Maxwell-Boltzmann speed distribution

A distribution of speeds

In an ideal gas at thermal equilibrium, molecules move randomly and collide frequently. They do not all have the same speed. Instead, their speeds follow a probability distribution called the Maxwell-Boltzmann speed distribution.

At any moment, some molecules move slowly, many move at moderate speeds, and some move very fast. Temperature controls the shape of the distribution.

The distribution function

The Maxwell-Boltzmann speed distribution is

ight)^{3/2}v^2e^{-mv^2/(2k_BT)}$$ The quantity $f(v)dv$ gives the probability that a molecule has speed between $v$ and $v+dv$. The distribution is normalized: $$int_0^infty f(v),dv=1$$ This means every molecule has some speed between zero and infinity. ## Why the shape looks the way it does The factor $$e^{-mv^2/(2k_BT)}$$ suppresses very high speeds because high kinetic energies are less probable. The factor $v^2$ comes from the number of velocity states with a given speed in three-dimensional velocity space. There are more ways to have a moderate speed than a near-zero speed. The result is a distribution that starts at zero, rises to a peak, then falls off with a long high-speed tail. ## Most probable speed The most probable speed is the speed at which $f(v)$ is maximum: $$v_p=sqrt{rac{2k_BT}{m}}$$ This is the peak of the distribution. It is not the same as the average speed or root-mean-square speed. ## Average and RMS speed The average speed is $$langle v angle=sqrt{rac{8k_BT}{pi m}}$$ The root-mean-square speed is $$v_{rms}=sqrt{langle v^2 angle}=sqrt{rac{3k_BT}{m}}$$ These speeds satisfy $$v_p<langle v angle<v_{rms}$$ because the high-speed tail raises the average and especially the root-mean-square value. ## Temperature effects As temperature increases, the distribution broadens and shifts toward higher speeds. More molecules occupy high-speed states. The area under the curve remains 1, so a broader distribution has a lower and wider peak. Higher temperature means larger average kinetic energy: $$langle K angle=rac{3}{2}k_BT$$ ## Mass effects At the same temperature, lighter molecules have higher characteristic speeds than heavier molecules. Hydrogen molecules move faster on average than oxygen molecules at the same temperature. This affects diffusion, effusion, atmospheric escape, and chemical reaction rates. ## Effusion and Graham's law Effusion is the escape of gas through a tiny hole. Lighter gases effuse faster because their average speeds are higher. Graham's law states that effusion rate is inversely proportional to the square root of molar mass: $$ratepropto rac{1}{sqrt{M}}$$ ## The big idea The Maxwell-Boltzmann distribution describes the spread of molecular speeds in a gas at equilibrium. Temperature and molecular mass determine its shape. The distribution explains why gases have characteristic but varied molecular speeds and connects microscopic randomness to macroscopic temperature.