Intuition The one core ideaA gas is billions of molecules flying at random speeds, and the Maxwell-Boltzmann distribution is just a hill-shaped graph telling you how common each speed is . Everything in the derivation is built from two facts: fast molecules are energetically expensive (they get rare), but there are more ways to be fast (a bigger "shell" in velocity space) — and the whole page below teaches you the exact symbols that encode those two facts.
This page assumes nothing . Before you touch the full derivation , every letter, squiggle, and picture it uses is built here from the ground up, in an order where each idea leans on the one before it.
Definition Velocity vectorv direction . We write its motion as an arrow v x y z components v x , v y , v z
Picture a molecule shooting across a room. Its shadow on the floor moving east-west is v x v y v z
v Speed is the length of the arrow — how fast, ignoring direction. If you know the three components, Pythagoras in 3D gives:
v = v x 2 + v y 2 + v z 2 .
Why the square root of a sum of squares? Because that is just the 3D version of the ruler-distance formula: the length of the diagonal of a box with sides v x , v y , v z
Intuition Why the topic needs BOTHThe derivation starts with components (they are independent — no force ties east-motion to up-motion) and only later assembles them into speed (the thing we actually measure and plot). Keeping the two separate is the whole trick.
Before we can build the energy-cost factor, we need the three physical quantities it is made from.
Definition The three physical inputs
T absolute temperature in kelvin (K). T = 0 T k B Boltzmann's constant = 1.38 × 1 0 − 23 J/K unit converter : it turns temperature (kelvin) into energy (joules), because k B T m mass of one single molecule in kilograms. Not molar mass! (Molar mass M mole ; divide by Avogadro's number to get m
m M Why it trips people: tables list molar mass M The fix: either use per-molecule m k B M R m R M k B
The combination k B T natural energy scale of the gas — roughly "how much kinetic energy one degree of freedom carries." That's the message of the Equipartition theorem .
e e − x e ≈ 2.718 e − x x placeholder input — any real number you feed in (x e − x
equals 1 x = 0
shrinks smoothly toward 0 x
never actually reaches 0 "how much is left" — like a hot cup cooling: it drops fast, then slower, forever approaching room temperature.
Why this exact curve and not, say, a straight line down? Because e − x it turns a sum in the exponent into a product of separate factors . That is, e − ( a + b ) = e − a ⋅ e − b split three axes bundled together in one exponent into three independent factors, one per axis.
Definition The Boltzmann factore − E / k B T x E / k B T k B T E e − E / k B T 2 1 m v 2 m e − m v 2 /2 k B T energy-cost half of the tug-of-war.
You'll meet this factor properly in Boltzmann factor and partition function — for now just read it as "big energy ⇒ small probability."
Intuition Why the sum in the exponent splits into three component factorsKinetic energy uses the sum v x 2 + v y 2 + v z 2 e − m ( v x 2 + v y 2 + v z 2 ) /2 k B T e − 2 k B T m ( v x 2 + v y 2 + v z 2 ) = e − 2 k B T m v x 2 ⋅ e − 2 k B T m v y 2 ⋅ e − 2 k B T m v z 2 . only one axis . So the joint density falls apart into three independent single-axis pieces — that is why the derivation can treat v x , v y , v z
Intuition You can't ask "what's the probability of EXACTLYv = 422 band of speeds, from v v + d v
f ( v ) d v
d v f ( v ) probability density : probability per unit speed . It is not itself a probability.f ( v ) d v [ v , v + d v ]
Picture a histogram of speeds. As you make the bars infinitely thin, the tops trace a smooth curve f ( v ) area of each thin bar (height f ( v ) d v
Definition The integral sign∫ ∫ 0 ∞ f ( v ) d v all the thin bars from speed 0 some speed, these fractions must total 1 100% ∫ 0 ∞ f ( v ) d v = 1. normalization — it fixes the leftover constants in the formula.
Read ∫ S um. It is the tool for adding infinitely many infinitely thin pieces — exactly what a continuous distribution demands.
Definition The single-component densityg ( v x ) g probability density of one velocity component : g ( v x ) d v x x [ v x , v x + d v x ] f ( v ) ∫ − ∞ ∞ g ( v x ) d v x = 1 g y z
Definition A Gaussian (bell curve)A e − b v x 2 e − m v x 2 /2 k B T 0 v x
b width parameter in the exponent. Comparing with §3, b = 2 k B T m b A height constant out front, chosen so the area equals 1 g ( v x ) = A e − b v x 2
Definition The spherical-shell factor4 π v 2 geometry half of the tug-of-war. Imagine all velocity arrows of the same length v v 4 π v 2 more room to be fast. This is why f ( 0 ) = 0 0
Intuition Putting the two heroes togetherMultiply three identical Gaussians (one per axis) by the shell factor. The three height constants combine into A 3 = ( 2 π k B T m ) 3/2 f ( v ) = more room (grows) 4 π v 2 ( 2 π k B T m ) 3/2 energy cost (shrinks) e − m v 2 /2 k B T . is Maxwell-Boltzmann, and the messy-looking ( 2 π k B T m ) 3/2 A
velocity components vx vy vz
Gaussian per component g of vx
exponential e to the minus x
sum in exponent splits into product
temperature T and kB and mass m
Gaussian integral fixes A
spherical shell 4 pi v squared
probability density f of v and integral
Notice the shape: independent components each get a Gaussian , the Gaussian integral and normalization nail down the constants, the shell factor converts components into speed , and the Boltzmann factor supplies the energy cost. Multiply — done.
Test yourself: cover the right side and answer before revealing.
What does the little arrow in v That v
How do you get speed v v = v x 2 + v y 2 + v z 2 — the 3D Pythagoras / diagonal length.
What is the value and role of k B 1.38 × 1 0 − 23 k B T
Why can't we ask for the probability of an exact speed? Speed is continuous, so any single value has probability zero; we use bands f ( v ) d v
What is the difference between f ( v ) f ( v ) d v f ( v ) per unit speed (a density); f ( v ) d v [ v , v + d v ]
What does g ( v x ) The probability density of one velocity component; g ( v x ) d v x x [ v x , v x + d v x ]
What does ∫ 0 ∞ f ( v ) d v = 1 Every molecule has some speed, so all fractions add to 100%.
What key property of the exponential does the derivation use? e − ( a + b ) = e − a ⋅ e − b
What are A b g ( v x ) = A e − b v x 2 b = m /2 k B T (width in the exponent);
A = b / π = m /2 π k B T (height, fixed by normalization).
How is A By the Gaussian integral
∫ e − b v x 2 d v x = π / b plus area = 1, giving
A π / b = 1 .
What is the full prefactor of f ( v ) ( 2 π k B T m ) 3/2 A 3
Where does 4 π v 2 The surface area of the sphere of velocity vectors all having length v
Why is f ( 0 ) = 0 e 0 = 1 The shell factor 4 π v 2 v = 0
What is the Gaussian integral result? m M k B Per-molecule mass m k B M R
Ready? Then head to the full derivation . Related tools live in Kinetic theory of gases , Effusion and Graham's law , and the comparison Maxwell-Boltzmann vs Fermi-Dirac vs Bose-Einstein .
