Gas Laws in Respiratory Medicine — Dalton, Boyle and PV = nRT Applied
Dalton's law of partial pressures is usually taught with a cylinder of two mixed gases and then left there. Its clearest real application is breathing: every step from the air in the room to the oxygen reaching the blood is a partial-pressure calculation, and the arithmetic is entirely within a first-year syllabus. It is also a context that turns up in application questions in IIT-JAM, GATE, CSIR-NET and CUET-PG, because it forces you to keep fractions and pressures apart.
This is a chemistry article written for exam preparation. It is not medical guidance. Oxygen therapy, altitude exposure and breathing-gas mixtures are set by qualified professionals.
The formulas
Applied to inspired gas, once air has been warmed and fully humidified in the airway:
What each term means
| Symbol | Meaning | Unit |
|---|---|---|
| xi or FiO2 | Mole fraction of the gas in the mixture. Dry air is 20.95% oxygen, so x(O2) = 0.2095 | dimensionless |
| PB | Barometric (total) pressure. 760 mmHg at sea level; 1 atm = 760 mmHg = 101.325 kPa | mmHg or kPa |
| PH₂O | Saturated water vapour pressure at body temperature, 47 mmHg at 37 °C. It is a property of temperature only | mmHg |
| PiO2 | Partial pressure of oxygen in the fully humidified inspired gas | mmHg |
| R | Gas constant: 0.082057 L·atm·mol−1·K−1, or 8.314 J·mol−1·K−1 in SI | — |
| T | Absolute temperature; body temperature 37 °C = 310.15 K | K |
Worked example 1 — oxygen in dry air at sea level
PO₂ = x(O2) × PB = 0.2095 × 760
0.2 × 760 = 152.0; 0.0095 × 760 = 7.22
PO₂ = 152.0 + 7.22 = 159.2 mmHg
That is the starting figure. Every subsequent step only reduces it.
Worked example 2 — the humidification step
Inhaled gas is warmed to 37 °C and saturated with water vapour. Water vapour occupies part of the total pressure, so the remaining gases share what is left:
Dry-gas pressure available = 760 − 47 = 713 mmHg
PiO2 = 0.2095 × 713
0.2 × 713 = 142.60; 0.0095 × 713 = 6.77
PiO2 = 142.60 + 6.77 = 149.4 mmHg
Humidification alone has cost about 10 mmHg. Note carefully that the water vapour term is subtracted before multiplying by the fraction — multiplying first and subtracting 47 afterwards gives 112 mmHg, which is a very common and completely wrong answer.
Worked example 3 — the alveolar gas equation
Oxygen is continuously taken up in the alveoli and carbon dioxide continuously added, so the gas there has a lower oxygen partial pressure than the inspired gas. A simplified balance gives:
RQ is the respiratory quotient, the ratio of CO2 produced to O2 consumed; a mixed-diet value of 0.8 is conventionally used. Take PACO2 = 40 mmHg.
PACO2 ÷ RQ = 40 ÷ 0.8 = 50.0 mmHg
PAO2 = 149.4 − 50.0 = 99.4 mmHg
So the chain 159 → 149 → 99 mmHg is entirely explained by two ideas from the gas-laws chapter: water vapour takes a share of the total pressure, and one gas is being swapped for another.
Worked example 4 — the same air at altitude
Suppose the barometric pressure is 460 mmHg, roughly what is found at around 4000 m. The composition of the atmosphere is unchanged — it is still 20.95% oxygen.
Dry-gas pressure = 460 − 47 = 413 mmHg
PiO2 = 0.2095 × 413 = 82.60 + 3.92 = 86.5 mmHg
If breathing increases so that PACO2 falls to 30 mmHg, with RQ still 0.8:
30 ÷ 0.8 = 37.5
PAO2 = 86.5 − 37.5 = 49.0 mmHg
Two conclusions follow directly. First, the oxygen fraction does not change with altitude; the pressure does. "There is less oxygen up there" is loose language for a lower partial pressure. Second, notice that the 47 mmHg water vapour term is a fixed subtraction — at 460 mmHg it removes a much larger share of the total than it does at 760 mmHg, so its effect gets proportionally worse as pressure falls.
Worked example 5 — Boyle's law and a compressed volume
A sealed gas volume of 1.00 L at 1.00 atm is taken to 2.50 atm at constant temperature.
