Intuition The one core ideaAt the same temperature, every gas carries the same average kicking energy per molecule — so a light molecule must move fast and a heavy one must move slow . Because "escaping through a hole" happens at the speed the molecules move, light gases leak and spread faster, and the whole of Graham's law is just this sentence turned into a square root.
This page assumes nothing . Before you can read the parent note's derivation, you need to own every letter and symbol in it. We build them one at a time, each on the picture of the one before.
Everything below lives inside one image: a box of gas with a tiny pinhole in the wall.
Gas ::= a huge swarm of tiny molecules flying in straight lines, bouncing off walls and each other, otherwise mostly empty space .Pinhole ::= a hole so small that a molecule slips through one at a time without bumping another molecule inside the hole.
Keep this box in your head. Every symbol we define is a number about the molecules in this box .
Definition Two masses, two scales
m single molecule (unimaginably tiny, kilograms).M molar mass — the mass of one mole (a fixed huge count) of those molecules. In SI it is kilograms per mole (kg/mol); chemists usually quote it in grams per mole (g/mol), which is just 1000 ×
Picture: m M count of grains. Same "heaviness" idea, just a convenient bookkeeping size.
Intuition Why the topic needsM m mole . So chemists work with M m units match.
M H 2 = 2 O 2 = 32 NH 3 = 17 HCl = 36.5
v speed of one molecule = how many metres it covers each second (m/s).The overline / angle brackets ⋅ ⟨ ⋅ ⟩ average of this over all molecules."
⟨ v ⟩ mean speed — average the speeds themselves.v 2 mean square speed — square every molecule's speed first , then average those squares.
Picture: a crowd of arrows of different lengths (each arrow = one molecule's speed). ⟨ v ⟩ v 2 lengths squared — a different, slightly larger flavour of "typical speed."
Why bother with two averages? Because energy needs v 2 flux through a hole needs ⟨ v ⟩ ∝ 1/ M ratio they behave identically, but they are not the same number, and honest physics keeps them distinct.
this formula, and why v 2 v four times as hard — energy grows with the square of speed, not the speed itself. That single fact, K E ∝ v 2 square root and not a plain 1/ M
Because different molecules move at different speeds, we take the average: K E = 2 1 m v 2 v 2 v 2
Definition Temperature and Boltzmann's constant
T absolute temperature in kelvin , a direct measure of average molecular jostling energy. T = 0 k B Boltzmann's constant , a tiny fixed conversion number that turns "degrees of temperature" into "joules of energy per molecule."
Common mistake The word "translational" is not decorationThe 2 3 k B T moving through space (its three directions x, y, z — hence the 3 O 2 NH 3 also stores energy in spinning and vibrating (internal degrees of freedom), and its total energy is larger than 2 3 k B T Why we can ignore that here: only translational motion carries a molecule toward the hole , so effusion depends on translational speed alone. The 2 3 k B T
The picture above is the whole story: same translational energy budget, so the light molecules must run fast to spend it and the heavy ones crawl.
Setting the two expressions for translational K E 2 1 m v 2 = 2 3 k B T ⟹ v 2 = m 3 k B T
Definition What "rms" means — read the name backwardsRoot-Mean-Square speed ::= take the square root of the mean of the squares of the speeds.
v r m s = v 2 = m 3 k B T Why take the root at the end? v 2 Maxwell-Boltzmann Distribution .
v r m s ⟨ v ⟩ ⟨ v ⟩ = π m 8 k B T little smaller than v r m s 92% Both carry the crucial factor 1/ m
See Root Mean Square Speed for the full treatment; here we only need its shape: speed ∝ m 1 at fixed T
The physics gave us v r m s = 3 k B T / m molecular mass m kilograms . Chemistry wants molar mass M
Definition Avogadro's number and the gas constant
N A Avogadro's number = how many molecules make one mole (a fixed count, per mole).R universal gas constant , and it is literally k B N A R = N A k B , M = N A m m N A M = N A m kg/mol — the SI molar mass. That is the version that keeps the equation dimensionally clean.
Common mistake The g/mol trapIf you grab M g/mol and drop it into 3 R T / M R 1000 ≈ 32 × Fix: convert to kg/mol first (e.g. O 2 32 g/mol = 0.032 kg/mol ratio the 1000 R Ideal Gas Equation , which is why all gas problems share this same unit hygiene.
Definition Rate and the "proportional to" sign
Rate r throughput passes through the hole per unit time. Physically it is an amount per time : moles/s or volume/s (both count molecules leaving).∝ is proportional to " — grows in step with, up to a fixed multiplier.
A diffusion tube instead reports a distance the gas front has travelled. That is not a throughput — it is really a speed — but since distance-covered scales with molecular speed exactly as throughput does, the same 1/ M
Intuition The collapse to Graham's lawSince n , A , 4 1 ⟨ v ⟩ ⟨ v ⟩ = 8 k B T / π m ∝ 1/ M T r ∝ ⟨ v ⟩ ∝ M 1 = M 1 r ∝ v r m s ⟨ v ⟩ v r m s same constant factor for every gas, the ratio is identical either way — so both routes give the exact same Graham's law.)
Definition Subscripts and the flip
Subscripts 1, 2 ::= just labels for "gas one" and "gas two." r 1 M 2 The fraction r 2 r 1
Common mistake Why the masses swap sidesRate is inverse to M bottom and mass 2 on the top — the labels cross:
r 2 r 1 = 1/ M 2 1/ M 1 = M 1 M 2 H 2 O 2 32/2 = 4
This exact skill — reading a tiny mass difference as a tiny rate difference — is what powers Isotope Separation (Uranium Hexafluoride) .
Molecule mass m and molar mass M
Mean speed and mean square speed
Kinetic energy = half m v squared
Equal translational energy = three halves kB T
mean speed = sqrt 8RT over pi M
R = NA kB and M = NA m in kg per mol
Effusion flux = one quarter n mean-v A
Graham law rate ∝ 1 over sqrt M
Test yourself — reveal only after answering out loud.
What is the difference between m M m M M = N A m m
What does the overline / bracket mean, and how do ⟨ v ⟩ v 2 Both mean "average over all molecules." ⟨ v ⟩ v 2
Write the kinetic energy of one molecule in terms of m v K E = 2 1 m v 2
Why does energy involve v 2 v Doubling speed quadruples the impact energy; this square is the source of the square root in Graham's law.
State the equal-energy law and its crucial qualifier. K E trans = 2 3 k B T translational energy only, depends only on T
Why can we ignore internal (rotation/vibration) energy for effusion? Only translational motion carries a molecule toward the hole, so only 2 3 k B T
Which average speed governs effusion flux, and via what formula? The mean speed ⟨ v ⟩ r = 4 1 n ⟨ v ⟩ A
Why does r ∝ v r m s r ∝ ⟨ v ⟩ ⟨ v ⟩ v r m s ratio is identical.
How do R k B M R = N A k B M
In r 2 r 1 = M 1 M 2 Rate is inverse to
M , so forming the ratio flips the mass labels top-for-bottom.
