A–a gradient = PAO₂ − PaO₂, where alveolar oxygen PAO₂ = FiO₂ × (760 − 47) − PaCO₂ ÷ 0.8 at sea level. On room air that is about 150 − PaCO₂ ÷ 0.8. A normal gradient is under about 10 mmHg in young adults and rises with age (expected ≈ 2.5 + 0.21 × age). A raised gradient points to V/Q mismatch, shunt or diffusion problems; low oxygen with a normal gradient points to hypoventilation.
This calculator works out alveolar oxygen, the A–a gradient, the expected value for age, the PaO₂/FiO₂ ratio and the full-form result, in mmHg or kPa and at any altitude. It is written for clinicians and students.
How to use this calculator
- Enter FiO₂ as a percentage: 21 for room air.
- Enter PaO₂ and PaCO₂ from the arterial blood gas, in mmHg or kPa.
- Enter age to get the expected normal gradient.
- If you are not at sea level, enter the altitude, or the measured barometric pressure if you have it.
- Compare the gradient with the expected value for age.
The alveolar gas equation
PAO₂ = FiO₂ × (Pb − PH₂O) − PaCO₂ ÷ R
As set out by Sarkar and colleagues (Lung India, 2017) and the MSD Manual: FiO₂ is the inspired oxygen fraction (0.21 in room air), Pb is barometric pressure (760 mmHg at sea level), PH₂O is water vapour pressure (47 mmHg at 37 °C), PaCO₂ stands in for alveolar CO₂ and R is the respiratory quotient, about 0.8. With a normal PaCO₂ of 40 mmHg on room air, PAO₂ = 0.21 × 713 − 40 ÷ 0.8 = 99.7 mmHg, the familiar figure of about 100.
The full form adds a small correction, + PaCO₂ × FiO₂ × (1 − R) ÷ R, for oxygen uptake and CO₂ output not being equal. On room air it adds about 2 mmHg; the Merck Manual calculator uses it, and our results panel shows both.
Expected A–a gradient by age (room air)
| Age | Expected upper normal, 2.5 + 0.21 × age (mmHg) | Shortcut, age ÷ 4 + 4 (mmHg) |
|---|---|---|
| 20 | 6.7 | 9.0 |
| 30 | 8.8 | 11.5 |
| 40 | 10.9 | 14.0 |
| 50 | 13.0 | 16.5 |
| 60 | 15.1 | 19.0 |
| 70 | 17.2 | 21.5 |
| 80 | 19.3 | 24.0 |
The gradient widens with age mainly because arterial oxygen falls as V/Q mismatch increases. Sarkar and colleagues put the fall in PaO₂ after 70 at about 0.43 mmHg a year. The Merck expected-gradient formula, based on Kanber (1968) and Mellemgaard (1966), assumes room air and is less accurate at higher FiO₂.
Normal vs raised gradient: causes of low oxygen
| Mechanism | A–a gradient | PaCO₂ | Response to oxygen | Examples |
|---|---|---|---|---|
| Hypoventilation | Usually normal | High | Good | Opioids or sedatives, neuromuscular disease, kyphoscoliosis |
| Low inspired O₂ (altitude) | Normal once local pressure is used | — | — | High altitude |
| V/Q mismatch | Raised | Variable | — | COPD, pulmonary embolism (most common cause overall) |
| Diffusion limitation | Raised | Usually normal | Good | Emphysema, interstitial lung disease |
| Right-to-left shunt | Raised | Normal | Poor | Pneumonia, pulmonary oedema, ARDS, alveolar collapse, AV malformation |
Characteristics as described by Sarkar and colleagues. They also note that long-standing hypoventilation can cause areas of collapse that then widen the gradient, and that a normal PaO₂ and A–a gradient do not rule out pulmonary embolism: in one series, 12% of 280 patients with acute PE had a gradient under 20 mmHg.
Worked examples
Six sample blood gases worked through with the calculator's own functions (mmHg).
| Case | FiO₂ | PaO₂ | PaCO₂ | Pb | PAO₂ | A–a gradient | Expected for age |
|---|---|---|---|---|---|---|---|
| Healthy 25-year-old, room air | 0.21 | 95 | 40 | 760 | 100 | 5 | ≤8 |
| Opioid overdose (hypoventilation) | 0.21 | 55 | 70 | 760 | 62 | 7 | ≤9 |
| Pneumonia, room air | 0.21 | 60 | 32 | 760 | 110 | 50 | ≤15 |
| Pulmonary embolism, room air | 0.21 | 70 | 30 | 760 | 112 | 42 | ≤12 |
| Pneumonia on 40% oxygen | 0.4 | 75 | 35 | 760 | 241 | 166 | ≤15 |
| Healthy hiker at 2,500 m | 0.21 | 65 | 34 | 560 | 65 | 0 | ≤9 |
The overdose patient has a PaO₂ of only 55, but the gradient is 7 mmHg, within normal: the high PaCO₂ has simply crowded oxygen out of the alveoli. The pneumonia patients have wide gradients, and on 40% oxygen the gradient is larger still, partly because oxygen therapy widens it on its own. The hiker's PaO₂ of 65 looks low, yet at about 560 mmHg barometric pressure the gradient is normal.
Barometric pressure by altitude
| Altitude (m) | Altitude (ft) | Pressure (mmHg) | Room-air PAO₂ at PaCO₂ 40 (mmHg) |
|---|---|---|---|
| 0 | 0 | 760 | 100 |
| 500 | 1,640 | 716 | 90 |
| 1,000 | 3,281 | 674 | 82 |
| 1,500 | 4,921 | 634 | 73 |
| 2,000 | 6,562 | 596 | 65 |
| 2,500 | 8,202 | 560 | 58 |
| 3,000 | 9,843 | 526 | 51 |
| 4,000 | 13,123 | 462 | 37 |
Values from the U.S. Standard Atmosphere (1976) troposphere model. Real pressure varies with weather by a few mmHg, so use a measured value when you have one.
