# Alveolar gas equation

The alveolar gas equation is the method for calculating partial pressure of alveolar oxygen (p$_A$O$_2$). The equation is used in assessing if the lungs are properly transferring oxygen into the blood. The alveolar air equation is not widely used in clinical medicine, probably because of the complicated appearance of its classic forms. The partial pressure of oxygen (pO$_2$) in the pulmonary alveoli is required to calculate both the alveolar-arterial gradient of oxygen and the amount of right-to-left cardiac shunt, which are both clinically useful quantities. However, it is not practical to take a sample of gas from the alveoli in order to directly measure the partial pressure of oxygen. The alveolar gas equation allows the calculation of the alveolar partial pressure of oxygen from data that is practically measurable. It was first characterized in 1946.

## Assumptions

The equation relies on the following assumptions:
* Inspired gas contains no carbon dioxide (CO$_2$)
* Nitrogen (and any other gases except oxygen) in the inspired gas are in equilibrium with their dissolved states in the blood
* Inspired and alveolar gases obey the ideal gas law
* Carbon dioxide (CO$_2$) in the alveolar gas is in equilibrium with the arterial blood i.e. that the alveolar and arterial partial pressures are equal
* The alveolar gas is saturated with water

## Equation

$p_{A}{\ce {O2}}=F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})-{\frac {p_{a}{\ce {CO2}}(1-F_{I}{\ce {O2}}(1-{\ce {RER}}))}{{\ce {RER}}}}$
If F$_i$O$_2$ is small, or more specifically if $F_{I}{\ce {O2}}(1-{\ce {RER}})\ll 1$ then the equation can be simplified to:
$p_{A}{\ce {O2}}\approx F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})-{\frac {p_{a}{\ce {CO2}}}{{\ce {RER}}}}$
where:
| Quantity | Description | Sample value |
| $p_{A}{\ce {O2}}$ | The alveolar partial pressure of oxygen ($p{\ce {O2}}$) | 107 mmHg (14.2 kPa) |
| $F_{I}{\ce {O2}}$ | The fraction of inspired gas that is oxygen (expressed as a decimal). | 0.21 |
| $P_{ATM}$ | The prevailing atmospheric pressure | 760 mmHg (101 kPa) |
| $p{\ce {H2O}}$ | The saturated vapour pressure of water at body temperature and the prevailing atmospheric pressure | 47 mmHg (6.25 kPa) |
| $p_{a}{\ce {CO2}}$ | The arterial partial pressure of carbon dioxide ($p{\ce {CO2}}$ ) | 40 mmHg (5.33 kPa) |
| ${\text{RER}}$ | The respiratory exchange ratio | 0.8 |
Sample Values given for air at sea level at 37 °C.
Doubling F$_i$O$_2$ will double p$_i$O$_2$.
Other possible equations exist to calculate the alveolar air.
${\begin{aligned}p_{A}{\ce {O2}}&=F_{I}{\ce {O2}}\left(PB-p{\ce {H2O}}\right)-p_{A}{\ce {CO2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\[4pt]&=p_{I}{\ce {O2}}-p_{A}{\ce {CO2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\[4pt]&=p_{I}{\ce {O2}}-{\frac {V_{T}}{V_{T}-V_{D}}}\left(p_{I}{\ce {O2}}-p_{E}{\ce {O2}}\right)\\[4pt]&={\frac {p_{E}{\ce {O2}}-p_{I}{\ce {O2}}\left({\frac {V_{D}}{V_{T}}}\right)}{1-{\frac {V_{D}}{V_{T}}}}}\end{aligned}}$

### Abbreviated alveolar air equation

$p_{A}{\ce {O2}}={\frac {p_{E}{\ce {O2}}-p_{i}{\ce {O2}}{\frac {V_{D}}{V_{T}}}}{1-{\frac {V_{D}}{V_{T}}}}}$ p$_A$O$_2$, p$_E$O$_2$, and p$_i$O$_2$ are the partial pressures of oxygen in alveolar, expired, and inspired gas, respectively, and VD/VT is the ratio of physiologic dead space over tidal volume.

### Respiratory quotient (R)

$R={\frac {p_{E}{\ce {CO2}}(1-F_{I}{\ce {O2}})}{p_{i}{\ce {O2}}-p_{E}{\ce {O2}}-(p_{E}{\ce {CO2}}*F_{i}{\ce {O2}})}}$

### Physiologic dead space over tidal volume (VD/VT)

${\frac {V_{D}}{V_{T}}}={\frac {p_{A}{\ce {CO2}}-p_{E}{\ce {CO2}}}{p_{A}{\ce {CO2}}}}$