Dalton's Law of Partial Pressures

How partial pressures work

Imagine a sealed flask containing only nitrogen at 0.78 atm. Now inject oxygen until it alone would reach 0.21 atm in that same flask, plus a sliver of argon at 0.01 atm. The pressure gauge now reads 1.00 atm — exactly the sum. The gases have not interacted; each species pushes the walls as if the others were absent. That is Dalton's law in one sentence.

The reason is mechanical. Gas pressure comes from billions of molecular collisions per second against the container walls. In an ideal gas each molecule travels independently — point-like, no forces between them. A nitrogen molecule colliding with the wall does not care whether oxygen molecules share the volume. Every species contributes its own collision rate, set only by its own concentration and the temperature.

From the ideal gas law applied to each component:

P_i = n_i · R · T / V          (each species, same V and T)
P_total = (n_1 + n_2 + ... + n_n) · R · T / V
        = P_1 + P_2 + ... + P_n

Dividing one by the other gives the mole-fraction form: P_i / P_total = n_i / n_total = X_i, so P_i = X_i · P_total. For ideal gases, mole fraction, volume fraction, and pressure fraction are interchangeable.

Worked example — three-component mixture

A 5.00 L tank at 298 K is loaded with 0.30 mol N₂, 0.20 mol O₂, and 0.10 mol CO₂. Find the total pressure and each partial pressure.

n_total = 0.30 + 0.20 + 0.10 = 0.60 mol
P_total = nRT / V
        = (0.60 mol)(0.0821 L·atm·mol⁻¹·K⁻¹)(298 K) / (5.00 L)
        = 14.68 / 5.00
        = 2.94 atm

X(N₂)  = 0.30 / 0.60 = 0.500   →  P(N₂)  = 0.500 · 2.94 = 1.47 atm
X(O₂)  = 0.20 / 0.60 = 0.333   →  P(O₂)  = 0.333 · 2.94 = 0.98 atm
X(CO₂) = 0.10 / 0.60 = 0.167   →  P(CO₂) = 0.167 · 2.94 = 0.49 atm

Check: 1.47 + 0.98 + 0.49 = 2.94 atm ✓

Notice that the result is independent of which gases are present. Replacing CO₂ with helium would give the same total pressure and the same numerical breakdown — the mole fraction, not the chemical identity, sets the partial pressure.

Real-world specs

Mixture	Conditions	Partial pressure of interest
Sea-level air	1.00 atm, 25 °C	pO₂ = 0.21 atm = 160 mmHg
Mt. Everest summit	0.33 atm, −36 °C	pO₂ = 0.069 atm = 53 mmHg
Arterial blood	Body, 37 °C	pO₂ = 100 mmHg, pCO₂ = 40 mmHg
Scuba air at 30 m	4.0 atm	pO₂ = 0.84 atm, pN₂ = 3.12 atm
Heliox (50/50) at 60 m	7.0 atm	pO₂ = 3.5 atm, pHe = 3.5 atm
Anaesthesia mix (sevoflurane)	1 atm	p(sevoflurane) = 0.02 atm
Hydrogen collected over water	22 °C, 750 mmHg	pH₂ = 750 − 19.83 = 730.17 mmHg

The atmospheric column itself is the world's biggest demonstration. Climbers above 8,000 m enter the "death zone" not because the air composition changes — it doesn't, the 21% O₂ ratio holds — but because total pressure drops, dragging pO₂ below the threshold tissues need.

Dalton's law vs other gas relations

	Dalton's law	Amagat's law	Raoult's law	Henry's law
What's additive	Pressures at fixed V, T	Volumes at fixed P, T	Vapor P over solution	Dissolved gas in liquid
Equation	P_total = ΣP_i	V_total = ΣV_i	P_i = X_i · P_i*	p_i = k_H · c_i
Phase	Gas mixture	Gas mixture	Liquid–vapor	Gas–liquid
Component independence	Yes (no IMFs)	Yes (additive volumes)	Ideal solution	Dilute solute
Year	1801	1880	1887	1803
Failure mode	Reactive / dense gas	High P deviations	Non-ideal mixing	High concentration

Amagat's law is the volumetric twin of Dalton's: at fixed total pressure each gas would occupy a "partial volume" V_i, and these add to V_total. For ideal gases the two laws contain the same information. Raoult and Henry are not gas-mixture statements at all — they describe equilibrium between a gas and a liquid solution — but they share the Dalton-style mole-fraction structure.

ASCII visualization

BEFORE MIXING                             AFTER MIXING (same V, T)

  N₂ alone        O₂ alone        Ar alone     N₂ + O₂ + Ar
  ┌────────┐     ┌────────┐     ┌────────┐   ┌──────────────┐
  │ N N N  │     │ O      │     │ A      │   │ N O A N N    │
  │ N N N  │ +   │  O O   │ +   │   A    │ = │  O N A  O    │
  │ N N    │     │   O O  │     │        │   │ N A N  O N   │
  └────────┘     └────────┘     └────────┘   └──────────────┘
  P = 0.78         P = 0.21       P = 0.01      P_total = 1.00

  Each species' contribution to wall collisions is unchanged.
  Total pressure = sum of independent partial pressures.

Variants and corrections

• Real-gas correction. When intermolecular forces matter (CO₂ above 50 atm, water vapor near saturation), each component's effective pressure is its fugacity f_i, not its raw P_i. The Lewis-Randall rule generalises Dalton: f_i(mixture) = X_i · f_i(pure). For modest deviations a virial expansion adds cross terms B_ij that couple species pairs.
• Chemical equilibrium. If the gases react (e.g. N₂ + 3H₂ ⇌ 2NH₃), Dalton's law still applies to whatever mixture exists at any instant — but the moles change as the reaction proceeds, so partial pressures shift along the way. The equilibrium constant Kp is written in those final partial pressures.
• Wet-gas measurements. Gas collected over water carries water vapor at the saturation pressure for that temperature. The measured pressure is P_total = P_dry + P(H₂O), and you subtract a tabulated P(H₂O) to recover the dry-gas value.
• Daltonian fraction in flowing systems. In gas chromatography and mass spectrometry, mole fractions in a flowing carrier are computed from peak areas; partial pressures inside the detector chamber are X_i · P_chamber. The same formula does double duty.

Common pitfalls and failure modes

• Forgetting water vapor. Lab samples collected over water are wet by definition. Reporting P_total as the dry-gas pressure overestimates the moles of analyte by a few percent at room temperature, more at higher temperatures.
• Applying it to reactive mixtures. Mix NH₃ and HCl at room temperature and the pressure drops as solid NH₄Cl forms. The "law" still holds at every instant, but the mole counts are now functions of time.
• Treating it as exact at high pressure. Above roughly 10 atm for small molecules, intermolecular forces between unlike species shift the partial pressures by a few percent. Industrial reactor design uses fugacity coefficients, not raw mole fractions.
• Ignoring temperature units. If you compute P_total from PV = nRT for the mixture and then check against the sum, every component must be evaluated at the same Kelvin temperature. Mixing 25 °C and 298 K gives nonsense.
• Confusing partial pressure with concentration. P_i is a pressure (atm, Pa, mmHg), not a concentration. Henry's-law solubilities use the partial pressure of the gas above the liquid — not the total pressure of the mixture.
• Assuming "air" is constant. The 21% O₂ figure is dry-air composition. Humid tropical air at 35 °C contains up to 6% water vapor, so its O₂ mole fraction drops to ~19.7%. Aviators and athletes in humid heat feel this directly.