Common mistakes
- Using 760 mmHg at altitude. This overestimates PAO₂ and makes the gradient look falsely wide.
- Applying room-air normals on oxygen. The age formula is for FiO₂ 0.21; any extra oxygen widens the gradient.
- Entering FiO₂ wrongly. Use 21%, not 0.21% or 2 L/min. Nasal cannula flow does not translate to an exact FiO₂.
- Mixing units. Enter all gases in mmHg or all in kPa.
- Forgetting age. A gradient of 20 mmHg is raised at 25 but expected at 80.
More blood gas and lab tools: Winter's formula, anion gap, corrected sodium, BUN/creatinine ratio and CHA₂DS₂-VASc.
For healthcare professionals and students only. The A–a gradient helps sort out why blood oxygen is low; it does not diagnose a condition, and this tool offers no medical advice. Severe shortness of breath, blue lips or confusion need emergency care. Patients should discuss blood gas results with their own clinician.
Frequently asked questions
What is the A–a gradient?
The alveolar–arterial (A–a) oxygen gradient is the difference between the oxygen pressure in the alveoli (PAO₂), which is calculated, and the oxygen pressure in arterial blood (PaO₂), which is measured on a blood gas. A wide gradient means oxygen is not getting from the air sacs into the blood as it should.
How do you calculate the A–a gradient?
First work out alveolar oxygen with the alveolar gas equation: PAO₂ = FiO₂ × (barometric pressure − 47) − PaCO₂ ÷ 0.8. At sea level on room air that is 0.21 × 713 − PaCO₂ ÷ 0.8. Then subtract the measured PaO₂: A–a gradient = PAO₂ − PaO₂.
What is a normal A–a gradient?
It depends on age. Sarkar and colleagues give under 10 mmHg for a young person, and the MSD Manual says it is typically under 20 mmHg but rises with age. The Merck Manual estimates the expected upper value on room air as 2.5 + 0.21 × age, which is 6.7 mmHg at 20 and 19.3 mmHg at 80.
What is the age/4 + 4 rule?
A bedside shortcut for the normal A–a gradient on room air: age ÷ 4 + 4. It gives similar numbers to the Merck formula (14 mmHg at 40 against 10.9). It is widely taught, but we could not trace it to a primary study, so the calculator shows it only as a secondary guide.
What does a high A–a gradient mean?
Low oxygen with a wide gradient points to a problem at the alveolar–capillary unit: ventilation–perfusion (V/Q) mismatch, right-to-left shunt or a diffusion limitation. Sarkar and colleagues list V/Q mismatch as the most common cause of hypoxemia; shunt causes include pneumonia, pulmonary oedema, ARDS, alveolar collapse and arteriovenous communications.
What causes low oxygen with a normal A–a gradient?
Hypoventilation, where the lungs are normal but breathing is not enough, raises PaCO₂ and lowers PAO₂ and PaO₂ together, so the gradient stays normal. Causes include sedative or opioid drugs, neuromuscular disease and chest wall problems. Low inspired oxygen at altitude also lowers both values, which the equation accounts for through barometric pressure.
Why does oxygen therapy widen the A–a gradient?
Raising FiO₂ increases alveolar oxygen more than arterial oxygen, so the gradient grows even in healthy lungs. The MSD Manual explains that oxygen dissolved in plasma keeps rising after haemoglobin is nearly fully saturated at a PaO₂ of about 150 mmHg. The age-based normal values apply to room air only.
Why is the respiratory quotient 0.8?
The respiratory quotient is carbon dioxide produced divided by oxygen used. Sarkar and colleagues and the MSD Manual take it as about 0.8 at rest on a normal mixed diet. A different diet or metabolic state changes it slightly.
Why subtract 47 mmHg?
Inspired air is fully saturated with water vapour by the time it reaches the alveoli. At body temperature (37 °C) water vapour exerts 47 mmHg, which takes up part of the total pressure and leaves less for oxygen.
How does altitude change the A–a gradient?
Barometric pressure falls with altitude, so alveolar and arterial oxygen both fall. The gradient itself stays near normal in healthy lungs, but only if you enter the local pressure; using 760 mmHg at altitude makes the gradient look falsely wide. This calculator estimates pressure from altitude with the U.S. Standard Atmosphere or takes a measured value.
What is the PaO₂/FiO₂ ratio?
PaO₂ divided by FiO₂ as a fraction. Sarkar and colleagues note that the Berlin ARDS definition grades severity by it (mild 200–300, moderate 100–200, severe 100 or less mmHg, with PEEP or CPAP of at least 5 cmH₂O), and that a ratio under 200 suggests a shunt fraction above 20%.
Can I enter values in kPa?
Yes. Choose kPa next to PaO₂; PaCO₂ and barometric pressure use the same unit. One kPa is 7.50062 mmHg, and results are shown in the unit you choose.
Sources & method
- Sarkar, Niranjan & Banyal (2017), Lung India — Mechanisms of hypoxemia (alveolar gas equation, A–a gradient, P/F ratio)
- MSD Manual Professional — Equations for calculating alveolar oxygen pressure and alveolar to arterial oxygen gradient
- Merck Manual Professional — A-a Gradient calculator (expected gradient 2.5 + 0.21 × age; Kanber 1968, Mellemgaard 1966)
- Mellemgaard (1966), Acta Physiol Scand — The alveolar-arterial oxygen difference: its size and components in normal man
- NOAA, NASA & USAF (1976) — U.S. Standard Atmosphere, 1976 (pressure vs altitude)
Results are estimates for general information. Found an error? It helps everyone — see our methodology.