The Maxwell-Boltzmann Distribution of Molecular Speeds
Molecules in a gas do not share one speed but a distribution with a peak, a mean and a longer tail at high speed. Raising the temperature flattens and widens the curve rather than shifting it rigidly.
What a learner can do afterwards
- Sketches the speed distribution at two temperatures and describes how the peak and tail change
- Distinguishes the most probable, mean and root mean square speeds and orders them
- Explains evaporation and reaction rates by appealing to the high-speed tail
1 · Read
Molecules in a gas do not share one speed. At one temperature they spread over a wide range of speeds, and temperature sets only the average kinetic energy. Lighter molecules move faster on average, so their whole curve shifts toward higher speeds. That spread is why puddles still dry on cool days.
The Maxwell-Boltzmann curve counts how many molecules sit at each speed. Its peak marks the most probable speed. Heating the gas shifts the peak toward higher speeds and flattens and widens the curve instead of shifting it rigidly. The area under any stretch of the curve is the share of molecules sitting in that stretch.
Three speeds describe the curve, and they always line up the same way. From slowest to fastest they are the most probable speed, then the mean speed, then the root mean square speed.
Only the rare fast molecules in the high speed tail can escape a liquid or clear the bar for a reaction. A small rise in temperature grows that tail fast, so evaporation and reaction rates jump.
One temperature means a spread of speeds, not one speed, and the fast tail runs evaporation and reactions.
Includes a reading credited to the academy collection. Rewritten readings were written for this dataset.
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Learn and try: What is the Maxwell-Boltzmann distribution?
Picture the air in this room. Even though it all sits at one temperature, its molecules refuse to march in step. A few crawl, most cruise near the middle, and a rare few sprint. The Maxwell-Boltzmann distribution is the graph of that spread, with speed along the bottom and number of molecules up the side. Its peak marks the most probable speed, the value you would bet on for a randomly picked molecule.
Warm the gas and the whole curve stretches. The peak slides toward higher speeds and the curve flattens, because the same molecules now spread over a wider range. Cool it and the curve squeezes into a tall narrow bump near zero. Sketching both curves side by side is the standard way to show what heating really does at the molecular level.
Three measures of molecular speed
A Maxwell-Boltzmann speed distribution has three useful measures. The most probable speed occurs at the curve's peak. The mean is the average molecular speed. To find the root mean square (rms), average the squared speeds, then take their square root. At any positive temperature, this distribution puts the most probable speed lowest, the mean next and the rms highest.
Temperature is directly related to rms speed through molecular kinetic energy. The average translational kinetic energy is half the molecular mass multiplied by the mean squared speed. In an ideal gas, this energy is proportional to absolute temperature. A thermometer therefore tells us about the molecules' motion.
Molecular impacts and energetic collisions
In a gas, pressure comes from molecules striking container walls and transferring momentum. Many tiny impacts combine into a steady force per unit area. Faster molecules generally transfer more momentum and hit walls more frequently. Heating a sealed, rigid container therefore increases its gas pressure. This molecular account explains the gas laws.
Energetic molecules also help explain evaporation and reactions. Some molecules at a liquid's surface have enough energy to escape, even when the liquid is cool. Reacting molecules must collide with sufficient energy to overcome the activation barrier. When that energy is high compared with typical energies, relatively few collisions qualify.
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