Oxygen Supply Cascade Equations
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A consolidated reference of the equations used across Weeks 1–4 for gas exchange, the oxygen supply cascade, and metabolic dynamics during exercise. Please see lecture notes on the units, and carefully check your work for consistency of units when solving problems.
Gas laws and partial pressures
| Equation |
Name |
Purpose |
| $PV = nRT$ |
Ideal gas law |
Relates pressure, volume, temperature, and number of moles through the universal gas constant R; the foundation for all gas behavior. |
| $P_1 V_1 = P_2 V_2$ |
Boyle’s law |
At constant T and n, pressure is inversely proportional to volume; explains how thoracic volume change drives airflow. |
| $V_1 / T_1 = V_2 / T_2$ |
Charles’ law |
At constant P and n, volume is directly proportional to temperature. |
| $P_1 / T_1 = P_2 / T_2$ |
Gay-Lussac’s law |
At constant V and n, pressure is directly proportional to temperature. |
| $\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}$ |
Combined gas law |
General form from which the individual gas laws are derived by holding variables constant. |
| $P_{\text{air}} = P_{O_2} + P_{CO_2} + P_{N_2} + \ldots$ |
Dalton’s law of partial pressures |
Total pressure of a gas mixture equals the sum of the partial pressures of its component gases. |
| $P_{O_2} = P_B \cdot F_{O_2}$ |
Partial pressure of a gas |
Partial pressure equals barometric pressure multiplied by fractional concentration. |
| $P_IO_2 = (P_B - P_{H_2O}) \cdot F_IO_2$ |
Inspired PO₂ |
Partial pressure of O2 in the airways after warming and saturation with water vapor (PH₂O = 47 mmHg at 37 °C); used to start the oxygen cascade and to quantify hypoxia at altitude. |
Pulmonary mechanics and ventilation
| Equation |
Name |
Purpose |
| $\dot{V} = \Delta P / R_{airway}$ |
Airflow equation |
Rate of airflow equals the pressure difference across the airways divided by airway resistance. |
| $\dot{V}_E = f_b \cdot V_T$ |
Minute (expired) ventilation |
Total volume of air moved in and out of the lungs per minute. |
| $\dot{V}_A = f_b \cdot (V_T - V_D)$ |
Alveolar ventilation |
Volume of air per minute that actually reaches the gas-exchange surfaces, after subtracting dead space. |
| $V_{D,\text{phys}} = V_{D,\text{anatomic}} + V_{D,\text{alveolar}}$ |
Physiological dead space |
Total non-gas-exchanging ventilation volume (conducting airways plus unperfused alveoli). |
Alveolar gas, diffusion, and respiratory exchange
| Equation |
Name |
Purpose |
| $P_ACO_2 = \dot{V}CO_2 / \dot{V}_A \cdot K$ |
Clinical alveolar ventilation equation |
Estimates alveolar PCO₂ from CO2 production and alveolar ventilation; $K = 863$ mmHg. |
| $P_AO_2 = P_IO_2 - P_ACO_2 / R$ |
Alveolar gas equation (simplified) |
Estimates alveolar PO₂ from inspired PO₂, accounting for CO2 dilution; explains how hyperventilation at altitude defends PAO2. |
| $R = \dot{V}CO_2 / \dot{V}O_2$ |
Respiratory exchange ratio |
Ratio of CO2 output to O2 uptake; reflects metabolic substrate (≈ 0.7 fat, 1.0 carbohydrate). |
| $\dot{V}O_2 = \frac{A \cdot D \cdot \Delta P_{O_2}}{T}$ |
Fick’s law of diffusion |
Rate of gas transfer is proportional to surface area and partial-pressure gradient and inversely proportional to barrier thickness. |
| $D_LO_2 = A \cdot D / T$ |
Pulmonary diffusion capacity |
Combined structural factors determining the lungs’ capacity for O2 transfer. |
| $\dot{V}O_2 = D_LO_2 \cdot \Delta P_{O_2}$ |
Simplified diffusion equation |
O2 flux across the blood–gas barrier equals diffusion capacity times the alveolar–capillary PO₂ gradient. |
| $\dot{V}O_2 = \dot{V}_A \cdot \beta_{gO_2} \cdot (P_IO_2 - P_EO_2)$ |
Alveolar O2 delivery (capacitance form) |
Ventilatory O2 delivery in terms of alveolar ventilation, the capacitance coefficient of O2 in air ($\beta_{gO_2}$), and the inspired–expired PO₂ difference. |
Fick principle across the cascade
| Equation |
Name |
Purpose |
| $M_x = M_{x,\text{in}} - M_{x,\text{out}}$ |
Fick principle (general) |
