy of the particles, not the kinetic energy of every single particle.
The graph is built by sorting molecules into small speed ranges. One range might contain molecules moving from one hundred to one hundred and ten metres per second. The height of the curve shows how common that range is.
A narrow speed range has a small fraction of all molecules, so probability is found from the area under the curve across that range. This explains why the total area is fixed.
The curve never says that every molecule travels at one chosen speed. It gives the likelihood of finding a molecule near a particular speed at one instant.
Speed is different from velocity. Speed has size only, while velocity includes direction. In a gas, molecules move in all directions with no preferred overall direction.
Their average velocity can therefore be zero, even though every molecule is moving. Pressure comes from molecules striking the walls of a container. Faster molecules make larger momentum changes during collisions.
A gas with more energetic molecular motion produces more pressure if its volume is held fixed. This connects the distribution to tyres, aerosol cans, bicycle pumps, and sealed containers that become more pressurised when warmed.
The high-speed tail has effects that are much larger than its small number of molecules suggests. Some molecules at a liquid surface have enough energy to escape into the air. This process is evaporation.
Heating a liquid increases the number of particles in the fast tail, so evaporation becomes faster. Lighter gas molecules have higher typical speeds at the same temperature because the same average kinetic energy gives a greater speed to a smaller mass. This helps explain why helium leaks through tiny gaps more readily than heavier gases.
When reading graphs, pay close attention to the horizontal scale, the vertical meaning, and whether the graph shows speed or velocity. Remember that the root-mean-square speed gives extra weight to very fast molecules, which is why it is useful when relating molecular motion to pressure and kinetic energy.
Key Facts
- Maxwell-Boltzmann speed distribution: f(v) = 4π(m/2πkT)^(3/2) v^2 e^(-mv^2/2kT)
- Most probable speed: v_p = sqrt(2kT/m)
- Average speed: v_avg = sqrt(8kT/πm)
- Root-mean-square speed: v_rms = sqrt(3kT/m)
- The three characteristic speeds are ordered as v_p < v_avg < v_rms.
- At higher temperature, the curve shifts right, becomes broader, and has a lower peak while the total area remains 1.
Vocabulary
- Maxwell-Boltzmann distribution
- A probability distribution that describes the speeds of particles in an ideal gas at a given temperature.
- Most probable speed
- The speed corresponding to the highest point on the Maxwell-Boltzmann speed distribution curve.
- Average speed
- The arithmetic mean of all molecular speeds in a gas sample.
- Root-mean-square speed
- The square root of the average of the squared molecular speeds, closely related to average kinetic energy.
- Boltzmann constant
- The constant k = 1.38 x 10^-23 J/K that links temperature to the average energy of particles.
Common Mistakes to Avoid
- Assuming all molecules move at the same speed. A gas at one temperature contains a range of molecular speeds described by a probability distribution.
- Thinking the peak height increases when temperature increases. Higher temperature spreads the molecules over a wider range of speeds, so the peak usually becomes lower while the area stays the same.
- Mixing up v_p, v_avg, and v_rms. These are different statistical measures, and for a Maxwell-Boltzmann gas they satisfy v_p < v_avg < v_rms.
- Forgetting to use absolute temperature in kelvins. The speed formulas require T in kelvins because particle kinetic energy is proportional to absolute temperature.
Practice Questions
- 1 Hydrogen molecules have mass m = 3.32 x 10^-27 kg. Calculate the most probable speed at T = 300 K using v_p = sqrt(2kT/m) and k = 1.38 x 10^-23 J/K.
- 2 For nitrogen molecules with m = 4.65 x 10^-26 kg at T = 300 K, calculate v_rms = sqrt(3kT/m).
- 3 Two gases are at the same temperature, helium and argon. Explain which gas has a speed distribution curve shifted farther to the right and why.