---
title: Kinetic Theory of Ideal Gases
module: Thermodynamics
moduleNumber: 8
lessonNumber: 1
order: 801
summary: >
  A gas has no springs and no gears, yet it pushes on its container with a definite
  pressure and stores energy in a lawful way. Kinetic theory explains both from the
  motion of the molecules alone: pressure is the accumulated recoil of countless
  elastic impacts, and temperature is the average translational kinetic energy each
  molecule carries. We derive $pV=\tfrac13Nm\overline{v^2}$ from momentum transfer,
  read off $\overline{K}_{\rm tr}=\tfrac32kT$, and use the Maxwell–Boltzmann
  distribution to separate the most probable, mean, and rms speeds — each the right
  average for a different question — while marking where the dilute, classical
  assumptions stop holding.
topics: [Thermodynamics]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 17 — Temperature and Kinetic Theory of Gases; §17-4 and The Distribution of Molecular Speeds"
---

## Molecular collisions and gas pressure

Kinetic theory models a dilute gas as a large collection of molecules in continual
random motion. Between collisions, each molecule moves approximately in a straight
line at constant velocity. Collisions with the container walls reverse the velocity
component normal to a wall and transfer momentum to that wall. Gas pressure is the
macroscopic effect of the enormous number of such momentum transfers per unit area
and per unit time. A molecule moving parallel to a wall does not transfer momentum
to that wall until a collision changes its direction.

An elastic wall collision changes the momentum of a molecule of mass $m$ approaching
a wall normal to the $x$ axis with velocity component $v_x$ from
$mv_x$ to $-mv_x$. The molecule therefore has momentum change $-2mv_x$, and the
wall receives the opposite impulse $+2mv_x$. In a cubical container of side $L$,
the same molecule reaches the same wall at intervals $2L/|v_x|$. Its average force
contribution is consequently proportional to $mv_x^2/L$. The square is essential:
molecules arriving from opposite directions both increase the average pressure.

Summing the contributions of all molecules gives

$$
pV=\frac13Nm\overline{v^2}=\frac23N\overline{K}_{\rm tr},
$$

where $N$ is the number of molecules, $V$ is volume, and the bar denotes an
average over molecules. The factor one third follows from isotropy. In a gas with
no preferred direction, the averages of $v_x^2$, $v_y^2$, and $v_z^2$ are equal,
so each contributes one third of the average squared speed. A gas streaming through
a tube or a strongly directional molecular beam does not satisfy this isotropic
assumption without further analysis.

$$
% caption: Gas pressure is the accumulated recoil of normal molecular impacts on the wall. A faster incident molecule, or a higher collision rate, raises the average force on the shaded wall.
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$$

## Temperature and translational kinetic energy

The ideal-gas law gives $pV=NkT$, where $k$ is Boltzmann's constant. Comparing
this relation with the kinetic-theory pressure equation gives

$$
\overline{K}_{\rm tr}=\frac32kT.
$$

Temperature on the kelvin scale therefore fixes the average translational kinetic
energy per molecule, not the kinetic energy of every individual molecule. Two gas
samples at the same temperature have the same average translational kinetic energy
per molecule even when their molecules have different masses. Their speed scales
are different: lighter molecules must move faster to carry the same average kinetic
energy. Temperature is not a measure of the total kinetic energy of a sample,
because the total also depends on the number of molecules.

The root-mean-square speed is defined by
$v_{\rm rms}=\sqrt{\overline{v^2}}=\sqrt{3kT/m}$. It is the relevant speed scale
because pressure depends on the average of $v^2$, but it is not the average speed
and it is not the most probable speed. For one mole of gas with molar mass $M$,
the equivalent form is $v_{\rm rms}=\sqrt{3RT/M}$. At the same temperature,
helium has a greater rms speed than oxygen because its molar mass is smaller.

## The distribution of molecular speeds

An ideal gas at equilibrium does not contain molecules all travelling at one speed.
The Maxwell--Boltzmann speed distribution places a fraction $f(v)\d v$ of the
molecules in the narrow speed interval from $v$ to $v+\d v$. Its functional form is

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

The distribution begins at zero speed because the number of distinct velocity
directions associated with a very small speed is small. It rises to a peak, then
falls with a long high-speed tail. The most probable speed is the peak location,
$v_{\rm mp}=\sqrt{2kT/m}$. The average speed and rms speed lie farther to the
right because rare fast molecules contribute strongly to averages, especially to
the squared-speed average. The ordering is
$v_{\rm mp}<v_{\rm av}<v_{\rm rms}$.

