This equation describes the concentration of gases in the alveolus, and thus allows us to make educated guesses as to the effectiveness of gas exchange. One can use this to calculate the tension-based indices of oxygenation, such as A-a gradient or the a/A ratio (which is expressed as a percentage). The ABG machine frequently does this work for you, provided you have entered the FiO2 and have specified that your sample is "arterial". The result is usually reported as pO2(a/A).

Alveolar gas equation

An excellent article exploring the history of this equation discusses the original 1946 paper by Fehn, Rahn, and Otis. The modern form of the equation is as follows:

Alveolar gas equation

Thus, on room air and at sea level, we can assume certain constants.

PAO2 = (0.21 x (760 - 47)) - (PaCO2 x 1.25)

Thus:

PAO2 = (149 - (PaCO2 x 1.25)

Thus, the patient with a relatively normal PaCO2 (say, 40) :

PAO2 = (149 - 50)

So, a normal person should have a PAO2 of around 99 mmHg.

Or, for a patient with normal PaCO2 and an increased FiO2:

PAO2 = (FiO2 x 713) - 50

Of course, it is possible to have a strange respiratory quotient, but for this we would need to measure the total body VO2 and VCO2, which can only be accomplished by means of indirect calorimetry.

So, what should your PAO2 be at any given FiO2? In mmHg, the values are as follows:

FiO2 21% 100
FiO2 30% 164
FiO2 40% 235
FiO2 50% 307
FiO2 60% 378
FiO2 70% 449
FiO2 80% 520
FiO2 90% 592
FiO2 100% 662

In a nutshell, one can say that for every 10% increase in FiO2, the PAO2 will rise by about 71-72 mmHg.

Atmospheric gas mixture

Of course it would be amiss for us not to ask: why does the Earth titrate its FiO2 to 21% (or, more precisely, 20.9%)?

Thankfully, this would not be the first ridiculous digression in physiology. For instance,  John F. Nunn has written a chapter (Chapter 1 of Nunn's Respiratory Physiology) about the atmosphere. In it,  he is grateful that greenhouse gases have allowed the existence of surface water for the last 4000 million years, and he laments that the sun "proceeds remorselessly towards becoming a red giant, which will ultimately envelop the inner planets". Unfortunately the rest of the textbook proceeds soberly along a straight and predictable path.

A better introduction into the subject would probably be afforded by The Chemical Evolution of the Atmosphere and Oceans, by Heinrich D. Holland.  The author confesses on page 2 that "the range of topics considered in the book is uncomfortably large", and that due to the mass of information "chaos was a continuous threat" during the assembly of the manuscript. In any case, for a monograph written between the years 1968 and 1981, this is a fine work. It is deserving of attention from anybody who has finished with their final CICM exams and still has some enthusiasm for the written word.

References

The Alveolar Gas Equation was first described in a famous work by Wallace Fenn, Hermann Rahn, and Arthur Otis.

Fenn, Wallace O., Hermann Rahn, and Arthur B. Otis. "A theoretical study of the composition of the alveolar air at altitude." American Journal of Physiology--Legacy Content 146.5 (1946): 637-653.

Curran-Everett, Douglas. "A classic learning opportunity from Fenn, Rahn, and Otis (1946): the alveolar gas equation." Advances in physiology education 30.2 (2006): 58-62.

Carroll, Gilbert C. "Misapplication of alveolar gas equation." The New England journal of medicine 312.9 (1985): 586-586.

Mellemgaard, Kresten. "The Alveolar‐Arterial Oxygen Difference: Its Size and Components in Normal Man." Acta physiologica scandinavica 67.1 (1966): 10-20.

Hills, B. A. "Analysis of relative contributions to the alveolar-arterial oxygen gradient." The Bulletin of mathematical biophysics 33.2 (1971): 259-280.

Bergman, Norman A. "Components of the alveolar-arterial oxygen tension difference in anesthetized man." Anesthesiology 28.3 (1967): 517-527.

Harris, E. A., et al. "The normal alveolar-arterial oxygen-tension gradient in man." Clin Sci Mol Med 46.1 (1974): 89-104.

Atwell, Robert J., et al. "Factors Influencing the Alveolar-Arterial Oxygen Pressure Gradient Role of the Left Heart." American Journal of Physiology--Legacy Content 183.3 (1955): 451-453.

Ravin, Mark B., Robert M. Epstein, and James R. Malm. "Contribution of thebesian veins to the physiologic shunt in anesthetized man." J Appl Physiol20 (1965): 1148-1152.

Gould, Michael K., et al. "Indices of hypoxemia in patients with acute respiratory distress syndrome: Reliability, validity, and clinical usefulness."Critical care medicine 25.1 (1997): 6-8.

Sganga, G. A. B. R. I. E. L. E., et al. "The physiologic meaning of the respiratory index in various types of critical illness." Circulatory shock 17.3 (1984): 179-193.