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4.2.5. Maxwell-Boltzmann Distribution

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What the distribution describes

In a gas, not all molecules move at the same speed. Even if the gas has one well defined temperature, some molecules move slowly, many move at medium speeds, and a few move very fast. The Maxwell-Boltzmann distribution describes how these molecular speeds are spread out.

This distribution is one of the central ideas of kinetic theory. It connects microscopic motion, meaning the motion of individual molecules, to macroscopic temperature, meaning the temperature we measure for the whole gas.

Why a distribution is needed

If every molecule in a gas had exactly the same speed, the gas would behave in a much simpler way. Real gases do not work like that. Molecules are constantly colliding and exchanging energy. Because of these collisions, their speeds keep changing.

At any moment, the gas contains a large range of speeds. The Maxwell-Boltzmann distribution tells us how likely it is to find molecules in different speed ranges.

For example, it can answer questions like these:

QuestionWhat the distribution tells us
Are slow molecules present?Yes, but not usually in the greatest number
Are very fast molecules present?Yes, but only a small fraction
What speed is most common?The speed at the peak of the curve
How does temperature affect speeds?Higher temperature shifts the distribution to larger speeds

The shape of the distribution

The Maxwell-Boltzmann speed distribution is not symmetric. It starts at zero, rises to a peak, and then falls off gradually toward high speeds.

Very low speeds are uncommon, because the number of possible molecular velocity states near zero is small. Very high speeds are also uncommon, because they require unusually large kinetic energy. Most molecules lie in the middle range.

A sketch of the curve looks like this.

Qualitative Maxwell-Boltzmann speed distribution

The peak marks the most probable speed, often written as $v_{mp}$.

Mathematical form

For molecular speeds in an ideal gas, the Maxwell-Boltzmann speed distribution is

$$
f(v) = 4\pi \left(\frac{m}{2\pi kT}\right)^{3/2} v^2 e^{-mv^2/(2kT)}
$$

where $f(v)\,dv$ is the fraction of molecules with speeds between $v$ and $v + dv$.

Here,

SymbolMeaning
$v$speed of a molecule
$m$mass of one molecule
$k$Boltzmann constant
$T$absolute temperature in kelvin

The factor $v^2$ makes the function rise from zero at small speed, and the exponential factor $e^{-mv^2/(2kT)}$ makes it fall at large speed.

Important interpretation:
$f(v)\,dv$ does not mean the speed of one molecule.
It means the fraction, or probability, of molecules with speeds in the small interval from $v$ to $v+dv$.

Most probable, average, and rms speeds

Because the distribution is not symmetric, there is more than one useful way to describe a "typical" speed.

The three most common are the most probable speed, the average speed, and the root mean square speed.

They are:

$$
v_{mp} = \sqrt{\frac{2kT}{m}}
$$

$$
\bar{v} = \sqrt{\frac{8kT}{\pi m}}
$$

$$
v_{rms} = \sqrt{\frac{3kT}{m}}
$$

These are not equal. Their order is

$$
v_{mp} < \bar{v} < v_{rms}
$$

This happens because the small number of very fast molecules pulls the average and rms values to the right.

For an ideal gas at temperature $T$,
$$
v_{mp} = \sqrt{\frac{2kT}{m}}, \qquad
\bar{v} = \sqrt{\frac{8kT}{\pi m}}, \qquad
v_{rms} = \sqrt{\frac{3kT}{m}}
$$
and always
$$
v_{mp} < \bar{v} < v_{rms}
$$

Comparison of the three characteristic speeds

Speed typeMeaningFormula
Most probable speed $v_{mp}$Speed at the peak of the distribution$\sqrt{\frac{2kT}{m}}$
Average speed $\bar{v}$Mean speed of all molecules$\sqrt{\frac{8kT}{\pi m}}$
RMS speed $v_{rms}$Square root of mean of $v^2$$\sqrt{\frac{3kT}{m}}$

The rms speed is especially useful because it connects directly to kinetic energy.

Effect of temperature

As temperature increases, the distribution changes in a clear way. The peak moves to higher speeds, the curve becomes broader, and the peak becomes lower.

This means molecules are moving faster on average, and the spread of speeds is larger.

A qualitative comparison is shown below.

Effect of temperature on Maxwell-Boltzmann distributions

At higher temperature, more molecules appear in the high speed tail of the curve.

Raising the temperature shifts the Maxwell-Boltzmann distribution toward higher speeds and makes it broader.

Effect of molecular mass

At the same temperature, lighter molecules move faster on average than heavier molecules. This follows directly from the formulas, since the characteristic speeds all contain $1/\sqrt{m}$.

For two gases at the same temperature,

$$
v \propto \frac{1}{\sqrt{m}}
$$

So hydrogen molecules move faster than oxygen molecules at the same temperature.

A qualitative comparison is shown below.

Effect of molecular mass at the same temperature

The lighter gas has its peak farther to the right.

Probability interpretation

The total area under the distribution curve is 1, because all molecules must have some speed.

Mathematically,

$$
\int_0^\infty f(v)\,dv = 1
$$

If we want the fraction of molecules with speeds between $v_1$ and $v_2$, we calculate

$$
\int_{v_1}^{v_2} f(v)\,dv
$$

This gives a probability or fraction of molecules in that speed interval.

Connection with kinetic energy

Since the kinetic energy of one molecule is

$$
K = \frac{1}{2}mv^2
$$

the spread in speeds also means there is a spread in molecular kinetic energies. Temperature is related to the average kinetic energy, but individual molecules do not all have the same kinetic energy.

A higher temperature means the distribution extends further into larger values of $v$, so higher kinetic energies become more common.

Physical meaning of the high-speed tail

The right side of the curve, called the high-speed tail, is very important. Even though only a small fraction of molecules are there, those molecules can strongly affect physical processes.

For example, the fastest molecules are the ones most likely to escape from a planet's atmosphere, or to participate in reactions that require a minimum energy. The Maxwell-Boltzmann distribution helps explain why rare but very energetic molecules can still matter.

Limits of the Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution works well for classical ideal gases, especially at ordinary temperatures and low enough densities. It assumes molecules behave like classical particles and that quantum effects are negligible.

At very low temperatures or very high densities, quantum statistics become important, and different distributions are needed.

The Maxwell-Boltzmann distribution is a classical result for gases. It is most accurate when quantum effects are unimportant.

Summary

The Maxwell-Boltzmann distribution describes how molecular speeds are distributed in a gas. Instead of one common speed, a gas contains a whole range of speeds. The distribution has a peak at the most probable speed and a long tail toward high speeds. Increasing temperature shifts the curve to higher speeds and broadens it. At the same temperature, lighter molecules move faster than heavier ones. The three important characteristic speeds are $v_{mp}$, $\bar{v}$, and $v_{rms}$, and they satisfy

$$
v_{mp} < \bar{v} < v_{rms}
$$

This distribution is one of the key links between microscopic molecular motion and the observable behavior of gases.

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