Increasing temperature shifts the peak to higher speed, lowers its height, and
broadens the curve. The total area remains one because every molecule belongs to
some speed interval. At a fixed temperature, increasing molecular mass shifts the
distribution toward lower speeds. These curve changes explain why a single
"molecular speed" cannot represent all collision rates, diffusion processes, or
escape probabilities. The rms speed is appropriate for pressure calculations; the
full distribution is needed when the high-speed tail matters.

$$
% caption: A Maxwell–Boltzmann speed distribution has a peak and a long high-speed tail. Higher temperature shifts the peak rightward and broadens the curve while the area under each stays one.
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$$

Kinetic theory describes dilute gases accurately when intermolecular forces and
molecular volume are small corrections. Near condensation, at high density, or in
strongly non-equilibrium flows, the ideal-gas assumptions may fail. The pressure
relation and speed distribution then require a model that includes molecular
interactions or a measured equation of state. Within its stated range, the theory
connects pressure, temperature, and molecular motion quantitatively.

The pressure formula contains an average over a very large molecular population.
An individual molecule exerts a sequence of short impulses separated by free
flights, so its force on a wall is not constant. The measured pressure becomes
nearly steady because a macroscopic wall receives collisions from many molecules
at many locations and times. Reducing the number density lowers the collision rate
at the wall and lowers pressure at fixed temperature. Reducing the volume at fixed
number of molecules raises the collision rate because the travel distance between
opposite walls is shorter. These two microscopic effects are contained together in
the inverse-volume dependence of the ideal-gas law.

Molecule--molecule collisions maintain the equilibrium distribution by continually
redistributing velocities. The mean free path is the average distance travelled
between such collisions. For molecules of effective diameter $d$ and number density
$n$, the hard-sphere estimate is $\lambda=1/(\sqrt2\pi d^2n)$. The factor
$\sqrt2$ accounts for the fact that the other molecules move as well. At ordinary
gas density, the mean free path is much larger than a molecular diameter, which
supports the idealization of long free flights interrupted by brief collisions.
At higher density, molecular separation and collision duration become less
negligible, and the simple ideal-gas picture loses accuracy.

The vertical axis on a speed-distribution graph is a fraction per unit speed, not
a fraction at one exact speed. A narrow interval has fraction approximately
$f(v)\,\Delta v$, represented by an area under the curve. Consequently, curve
height alone does not compare populations unless the speed intervals have equal
width. The broad high-temperature curve can have a lower peak than the
low-temperature curve while containing the same total fraction of
molecules. Integrating over any chosen speed range gives the fraction relevant
to a detector threshold, an escape-speed estimate, or a collision-rate calculation.

### Pressure from momentum transfer

The pressure derivation can be written as a sequence of momentum and timing
statements. One molecule occupies a cube of side $L$; take the wall
normal to the $x$ direction. Immediately before an elastic collision its momentum
component is $mv_x$; immediately after, it is $-mv_x$. The molecule's change is
$-2mv_x$, so the wall receives impulse $2mv_x$ in the positive direction. The
approaching molecule has the velocity sign appropriate to the selected wall, and
the transferred momentum always
pushes outward on that wall.

The molecule must travel to the opposite wall and back before striking the same
wall again. Its return time is $2L/|v_x|$. Dividing the impulse magnitude by this
time gives its time-averaged force contribution $mv_x^2/L$. Dividing again by the
wall area $L^2$ gives a pressure contribution $mv_x^2/V$. A slow molecule pushes
less strongly and collides less often; both effects appear in the squared velocity.
The derivation uses a time average for one molecule, then a sum over the molecular
population. A pressure gauge effectively performs both averages over an enormous
number of impacts.

For all molecules, the selected-wall result is
$pV=\sum_i mv_{xi}^2$. Random equilibrium motion has no preferred direction, so
the sums of $v_x^2$, $v_y^2$, and $v_z^2$ are equal. Their sum is the total
$v^2$, yielding $pV=(1/3)Nm\overline{v^2}$. Momentum impulse and collision rate
therefore make pressure depend on squared speed rather than average velocity. The vector average velocity
of a stationary gas is zero, yet every wall receives positive momentum transfer.
Bulk flow adds a nonzero average velocity and requires separating the ordered flow
motion from the random thermal motion.

