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# The alveolar gas equation and the A–a gradient

- Measurement
- Exam priority: high

A formula estimates how much oxygen is in the alveoli. If arterial blood has much less, the problem is in the lung.

Updated 2026-11-01

Also called: Alveolar–arterial gradient, PAO₂

## Key points

- PAO₂ = FiO₂ × (PB − 47) − PaCO₂/0.8.
- At sea level on room air: PAO₂ ≈ 100 mmHg; normal A–a gradient ≈ 5–15 mmHg (rises with age).
- In Bogotá (PB ≈ 560 mmHg): PAO₂ ≈ 65–70 mmHg and normal PaO₂ ≈ 60–70 mmHg.
- Normal gradient + hypoxemia: hypoventilation or low inspired PO₂. Wide gradient: V/Q mismatch, shunt or diffusion.
- Quick rule for the expected gradient: ≈ age/4 + 4 mmHg.

## What it is

The alveolar gas equation calculates alveolar PO₂ (PAO₂) from barometric pressure, inspired O₂ fraction and PaCO₂. Its difference from the measured PaO₂ is the A–a gradient: if normal, hypoxemia comes from breathing too little or from O₂-poor air; if widened, there is a gas-exchange defect. In Bogotá (≈ 560 mmHg) normal PAO₂ is far lower than at sea level.

## Equations

**Alveolar gas equation**PAO2 = FiO2 × (PB − PH2O) − PaCO2/R

**FiO 2**
inspired O₂ fraction (0.21 on room air)

**P B**
barometric pressure

**P H 2O**
water vapour pressure at 37 °C, 47 mmHg

**R**
respiratory quotient, ≈ 0.8

FiO₂ is 0.21 at any altitude: what falls in the mountains is barometric pressure.

**A–a gradient**Gradient = PAO2 − PaO2

**PaO 2**
arterial PO₂ measured on the blood gas

Use it on room air; on high FiO₂ the normal gradient widens.

## Reference values

| Parameter | Value |
| --- | --- |
| Water vapour pressure (37 °C) | 47 mmHg |
| Respiratory quotient (R) | ≈ 0.8 |
| A–a gradient, young adult | ≈ 5–15 mmHg |
| A–a gradient, older adult | up to ≈ 20–25 mmHg |
| Barometric pressure in Bogotá | ≈ 560 mmHg |

Approximate values for a healthy adult at rest; each laboratory sets its own ranges.

## Same calculation, two cities (room air)

| Step | Sea level | Bogotá (~2600 m) |
| --- | --- | --- |
| Barometric pressure | 760 mmHg | ≈ 560 mmHg |
| Inspired PO₂: 0.21 × (PB − 47) | ≈ 150 mmHg | ≈ 108 mmHg |
| Usual PaCO₂ | 40 mmHg | ≈ 32–35 mmHg |
| PAO₂ (subtracting PaCO₂/0.8) | ≈ 100 mmHg | ≈ 65–68 mmHg |
| Expected normal PaO₂ | ≈ 85–100 mmHg | ≈ 60–70 mmHg |

## Clinical notes

- Opioid overdose: hypoxemia with high PaCO₂ and a normal A–a gradient (pure hypoventilation).
- Pulmonary embolism: hypoxemia with low PaCO₂ and a widened A–a gradient.
- A Bogotá patient with PaO₂ 62 mmHg may have normal lungs; an A–a gradient calculated with 560 mmHg settles it.

## Memory aid

Up the mountain the barometer drops, not the 21%: always correct PB before judging a PaO₂.

## Test yourself

Think of the answer, then tap the question to see it.

## In Bogotá (PB 560 mmHg), room air, PaCO₂ 32 mmHg and PaO₂ 64 mmHg. What is the A–a gradient?

PAO₂ = 0.21 × (560 − 47) − 32/0.8 ≈ 108 − 40 = 68 mmHg; gradient ≈ 4 mmHg: normal for the altitude.

## Hypoxemia with a normal A–a gradient and a high PaCO₂: which mechanism?

Alveolar hypoventilation: the alveolus holds little O₂ because little fresh air enters, but exchange with the blood is normal.

## Why is normal PaO₂ lower in Bogotá than in Cartagena if FiO₂ is the same?

Barometric pressure is lower, so inspired and alveolar PO₂ fall even though the air is still 21% O₂.

## Why does PAO₂ rise when someone hyperventilates?

PaCO₂ falls, so less CO₂ is subtracted (PaCO₂/0.8): alveolar gas becomes more like inspired gas.

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## Related topics

- [The causes of hypoxemia](https://durgical.com/en/physiology/the-causes-of-hypoxemia/)
- [Oxygenation: SpO₂, PaO₂, cyanosis and oxygen therapy](https://durgical.com/en/physiology/oxygenation-spo-pao-cyanosis-and-oxygen-therapy/)
- [Dead space and alveolar ventilation](https://durgical.com/en/physiology/dead-space-and-alveolar-ventilation/)
- [High altitude and acclimatization](https://durgical.com/en/physiology/high-altitude-and-acclimatization/)
- [Reading an arterial blood gas step by step](https://durgical.com/en/physiology/reading-an-arterial-blood-gas-step-by-step/)
- [Ventilation–perfusion matching and West zones](https://durgical.com/en/physiology/ventilation-perfusion-matching-and-west-zones/)

## Related conditions

- [Pulmonary embolism](https://durgical.com/en/conditions/pulmonary-embolism/)
- [Acute respiratory distress syndrome (ARDS)](https://durgical.com/en/conditions/acute-respiratory-distress-syndrome/)
- [Chronic obstructive pulmonary disease (COPD)](https://durgical.com/en/conditions/chronic-obstructive-pulmonary-disease/)

## In Pharmacology

- [Opioids and antagonists](https://durgical.com/en/pharmacology/classes/opioids-and-antagonists/)

[← All of physiology](https://durgical.com/en/physiology/)

Educational physiology content; not a diagnostic or treatment guide and no substitute for a health professional's advice. Values are approximate references. No doses are given.

Sources: [22.4 Gas Exchange](https://openstax.org/books/anatomy-and-physiology-2e/pages/22-4-gas-exchange) (OpenStax (Rice University), 2022); [Arterial Blood Gas (ABG) Test: MedlinePlus Medical Test](https://medlineplus.gov/lab-tests/arterial-blood-gas-abg-test/) (MedlinePlus, US National Library of Medicine, 2024); [22.6 Modifications in Respiratory Functions](https://openstax.org/books/anatomy-and-physiology-2e/pages/22-6-modifications-in-respiratory-functions) (OpenStax (Rice University), 2022); Hall JE. Guyton and Hall Textbook of Medical Physiology, 13th ed. (ch. 40). Reference book used to check facts; its text and figures are not reproduced (Elsevier, 2016).

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