Partial Pressure (Dalton's Law): Diving & Altitude

Set the oxygen in a breathing gas, the air pressure at the surface and the depth. The page splits the pressure between the gases, works out what reaches the airways once the gas is warmed and wetted, and gives the depth at which the mix's oxygen becomes a hazard.

Worked example — Partial Pressure (Dalton's Law): Diving & Altitude

Total pressure, surface plus seawater	4.0288 bar
Oxygen partial pressure	0.8436 bar
Nitrogen partial pressure, with air's argon	3.1852 bar
Oxygen, against sea-level air	397.6%
Oxygen in the airways, against sea-level air	417.3%
Depth limit at 1.4 bar, in seawater	56.43 m
Depth limit at 1.6 bar, in seawater	65.93 m

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Partial Pressure (Dalton's Law): Diving & Altitude

In a mixture of gases that do not react, each gas exerts the pressure it would exert if it filled the space alone, its partial pressure, and the partial pressures add up to the total: P = p₁ + p₂ + …. That is Dalton's law. A gas that makes up a mole fraction f of the mixture has p = f·P. Breathing, diving and anaesthesia all respond to p and not to f, so the same gas can be harmless in one place and dangerous in another.

What each symbol means

p: the partial pressure of one gas, in bar
f: its mole fraction in the mixture: 0.2094 for the oxygen in air
P: the total pressure: whatever the atmosphere puts on the surface, plus 0.1005 bar for every metre of seawater
MOD: maximum operating depth, the depth at which a mix's oxygen reaches the chosen limit

Assumes: The gases mix ideally, which oxygen and nitrogen do closely at a few bar and room temperature. Seawater weighs 1025 kg/m³; real surface seawater runs from about 1020 to 1029, which moves air's 56.43 m by about a quarter of a metre either way. Everything in a mix that is not oxygen is counted as nitrogen, with air's 0.93% argon folded in, as dive tables do. The airway figure is for gas warmed to 37 °C and fully saturated with water; in the alveoli the oxygen is lower again, because carbon dioxide leaving the blood takes its own share, and this page does not model that.
Breaks when: The limits are conventions, not physics. 1.4 bar is the limit PADI proposed for open-circuit nitrox, and NOAA's limit for nitrox at 1.6 bar is 45 minutes of normal diving. Oxygen toxicity depends on time as well as pressure, and nothing happens to the body at the moment a line is crossed. The page is also a snapshot. The nitrogen a diver carries home depends on how long they spent at each partial pressure, which is what a decompression table works out and what this page does not.

More oxygen in the cylinder, a shallower limit on the dive 🖖

Press Air at 30 m, then Nitrox 32 and Nitrox 36. The depth stays put and the oxygen partial pressure climbs from 0.8436 to 1.2892 to 1.4504 bar, and the last is past the 1.4 bar limit PADI proposed for open-circuit nitrox. A mix reaches that limit where f·P = 1.4, so at P = 1.4/f, and the richer the gas the less pressure it needs. Air gets to 56.43 m, nitrox 32 to 33.44 m, nitrox 36 to 28.61 m and pure oxygen to 3.85 m, which is the table below read top to bottom. On the depth chart every mix is a straight line out of the surface, and the steeper ones hit the red line sooner. A beginner expects the gas with more oxygen in it to be the safer one to take deep, and it is the other way round.

Everest's air is 20.94% oxygen, the same as the beach's 🖖

Press Summit of Everest. The oxygen box still reads 20.94, because it is the same air: oxygen, nitrogen and argon stay mixed in the same proportions far higher than any mountain. What changed is the total. John West's expedition took a single barometer reading on the summit, published in 1983: 253.0 Torr, 0.3373 bar. Dalton's law does the rest. 20.94% of 0.3373 bar is 0.0706 bar of oxygen, against 0.2122 at sea level, which is 33.3%. What the blood makes of that pressure is the subject of the oxygen dissociation curve.