$$
% caption: One molecule strikes the same wall only after a round trip across the container. Its average force combines the momentum impulse at impact with the interval between impacts.
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$$

### Most probable, mean, and rms speeds

Three speed measures occur naturally in kinetic theory. The most probable speed
$v_{\rm mp}=\sqrt{2kT/m}$ marks the peak of the speed distribution. The mean
speed is the distribution-weighted average,
$v_{\rm av}=\sqrt{8kT/(\pi m)}$. The rms speed,
$v_{\rm rms}=\sqrt{3kT/m}$, is the square root of the average squared speed and
is the speed measure that enters the pressure derivation. Their numerical order is
$v_{\rm mp}<v_{\rm av}<v_{\rm rms}$. The high-speed tail shifts the mean above
the peak and affects the rms speed most strongly because fast speeds are squared.

For nitrogen gas at $300\ \mathrm K$, using molar mass
$0.028\ \mathrm{kg\,mol^{-1}}$, the three values are approximately
$422\ \mathrm{m\,s^{-1}}$, $476\ \mathrm{m\,s^{-1}}$, and
$517\ \mathrm{m\,s^{-1}}$, respectively. These are not three competing answers
to one question. Each summarizes a different feature of the same distribution.
A molecular-beam flux, a diffusion estimate, and a pressure calculation can call
for different averages. Stating only that molecules move "at the gas speed" loses
the information contained in the distribution.

At fixed temperature, all three scales vary as inverse square root of molecular
mass. A helium atom consequently has a much larger characteristic speed than an
oxygen molecule at the same temperature. At fixed mass, all three vary as square
root of absolute temperature, not Celsius temperature. Doubling kelvin temperature
increases each characteristic speed by a factor of $\sqrt2$, while quadrupling it
doubles the speed scale. These proportionalities follow directly from the balance
between molecular mass and average kinetic energy.

$$
% caption: The three standard speed measures lie at different locations on one equilibrium distribution, with $v_{\rm mp}<v_{\rm av}<v_{\rm rms}$. The rms speed is farthest right because high-speed molecules receive extra weight in the squared-speed average.
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$$

### Translational degrees of freedom

A freely moving molecule has three independent translational velocity components:
one along each Cartesian axis. These are the three translational degrees of freedom
in the kinetic-theory model. Its translational kinetic energy is the sum
$K_{\rm tr}=(1/2)mv_x^2+(1/2)mv_y^2+(1/2)mv_z^2$. Isotropy gives equal averages
for the three squared components. Comparison with the ideal-gas law therefore
gives average energy $(1/2)kT$ to each translational quadratic component and
gives the total $(3/2)kT$ per molecule.

This restricted result is often called translational equipartition. It does not
assert that every molecule has equal component energies at every instant. One
molecule can temporarily carry most of its kinetic energy in the $x$ direction,
while another moves mainly in the $z$ direction. The equality concerns the
ensemble average after collisions have randomized directions. It also does not add
rotational or vibrational energy storage to the present ideal translational model;
those degrees require a broader molecular-energy treatment.

For $n$ moles, the total translational kinetic energy is
$(3/2)nRT$. The result is proportional to amount of gas and to absolute
temperature. This molecular result explains why an ideal gas at fixed
volume has pressure proportional to temperature: increasing temperature raises the
average momentum transfer along each of the three directions. The factor three in
the energy and the factor one third in the pressure derivation arise from the same
three-dimensional motion.

The distinction among speed measures can be expressed directly as different
weightings of the same distribution. The most probable speed is selected by the
maximum of $f(v)$. The mean speed uses the area-weighted quantity
$\int_0^\infty vf(v)\d v$. The rms speed uses
$\sqrt{\int_0^\infty v^2f(v)\d v}$, which weights a molecule travelling twice as
fast four times as strongly before the square root is taken. This weighting is why
the rms value is linked to pressure: wall momentum transfer depends on squared
normal speed after collision rate and impulse are combined.

The distribution is a speed distribution, not a distribution of one Cartesian
velocity component. Molecules with the same speed can travel in many directions.
Isotropy makes positive and negative components equally common, so the average of
each velocity component is zero while the average of its square is positive. A
pressure calculation uses these positive squared components. A net flow calculation
uses the signed component average. Keeping these averages separate prevents the
incorrect conclusion that a gas with zero bulk velocity has zero kinetic energy or
zero pressure.