Conservation of mass applied to any cascade step: net uptake equals input minus output. |
| $\dot{V}O_2 = \dot{V}_E \cdot (F_IO_2 - F_EO_2)$ |
Fick principle (ventilation, gas fractions) |
Whole-body O2 consumption from minute ventilation and the inspired–expired O2 fraction difference; basis for indirect calorimetry. |
| $\dot{V}O_2 = \dot{V}_A \cdot \beta_{gO_2} \cdot (P_IO_2 - P_EO_2)$ |
Fick principle (ventilation, partial-pressure form) |
Equivalent form using the capacitance coefficient of O2 in air, $\beta_{gO_2}$. |
| $\dot{V}O_2 = \dot{Q} \cdot B_{blood} (P_aO_2 - P_{\bar{v}}O_2)$ |
Fick principle (circulation, partial-pressure form) |
O2 uptake from cardiac output, the blood O2-carrying coefficient, and the arterio-venous PO₂ difference. |
| $\dot{V}O_2 = \dot{Q} (C_aO_2 - C_{\bar{v}}O_2)$ |
Fick principle (circulation, content form) |
O2 uptake from cardiac output and the a–v O2 content difference; the central cardiovascular form used for exercise, altitude, and diving. |
| $\dot{Q} = \dot{V}O_2 / (C_aO_2 - C_{\bar{v}}O_2)$ |
Fick principle rearranged for $\dot{Q}$ |
Clinical inference of cardiac output from measured $\dot{V}O_2$ and the a–v O2 content difference. |
Cardiovascular transport
| Equation |
Name |
Purpose |
| $\dot{Q} = HR \cdot SV$ |
Cardiac output |
Volume of blood pumped per minute by one ventricle (L/min). |
| $SV = \dot{Q} / HR$ |
Stroke volume |
Volume ejected per heartbeat (L/beat); rises with training-induced cardiac hypertrophy. |
| $HR_{\max} \approx 208 - 0.7 \cdot \text{Age}$ |
Age-predicted maximum heart rate |
Estimates HRmax (beats/min) from age; used to prescribe exercise intensity. |
Blood O2 content and extraction
| Equation |
Name |
Purpose |
| $C_aO_2 = 1.39 \cdot [Hb] \cdot S_aO_2 + 0.003 \cdot P_aO_2$ |
Arterial O2 content |
Total O2 per unit blood as the sum of Hb-bound and dissolved O2. |
| $\%\text{Sat} = \frac{[O_2]}{O_2\,\text{capacity}} \cdot 100$ |
Hemoglobin percent saturation |
Fraction of Hb binding sites occupied by O2. |
| $\Delta C_{O_2} = C_aO_2 - C_{\bar{v}}O_2$ |
a–v O2 difference |
Amount of O2 extracted from each liter of blood passing through the tissues; widens from rest to maximal exercise. |
Ventilation–perfusion matching
| Equation |
Name |
Purpose |
| $\dot{V} / \dot{Q} \approx 1.0$ |
Ideal V/Q ratio |
Optimal matching of alveolar ventilation to pulmonary perfusion; V/Q < 1 = shunt, V/Q > 1 = dead space. |
| Equation |
Name |
Purpose |
| $\dot{V}O_2 = \dot{V}_E \cdot (F_IO_2 - F_EO_2)$ |
Fick principle (minute ventilation and gas fractions) |
Whole-body $\dot{V}O_2$ measured from minute ventilation and the inspired–expired O2 fraction difference; the standard form used for indirect calorimetry. |
| $\text{Mass-specific } \dot{V}O_2 = \dot{V}O_2 / m_{\text{body}}$ |
Mass-specific VO2 |
Normalizes O2 consumption to body mass (mL/kg/min) for comparisons across individuals and species. |
| $\dot{V}O_2 = 0.1 x + 3.5$ |
ACSM walking equation |
Estimated $\dot{V}O_2$ (mL/kg/min) for walking at speed $x$ (m/min). |
| $\dot{V}O_2 = 0.2 x + 3.5$ |
ACSM running equation |
Estimated $\dot{V}O_2$ (mL/kg/min) for running at speed $x$ (m/min). |
| $fAS = \dot{V}O_2\text{max} / SMR$ |
Factorial aerobic scope |
Ratio of maximum to standard (or basal) metabolic rate; typically 5–10× in vertebrates, > 50× in flying birds. |
Cellular energetics
| Equation |
Name |
Purpose |
| $H^+ + ADP + PCr \rightarrow ATP + \text{creatine}$ |
Phosphocreatine (PCr) pathway |
Fastest ATP-regeneration pathway; buffers ATP at exercise onset. |
| $C_6H_{12}O_6 + 6 O_2 \rightarrow 6 CO_2 + 6 H_2O + 36\,ATP$ |
Aerobic metabolism of glucose |
Complete oxidation of glucose in the mitochondria; actual ATP yield ≈ 29–32. |
Whole-body O2 stores (diving physiology)
| Equation |
Name |
Purpose |
| $\text{Total O}_2 = V_{\text{lung}} \cdot F_{\text{lung},O_2} + V_{\text{blood}} \cdot C_aO_2 + V_{\text{muscle}} \cdot [Mb] \cdot 1.34$ |
Whole-body O2 stores |
Sum of O2 in lung gas, blood (Hb-bound), and muscle (Mb-bound); diving specialists shift the balance toward blood and muscle. |