The lungs turn a third into less than three tenths 🖖

Air is warmed to 37 °C and saturated with water on its way down the airways, and water vapour at 37 °C has a partial pressure of 0.0628 bar, the 47 mmHg of physiology books, on a mountain or at sea level alike. It is set by the temperature of the airway and nothing else. The other gases share what is left. At sea level the water takes 6.2% of the pressure and the oxygen in the airways is 0.1990 bar. On the summit the same 0.0628 bar is 18.6% of a much smaller total, and the oxygen falls to 0.0575 bar: 28.9% of sea level, where the dry air promised 33.3%. The second panel draws both splits side by side. At 30 m the water is 1.6% of the pressure and hardly matters.

Problem solved in full

Choosing a nitrox for a 30 m dive, and what it buys 4 steps
A dive is planned to 30 m in the sea. What is the richest nitrox that keeps the oxygen at or under 1.4 bar there, and how much nitrogen does it save compared with air? Press Air at 30 m for the pressures to start from.

Open this problem in the tool
1. The page goes from a mix to its limit, and a dive plan goes the other way. At 30 m the total pressure is 4.0288 bar, and the fraction of oxygen that makes 1.4 bar of it is 1.4 divided by that.
  
  $P = 1.01325 + 0.100518 \times 30 = 4.0288 bar f_{m a x} = \frac{1 . 4}{4 . 0288} = 0.3475$
2. Mixes are named by the whole percent. Round down, never up, because rounding up would put 30 m past the limit: the gas is nitrox 34, and its working limit lands just below the planned depth.
  
  $f = 0.34 : d = \frac{1 . 4/0 . 34 - 1 . 01325}{0 . 100518} = 30.88 m$
3. Now the nitrogen. At 30 m nitrox 34 carries 0.66 of 4.0288 bar, where air carries 0.7906 of it.
  
  $p_{N_{2}} = 0.66 \times 4.0288 = 2.6590 bar air: 0.7906 \times 4.0288 = 3.1852 bar$
4. Ask at what depth air would carry that same nitrogen. The answer is the equivalent air depth, and an air decompression table is read at it when the diver breathes nitrox.
  
  $P_{air} = \frac{2 . 6590}{0 . 7906} = 3.3633 bar \Rightarrow d = \frac{3 . 3633 - 1 . 01325}{0 . 100518} = 23.38 m$
Answer

Nitrox 34, with a working limit of 30.88 m and the nitrogen of air at 23.38 m. The diver goes to 30 m and, as far as their tissues' nitrogen is concerned, dives 6.62 m shallower, which is what nitrox is for. The margin that pays for it is 0.88 m. Fill the cylinder one percent rich by mistake, at 35%, and the limit rises to 29.71 m, above the depth of the dive, which is why a diver checks the mix with an analyser before breathing it.

$nitrox 34 : 30 - 23.38 = 6.62 m f = 0.35 : d = 29.71 m < 30$

Learning path

Mixtures and solutions

References (3)

Air's 20.94% oxygen, from the composition of dry air the International Committee for Weights and Measures adopted for its air-density formula: A. Picard, R. S. Davis, M. Gläser and K. Fujii, "Revised formula for the density of moist air (CIPM-2007)." Metrologia 45(2), 2008, pp. 149-155. DOI 10.1088/0026-1394/45/2/004
The summit's 253.0 Torr, a single reading taken on the American Medical Research Expedition to Everest, and the base-camp and camp pressures beside it: J. B. West, S. Lahiri, K. H. Maret, R. M. Peters Jr and C. J. Pizzo, "Barometric pressures at extreme altitudes on Mt. Everest: physiological significance." Journal of Applied Physiology 54(5), 1983, pp. 1188-1194. DOI 10.1152/jappl.1983.54.5.1188
The two oxygen limits: PADI's 1.4 bar for open-circuit nitrox and NOAA's 45 minutes at 1.6 bar, as a diving physician summarises them for the Divers Alert Network: Ed Thalmann, "Oxygen Toxicity." Divers Alert Network, dan.org.

By the 42Math team · Reviewed against published sources. Corrections welcome.