Probability intervals provide a practical way to use the curve. The fraction of
molecules between two speeds is the area under the curve between the corresponding
horizontal coordinates. A high-energy tail may occupy a small fraction of the
area, yet it can control a process that has a threshold speed, such as escape from
a weak gravitational field or passage over an activation barrier. Raising
temperature enlarges that tail fraction even when the change in the most probable
speed is modest. A single rms value is therefore sufficient for the ideal-gas
pressure equation but insufficient for every molecular-rate question.

| distribution quantity | mathematical weighting | information retained | appropriate use |
|---|---|---|---|
| most probable speed $v_{\rm mp}$ | location of the maximum of $f(v)$ | the modal speed interval | Locate the peak of a measured speed histogram. |
| mean speed $v_{\rm av}$ | $\int_0^\infty vf(v)\d v$ | ordinary average of speed magnitude | Estimate a characteristic molecular transport speed when the relevant model calls for a first speed moment. |
| rms speed $v_{\rm rms}$ | $\sqrt{\int_0^\infty v^2f(v)\d v}$ | squared-speed average, weighted toward the fast tail | Relate random molecular motion to pressure and translational kinetic energy. |
| component mean $\overline{v_x}$ | signed average over positive and negative $v_x$ | bulk motion along one axis | Determine flow velocity; it vanishes in an isotropic stationary gas. |
| tail fraction | $\int_{v_*}^{\infty}f(v)\d v$ | population above a specified threshold $v_*$ | Predict a threshold-controlled molecular process rather than replacing the distribution by one representative speed. |

**Collision time and the pressure model.**

The kinetic-theory derivation treats a wall collision as brief compared with the
time between successive impacts. During the short contact interval, the wall force
is large and the molecular momentum changes rapidly. During the much longer free
flight, the idealized molecule experiences no force and carries constant momentum.
The average force in the pressure calculation is the impulse from one brief event
divided by the full interval from that event to the next collision with the same
wall. Replacing the intermittent interaction by a steady average is legitimate
only after specifying the time interval over which the pressure is observed.

The wall is assumed to be much more massive than one molecule and fixed in the
laboratory frame. Under this approximation, the normal molecular velocity reverses
with nearly unchanged magnitude, while tangential velocity is unchanged for a
smooth wall. A moving piston or a rough wall changes the molecular energy as well
as its momentum direction. The same momentum-transfer method still applies, but
the collision rule must include the wall velocity or surface interaction. A
constant-temperature ideal-gas description presumes that the container and gas
have had sufficient interactions to reach thermal equilibrium.

Molecule--molecule collisions are not included explicitly in the wall-force sum,
yet they are essential to the equilibrium assumptions. They redistribute speed and
direction, preventing a persistent beam of molecules from crossing the container.
The mean time between molecular collisions can be far shorter than the return time
to a particular wall in a macroscopic vessel. That does not invalidate the
pressure calculation: the sum uses the instantaneous distribution of velocity
components at the wall, and equilibrium establishes the isotropic average required
for the factor one third.

Pressure measurements also have a time-resolution limit. At very short time or
area scales, the count of collisions fluctuates and the inferred force is noisy.
Increasing the wall area or averaging time includes more impacts and narrows the
relative fluctuation. A macroscopic gauge reports a stable pressure because it
integrates over immense numbers of molecular events. In a rarefied gas or a
microscopic chamber, collision statistics can no longer be hidden behind a smooth
continuum pressure without estimating the event count.

$$
% caption: Kinetic theory separates a brief impact from the much longer interval between impacts. Average force is the impulse divided by the repeat interval, while a real pressure reading averages over many intervals and molecules.
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$$

### Mean free path and the dilute-gas limit

The mean free path is an average distance, not a fixed spacing between collisions.
Some molecules travel only a short distance before meeting another molecule;
others travel much farther through an unusually open region. For hard-sphere
molecules of diameter $d$ and number density $n$, the estimate
$\lambda=1/(\sqrt2\pi d^2n)$ follows from a collision cross-section argument.
A moving molecule sweeps out a cylindrical collision volume, and a collision is
likely when another molecular centre lies within one diameter of its path. The
relative motion of both collision partners produces the factor $\sqrt2$.

At ordinary atmospheric pressure, the mean free path is small on human scales but
large compared with molecular size. Taking $n=2.45\times10^{25}\ \mathrm{m^{-3}}$
and an effective diameter $d=3.7\times10^{-10}\ \mathrm m$ gives roughly
$6.7\times10^{-8}\ \mathrm m$, or several dozen nanometres. This is hundreds of
molecular diameters. The molecules are numerous, but most of the volume is empty
space on the molecular scale. Decreasing pressure at fixed temperature decreases
number density and increases the mean free path in inverse proportion.