P1V1 = P2V2
V2 = (1.00 × 1.00) ÷ 2.50 = 0.400 L
And converting a gas volume to an amount of substance at body temperature, with V = 0.500 L, P = 1.00 atm, T = 310.15 K:
RT = 0.082057 × 310.15 = 25.450 L·atm·mol−1
n = PV ÷ RT = (1.00 × 0.500) ÷ 25.450 = 0.01965 mol = 19.6 mmol
Boyle's law is the reason any trapped gas volume in the body changes with ambient pressure — the basis of the caution about pressure changes in diving and in flight.
Where this is actually used
The partial-pressure chain above is the framework behind supplemental oxygen (raising FiO2 raises PiO2 at the same barometric pressure), aircraft cabin pressurisation, altitude acclimatisation, hyperbaric and diving gas mixtures where the total pressure is raised so that each component's partial pressure must be recalculated, and anaesthetic delivery, where a vaporiser output is specified as a concentration but acts through its partial pressure.
It also underlies the routine corrections applied to measured gas volumes. A volume measured at ambient temperature and pressure is not the same volume inside the body at 37 °C and saturated with water vapour; converting between the two is a straight application of the combined gas law, and forgetting it produces a systematic error of several per cent.
Where the simple formula stops being valid
- The gas laws describe the gas phase, not oxygen in blood. This is the big one. Henry's law gives the dissolved oxygen, and that dissolved amount is a very small part of the total oxygen the blood carries — nearly all of it is bound to haemoglobin. Haemoglobin binding follows a sigmoid saturation curve, not a linear law, so doubling the partial pressure does not double the oxygen content. Any argument that applies a gas law directly to oxygen delivery is wrong at this step.
- Fraction and partial pressure are different quantities. FiO2 is a ratio and stays at 0.2095 in air anywhere on Earth; PiO2 is a pressure and falls with altitude. Questions are written specifically to catch students who treat them as interchangeable.
- Subtract the water vapour first. PiO2 = FiO2(PB − 47), never FiO2·PB − 47. The two differ by about 37 mmHg at sea level.
- PH₂O = 47 mmHg is a 37 °C value. It is the saturated vapour pressure of water at body temperature. At room temperature it is much smaller, and gas measured at room temperature must be handled differently.
- The alveolar equation is a simplified steady-state model. The RQ term is an approximation to a fuller expression, and RQ itself depends on metabolism. The equation gives an idealised alveolar value, and it assumes a steady state that does not hold during a rapid change in breathing.
- Real lungs are not one well-mixed compartment. Ventilation and blood flow are not perfectly matched across the lung, so the oxygen partial pressure actually measured in arterial blood is always somewhat lower than the calculated alveolar value. A single-compartment model cannot capture that.
- Kelvin, and consistent pressure units. T in PV = nRT is absolute. If you work in kPa, remember 760 mmHg = 101.325 kPa and that the 47 mmHg term becomes about 6.3 kPa — convert every pressure in the expression, not just one.
- The ideal gas law is an approximation. At the pressures and temperatures here it is an excellent one, but at the high pressures used in diving, real-gas deviations become measurable.
Where this appears in exams
| Exam | Typical question |
|---|---|
| IIT-JAM | Partial pressures from mole fractions; PV = nRT; Dalton's and Graham's laws applied together |
| CUET-PG | Direct substitution into the ideal gas law; Boyle's and Charles's law problems |
| GATE | Gas mixtures, real-gas corrections, and unit conversion between mmHg, atm and kPa |
| CSIR-NET | Kinetic theory, distribution of molecular speeds, and Henry's law for dissolved gases |
Check the substitution. The Ideal Gas Law calculator solves PV = nRT for whichever quantity you leave blank, so you can verify the amount-of-substance step in example 5 and any partial-pressure problem you convert into moles. Osmotic-pressure problems use the same algebra, πV = nRT, so the same tool handles those too — remembering to apply the van't Hoff factor yourself.
Open the Ideal Gas Law (PV = nRT) Calculator →Preparing for IIT-JAM, GATE, CSIR-NET or CUET-PG? ABC Chemistry runs dedicated competitive-exam batches at its coaching centre and online for students across India — details at abcchemistry.in.