Bohr equation

Derivation

Its derivation is based on the fact that only the ventilated gases involved in gas exchange (${\displaystyle V_A}$) will produce CO₂. Because the total tidal volume (${\displaystyle V_T}$) is made up of ${\displaystyle V_A+V_d}$ (alveolar volume + dead space volume), we can substitute ${\displaystyle V_A}$ for ${\displaystyle V_T-V_d}$.

Initially, Bohr tells us V_T = V_d + V_A. The Bohr equation helps us find the amount of any expired gas, CO₂, N₂, O₂, etc.

In this case we will focus on CO₂.

Defining F_e as the fraction of CO₂ in the average expired breath, F_A as the fraction of CO₂ in the perfused alveolar volume, and F_d as the CO₂ makeup of the unperfused (and thus 'dead') region of the lung;

V_T x F_e = ( V_d x F_d ) + (V_A x F_A ).

This states that all of the CO₂ expired comes from two regions, the dead space volume and the alveolar volume.
If we suppose that F_d = 0 (since carbon dioxide's concentration in air is normally negligible), then we can say that:

${\displaystyle V_T\times F_e=V_A\times F_A}$ Where F_e = Fraction expired CO₂, and F_A = Alveolar fraction of CO₂.

${\displaystyle V_T\times F_e=(V_T-V_d)\times F_A}$ Substituted as above.

${\displaystyle V_T\times F_e=V_T\times F_A-V_d\times F_A}$ Multiply out the brackets.

${\displaystyle V_d\times F_A=V_T\times F_A-V_T\times F_e}$ Rearranging.

${\displaystyle V_d\times F_A=V_T\times (F_A-F_e)}$

${\displaystyle V_d/V_T=\frac{F_A-F_e}{F_A}}$ Divide by V_T and by F_A.

The only source of CO₂ is the alveolar space where gas exchange with blood takes place. Thus the alveolar fractional component of CO₂, F_A, will always be higher than the average CO₂ content of the expired air because of a non-zero dead space volume V_d, thus the above equation will always yield a positive number.

Where P_tot is the total pressure, we obtain:

• ${\displaystyle F_A\times P_{tot}=P_A{\text{CO}}_2^{}}$ and
• ${\displaystyle F_e\times P_{tot}=P_e{\text{CO}}_2^{}}$

Therefore:

${\displaystyle \begin{array}{cc}{V}_d/V_T& =\frac{(F_A{\text{CO}}_2^{}-F_e{\text{CO}}_2^{})\times P_{tot}}{F_A{\text{CO}}_2^{}\times P_{tot}}\\ & =\frac{P_A{\text{CO}}_2^{}-P_e{\text{CO}}_2^{}}{P_A{\text{CO}}_2^{}}\end{array}}$

A common step is to then presume that the partial pressure of carbon dioxide in the end-tidal exhaled air is in equilibrium with that gas' tension in the blood that leaves the alveolar capillaries of the lung.