The ideal-gas approximation is strongest when the molecular diameter is much less
than the typical separation and when collision duration is short compared with
free-flight time. The dimensionless density measure is $nd^3$. When it is
much less than one, excluded volume and sustained intermolecular forces are small
corrections. Raising pressure or approaching condensation increases the importance
of those corrections. The observed pressure can then differ from $nkT$ because
repulsive volume exclusion and attractive forces alter the wall-collision picture.

Mean free path also determines whether a gas behaves as a continuum near a small
object or channel. If the channel size is much larger than the mean free path,
many collisions occur across it and pressure varies smoothly. If the sizes are
comparable, molecules can travel from wall to wall with few intermolecular
collisions. The pressure remains definable as a momentum flux, but standard
continuum-flow assumptions need modification. This is a limit of the model's
coarse-grained description, not a failure of momentum conservation.

$$
% caption: A mean free path is the average distance a molecule covers between collisions in a dilute gas. Molecular diameters are small compared with the path length, which supports the point-particle and brief-collision approximations.
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$$

**Inferring number density from pressure and temperature.**

A constant-volume gas thermometer permits a direct molecular-scale inference. At
fixed molecule number and volume, $p/T=(N/V)k=nk$. A
graph of absolute pressure against kelvin temperature is a
straight line through the origin in the ideal limit. Its slope is $nk$, so a
measured pressure--temperature ratio determines number density without first
weighing the gas. Celsius temperature cannot be used in this ratio because the
line is not proportional to a shifted temperature scale.

At $p=1.013\times10^5\ \mathrm{Pa}$ and $T=300\ \mathrm K$, the ratio is
$338\ \mathrm{Pa\,K^{-1}}$. Division by Boltzmann's constant gives
$n=2.45\times10^{25}\ \mathrm{m^{-3}}$. A one-litre vessel at those conditions
therefore contains about $2.45\times10^{22}$ molecules. The reciprocal number
density gives volume per molecule; taking its cube root gives a characteristic
centre-to-centre spacing of about $3.4\ \mathrm{nm}$. This spacing is not a
molecular diameter and not a mean free path. It is a density-derived length scale
for a three-dimensional molecular arrangement.

The same measurement can infer average translational kinetic energy. The
temperature obtained from the absolute pressure ratio gives
$\overline K_{\rm tr}=(3/2)kT$, which is about
$6.2\times10^{-21}\ \mathrm J$ per molecule at $300\ \mathrm K$. Pressure alone
does not determine this energy without number density or temperature; a dense cold
gas and a sparse hot gas can have the same pressure. Measuring both pressure and
temperature separates collision frequency from molecular kinetic-energy scale.

Experimental departures from a straight pressure--temperature line are
diagnostic. A changing container volume, a gas leak, a nonuniform temperature, or
nonideal molecular interactions can change the apparent slope. Before interpreting
the slope as $nk$, the apparatus must be allowed to equilibrate and the pressure
must be absolute rather than gauge pressure. The molecular inference is only as
reliable as those macroscopic controls.

$$
% caption: At fixed volume and molecule number, absolute pressure is proportional to kelvin temperature. The slope is $nk$, so one pressure–temperature measurement fixes the number density and a molecular spacing scale.
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$$

**A molecular-speed measurement model.**

Molecular speeds can be inferred from a time-of-flight measurement when collisions
along the flight path are rare. A short source pulse releases molecules into an
evacuated tube of known length $L$. A detector at the far end records the arrival
time after the release trigger. For molecules travelling approximately along the
tube axis, the speed estimate is $v=L/t$. Repeating the pulse produces an arrival
time distribution rather than one time, because the source contains molecules with
many speeds. Early arrivals correspond to faster molecules; later arrivals sample
the slower portion of the same distribution.

The measurement must distinguish flight time from instrument delay. If the source
trigger and detector have a fixed electronic offset, it must be measured or
subtracted before using $L/t$. The tube length is the distance between the defined
release and detection planes, not the outside length of the apparatus. A fractional
uncertainty in length contributes directly to speed uncertainty, while a timing
uncertainty contributes approximately $\Delta t/t$ to the fractional speed
uncertainty. Fast molecules have short flight times and can therefore be more
sensitive to a fixed timing error.

An aperture selects directions as well as position. Without collimation, a
molecule travelling at an angle follows a longer path than the axial separation,
and the simple ratio underestimates its actual speed. A narrow source and detector
acceptance angle reduce this geometric spread, at the cost of fewer detected
molecules. A detector also has finite response time, so a very narrow arrival peak
can be broadened by the instrument rather than by the gas. The measured histogram
is consequently a convolution of the molecular distribution with source duration,
path geometry, and detector response.

The gas must be sufficiently rarefied that a molecule normally reaches the
detector without a collision. If the tube length is much greater than the mean free
path, collisions randomize directions and times before detection, and the result
no longer represents a simple source speed distribution. A pressure reduction
increases the mean free path and restores ballistic transport. The required vacuum
level depends on tube length and molecular diameter, connecting a laboratory
measurement directly to the mean-free-path model.

$$
% caption: A pulsed molecular beam converts flight time into a speed estimate $v=L/t$. Collimation fixes the path geometry, and a low pressure makes collisions inside the tube unlikely.
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$$

**Conditions behind the ideal-gas approximation.**

The ideal-gas equation is not a statement that molecules have no size or no
interactions. It states that, under dilute conditions, those effects make small
corrections to the pressure and energy relations. Molecular volume is negligible
when the typical separation greatly exceeds the molecular diameter. Intermolecular
forces are negligible for most of a free flight when molecules are far apart and
act only briefly during a collision. Under those conditions, the container volume
is nearly all available free volume and wall momentum transfer dominates the
pressure description.

Two different departures appear as density rises. Finite molecular size reduces
the volume available for translational motion, tending to make the measured
pressure larger than a point-particle estimate at the same number density and
temperature. Attractive forces can pull molecules inward before they reach a wall,
reducing their wall momentum transfer and tending to reduce pressure. Which effect
dominates depends on temperature and density. The simple relation $p=nkT$ cannot
separate these mechanisms because it omits both from the model.

The ratio of mean free path to apparatus size is a separate issue from nonideal
molecular forces. A very low-density gas can be close to ideal yet fail a
continuum-flow approximation because molecules cross a narrow channel without
many collisions. Conversely, a dense gas can behave as a smooth continuum while
showing clear nonideal pressure corrections. One condition concerns the validity
of treating molecules as independent point particles; the other concerns whether
a spatially averaged pressure and velocity field describes a device.

Temperature matters because kinetic energy competes with intermolecular attraction.
At high temperature, molecules pass through the attractive region rapidly and
thermal motion dominates. At low temperature or near condensation, attraction
becomes comparatively important and a gas can no longer be represented reliably by
independent free flights. The warning signs are systematic curvature in a
pressure--temperature plot at fixed volume, strong density dependence of the
pressure ratio, or a measured compressibility inconsistent with the dilute model.

> **Worked example.** A sealed rigid vessel holds an ideal gas at absolute pressure
> $1.20\times10^5\ \mathrm{Pa}$ and temperature $300\ \mathrm K$. It is heated
> uniformly to $450\ \mathrm K$ with no gas added or removed. Both molecule number and
> volume are fixed, so $p/T$ is constant and $p_2=p_1T_2/T_1$; therefore
> $p_2=(1.20\times10^5\ \mathrm{Pa})(450/300)=1.80\times10^5\ \mathrm{Pa}$. The
> pressure rises by fifty percent because the kelvin temperature does.
>
> The ratio needs absolute pressure. If a gauge instead reads
> $1.20\times10^5\ \mathrm{Pa}$ above a one-atmosphere surrounding, the absolute
> pressure is $2.21\times10^5\ \mathrm{Pa}$; heating to $450\ \mathrm K$ raises it to
> $3.32\times10^5\ \mathrm{Pa}$, a gauge reading of about
> $2.31\times10^5\ \mathrm{Pa}$ if the atmosphere is unchanged. Using gauge values in
> the ratio predicts the wrong final reading.

In the collision model, heating raises the average translational kinetic energy per
molecule from $\tfrac32k(300\ \mathrm K)$ to $\tfrac32k(450\ \mathrm K)$. The rms
speed rises only by $\sqrt{450/300}=1.225$, not by $1.5$; pressure rises by $1.5$
because impulse per collision and collision frequency both grow with molecular speed,
and together they scale with mean squared speed and hence with temperature.

The fixed-volume assumption is testable: a flexible vessel expands during heating and
the pressure rise falls below the rigid-vessel prediction, and a leak lowers molecule
number with the same qualitative effect. Several equilibrated pressure--temperature
points distinguish a straight ideal trend from changes caused by volume, gas loss, or
nonideal interactions.

### Full microscopic derivation of the pressure equation

Take a cubical container with side length $L$ and select the wall perpendicular to
the positive $x$ axis. Write the velocity of molecule $i$ as
$(v_{xi},v_{yi},v_{zi})$. Only $v_{xi}$ enters the momentum transferred to this
particular wall. The tangential components can be nonzero, but a smooth elastic
collision leaves them unchanged and they provide no normal impulse. Just before a
collision with the selected wall, the normal molecular momentum is $mv_{xi}$;
just after it, the normal momentum is $-mv_{xi}$. Momentum conservation gives the
wall an impulse $2mv_{xi}$ directed outward from the gas.

A molecule moving toward that wall travels $2L$ between successive collisions with
the same wall. The recurrence interval is therefore
$2L/v_{xi}$ when the relevant velocity component is positive. Dividing impulse by
recurrence interval gives the average normal force from that molecule,
$mv_{xi}^2/L$. Molecules that are currently travelling away from the selected wall
give the same positive time-averaged contribution when their next round trip is
included. Squaring the component removes any dependence on which side of the cube
the molecule is moving toward at a particular instant.

Summing over the $N$ molecules gives the average force
$F_x=(m/L)\sum_i v_{xi}^2$. The wall area is $A=L^2$, so division by area gives
$p=(m/V)\sum_i v_{xi}^2$. This expression is already a microscopic pressure
equation, but it describes one wall direction. Equilibrium without a preferred
direction gives the additional relation
$\sum_i v_{xi}^2=\sum_i v_{yi}^2=\sum_i v_{zi}^2$. Adding the three equal sums
produces $\sum_i v_i^2$, hence
$pV=(1/3)m\sum_i v_i^2=(1/3)Nm\overline{v^2}$.

The final average is an average of squared speeds, not the square of average speed.
A stationary sample has zero signed average velocity because every direction is
equally represented. The average of $v^2$ remains positive and sets
the pressure. This distinction is visible even for two molecules of equal speed
moving in opposite directions: their vector momenta cancel, while their two wall
impulses add. A pressure gauge measures momentum flux, not net momentum stored in
the gas.

The dimensions check directly. Mass times squared speed has units
$\mathrm{kg\,m^2\,s^{-2}}$, which is energy. Division by volume gives
$\mathrm{kg\,m^{-1}\,s^{-2}}$, the pascal. The relation also gives correct
qualitative limits. At fixed volume and molecule number, doubling every molecular
speed quadruples pressure. At fixed average squared speed, doubling molecule number
doubles the collision rate and pressure. Both conclusions follow before inserting
the ideal-gas temperature relation.

The derivation has several stated assumptions. The wall is rigid on molecular
scales, collisions are elastic, the gas is spatially uniform, and molecular
velocity directions are isotropic. A moving piston changes the collision impulse;
a temperature gradient makes the velocity distribution position-dependent; a
directed beam produces unequal component averages. The formula can be generalized
in each case, but the simple scalar pressure relation then no longer follows from
the symmetric cube argument alone.

> **Worked example.** Argon is monatomic with a well-known molar mass
> $M=39.95\times10^{-3}\ \mathrm{kg\,mol^{-1}}$, so one atom has mass
> $m=M/N_A=6.63\times10^{-26}\ \mathrm{kg}$. At $T=300\ \mathrm K$ its three
> characteristic speeds use the same $T$ and $m$ but different averages:
> $v_{\rm mp}=\sqrt{2kT/m}$, $v_{\rm av}=\sqrt{8kT/(\pi m)}$, and
> $v_{\rm rms}=\sqrt{3kT/m}$. Substituting gives
> $v_{\rm mp}=353\ \mathrm{m\,s^{-1}}$, $v_{\rm av}=399\ \mathrm{m\,s^{-1}}$, and
> $v_{\rm rms}=433\ \mathrm{m\,s^{-1}}$, in the required order
> $v_{\rm mp}<v_{\rm av}<v_{\rm rms}$. The ratios are independent of $T$ and $m$:
> $v_{\rm av}/v_{\rm mp}=2/\sqrt\pi$ and $v_{\rm rms}/v_{\rm mp}=\sqrt{3/2}$. A result
> that reverses the order or makes all three equal has mixed the formulas or used an
> inconsistent molecular mass.

Kinetic energy checks the rms value. The average translational energy at
$300\ \mathrm K$ is $\tfrac32kT=6.21\times10^{-21}\ \mathrm J$, and
$\tfrac12mv_{\rm rms}^2$ with the calculated rms speed returns the same number. The
check works because rms speed is defined from the average of $v^2$; substituting the
mean speed into $\tfrac12mv^2$ does not give the average kinetic energy, since the
square must be averaged before the root is taken.

Heating the argon to $1200\ \mathrm K$ multiplies each characteristic speed by
$\sqrt{1200/300}=2$, so the rms speed becomes $866\ \mathrm{m\,s^{-1}}$ at unchanged
mass. At fixed temperature, replacing argon with helium raises the speeds by the
square root of the inverse mass ratio. The square-root scalings carry more than a
memorized room-temperature value does.

The speed scale is much larger than familiar bulk flow speeds. A sealed room can
contain air with zero average velocity while individual molecules move hundreds of
metres per second. Collisions randomize direction so efficiently that the vector
average vanishes. This separation between rapid molecular motion and slow or zero
bulk motion is one reason a gas can exert a steady pressure on a stationary wall.

### Assumptions and limits of Maxwell--Boltzmann statistics

The Maxwell--Boltzmann speed distribution describes a classical ideal gas in
thermal equilibrium. Thermal equilibrium fixes one well-defined temperature
and a time-independent distribution. Molecular collisions are frequent enough to
randomize directions and redistribute energy, yet the gas is dilute enough that
the molecules spend most of their time in free flight. The distribution is not a
trajectory law for one molecule. A single molecule continually changes speed and
direction through collisions; the curve describes the fraction of a very large
population found in each speed interval at an instant or over a long average.

Random directions are as important as random speed magnitudes. The distribution
of speeds is nonnegative and has a peak, whereas each Cartesian velocity component
has positive and negative values and a symmetric distribution about zero. A gas
with a nonzero bulk flow is best described by a thermal distribution around the
flow velocity, not by treating the laboratory-frame speed distribution as isotropic
without modification. In the rest frame of a uniform equilibrium sample, the three
component variances are equal and the standard speed formulas apply directly.

The distribution function is normalized: adding the fractions in all possible
speed intervals gives one. Temperature changes shift both the peak position and
the width; one speed parameter cannot characterize the distribution. A hotter curve has more molecules
in the high-speed tail even though many molecules remain slower than the rms speed.
This tail is physically important whenever an outcome requires exceeding a speed
threshold. A curve label at its peak cannot estimate a tail fraction; the relevant
quantity is the area over the specified speed interval.

The ideal-gas form loses accuracy when molecular interactions alter free flights
or when density makes excluded volume important. Near condensation, molecules are
not independent between brief collisions and a simple Maxwell--Boltzmann speed
description with $p=nkT$ is incomplete. Strong spatial gradients pose another
limit. A gas next to a hot wall, a rapidly expanding jet, or a narrow rarefied
channel can have different local distributions at different positions or a
directionally biased distribution. One temperature and one isotropic curve then
cannot represent the entire apparatus.

Statistical predictions also require a sufficient number of molecules in the
sampled interval. In a macroscopic gas, the high-speed tail still contains an
enormous count, so measured fractions are reproducible. In a tiny sample or a
short-time detector count, the same tail can be represented by only a few events
and fluctuate strongly. The distribution remains the expected ensemble result, but
the observed histogram needs uncertainty estimates based on the number of detected
molecules. Smooth curves summarize large-number behavior; they do not eliminate
counting variation in a finite experiment.

Within its domain, Maxwell--Boltzmann statistics joins the pressure derivation to
measurable molecular-speed data. The squared-speed average determines pressure and
rms speed, the ordinary speed average determines a different transport scale, and
integrated tail areas determine threshold probabilities. Each use selects a distinct
average from the same distribution. The model should therefore be stated together
with its equilibrium, dilute-gas, and frame assumptions rather than applied as a
universal curve for every gas flow.

Three checks catch the usual errors in the derivation. Pressure must rise when
molecular mass, collision frequency, or normal speed rises; reversing every molecular
velocity must leave pressure unchanged, which forces squared components rather than a
signed average; and the three Cartesian directions contribute equally only once
isotropy is granted. The corresponding mistakes are a one-way crossing time (missing
the round trip), a dropped momentum-reversal factor of two, and replacing
$\overline{v^2}$ with $\overline{v}^{\,2}$.