Gas Equations

Avogadro’s Law¹

• Avogadro’s law states that the volume (V) of an ideal gas is directly proportional to the number of moles (n) at constant pressure (P) and temperature (T).
• In other words, the law states that equal volumes of any gas, measured at the same temperature and pressure, contain the same number of molecules. This specific quantity of molecules is known as Avogadro’s number.

• Avogadro’s number (6.023 * 10²³): The number of atoms, molecules, or particles contained in one mole of any substance.
• Molar volume: One mole of any ideal gas occupies a standard volume of 22.4L at Standard Temperature and Pressure (STP).

Clinical Application I: Calibration and Concentration of Volatile Anesthetics²

• Avogadro’s law is the underlying principle used to ensure the precise concentration output of volatile anesthetic vaporisers (e.g., sevoflurane), relating the mass of liquid agent to the resulting volume of saturated vapor.
• The law links the mass of the agent (via molecular weight) to the volume it occupies when fully vaporized.
• Example calculation (sevoflurane):
  • The molecular weight of sevoflurane is 200g/mol. Therefore, 200g equals 1 mole, which occupies 22.4L at standard temperature and pressure (STP).
  • If a sample contains 20mL of liquid sevoflurane, this is 20 g or 0.1 mole (assuming a density of 1 g/mL; 20mL of liquid sevoflurane has a mass of 20 g. This liquid will produce 2.24 L of vapor at STP.
  • If this 2.24L of sevoflurane vapor is mixed into 224L of fresh gas flow(oxygen), the resulting gas concentration is calculated as 2.24 / 224 = 0.01, or 1%.

Universal Gas Equation¹

• The Universal gas equation, also known as the ideal gas law, is the fundamental equation that mathematically combines Boyle’s law, Charles’s law, Gay-Lussac’s law, and Avagadro’s law into a single relationship. It is given by the formula:

R = universal gas constant = 8.314 J/(K·mol)

• This law describes the behaviour of an ideal gas (a theoretical gas whose molecules have no volume and exert no intermolecular forces) under standard conditions.
• All real gases, particularly medical gases, closely approximate this behaviour under typical clinical conditions.

Clinical Application I: Calculating the Volume of Gas Available from an Oxygen Cylinder

• The universal gas equation (PV=nRT) is the fundamental principle used to calculate the total volume of oxygen gas available from a high-pressure cylinder.
• Since the cylinder is a sealed system containing a fixed amount of gas (n is constant) and the process is assumed to occur at a constant room temperature (T is constant), the constant terms (n, R, and T) can be removed from the equation.
• This simplification yields a comparison of the initial state (inside the cylinder) and the final state (outside the cylinder at ambient pressure), which is the definition of Boyle’s Law:

Alveolar Gas Equation³

• The rate of oxygen (O₂) diffusion across the lung barrier is primarily driven by the O₂ partial pressure gradient between the alveolus and the pulmonary capillary.
• Since the alveolar oxygen pressure (PAO₂) cannot be measured directly, it is estimated using the alveolar gas equation (AGE) to assess gas exchange efficiency.
• The alveolar gas equation allows the PAO₂ to be estimated using variables that are easily measured.

• • PAO₂: Partial pressure of oxygen in the alveolus (the estimated variable)
  • PIO₂: Partial pressure of inspired oxygen
  • P_B represents barometric pressure
  • P_{SVP water} represents the saturated vapour pressure of water
  • FiO₂ represents the inspired fraction of oxygen.
  • PaCO₂: Arterial partial pressure of carbon dioxide (reflects alveolar ventilation).
  • R: The respiratory quotient (R) is defined as the ratio of carbon dioxide (CO₂) production to oxygen (O₂) consumption.

• • • The value of R differs based on the primary dietary metabolic substrate being consumed (fat, protein, or carbohydrate). R is calculated for each substrate based on the stoichiometry of its aerobic metabolism reaction.
    • Carbohydrate (e.g., Glucose): R = 1.0

• • • In the above reaction, six molecules of CO₂ are produced for every six molecules of O2 consumed; the ratio is 6/6 = 1.0.
    • Fat: R = 0.7.
    • Protein: R = 0.9
    • For a typical patient on a balanced Western diet, the respiratory quotient is usually assumed to be 0.8.
    • See the OA summary on respiratory quotient for more details. Link
• The AGE demonstrates that PAO₂ is primarily dependent on three key variables:

1. Inspired fraction of O₂ (FiO₂):

• • Increasing FiO₂ will increase PAO₂.
  • This increases the pressure gradient across the alveolar-capillary barrier, thus increasing the rate of O₂ diffusion (according to Fick’s law).

2. Barometric Pressure (PB):

• • PB decreases exponentially with ascent to altitude.
  • A fall in PB results in a lower PAO₂ and a reduced rate of O₂ diffusion.

3. Alveolar Ventilation (VA):

• • This is represented by PaCO₂, which is inversely proportional to VA.
  • Hyperventilation (increase in VA) results in a decrease in PaCO₂, which, according to the alveolar gas equation, increases PAO₂ and the rate of O₂ diffusion.
  • Hypoventilation (decrease in VA) results in an increase in PaCO₂, which decreases PAO₂ and reduces the rate of O₂ diffusion.

Clinical Application I: Diagnosing the Cause of Hypoxemia (A-a gradient)⁴

• The primary clinical use of the alveolar gas equation is to calculate the alveolar-arterial oxygen gradient (A-a gradient).
• A-a gradient is the difference between the calculated alveolar oxygen tension (PAO₂) and the measured arterial oxygen tension (PaO₂):

• An increased A-a gradient suggests a true defect in gas exchange.
• The A-a gradient is normally less than 15 mmHg.
• It progressively increases with age, potentially reaching 20 to 30 mmHg. This increase is likely due to a rise in closing capacity relative to the functional residual capacity (FRC).
• Causes of increased A-a gradient:
  • V/Q mismatch: This is the most common cause and is observed in conditions such as pneumonia, acute respiratory distress syndrome, and atelectasis.
  • Right-to-left shunt: Seen in conditions such as pulmonary arteriovenous malformation or intracardiac shunts.
  • Severe diffusion impairment: Seen in conditions such as end-stage pulmonary fibrosis
• Causes of normal A-a gradient:
  • Alveolar hypoventilation: The alveolar gas equation fully accounts for hypoventilation (via the PaCO₂), so a normal gradient with low PaO₂ points directly to a ventilation problem, such as airway obstruction, neuromuscular disease, or drug-induced respiratory depression.
  • Hypoxic gas mixtures: Such as a faulty anesthetic machine.

Clinical Application II: Alveolar Gas Equation and Preoxygenation⁵

• Preoxygenation is a critical safety maneuver in anesthesia and critical care, used to maximize the oxygen reserves in the patient’s lungs prior to the temporary cessation of breathing (apnea), such as during intubation by administering 100% oxygen.
• The following analysis utilises the simplified alveolar gas equation to demonstrate the significant difference in alveolar oxygen partial pressure achieved by administering room air (FiO₂=0.21) versus pure oxygen (FiO₂=1.0).

1. Baseline condition: respiration of room air (FiO₂ = 0.21)

• We perform the calculation using standard sea-level values (Patm = 760 mmHg) and normal physiological parameters (P_H2O = 47 mmHg, PaCO₂ = 40 mmHg, R = 0.8).
• Under normal respiratory conditions, the PAO₂ is calculated as follows:

2. Intervention: administration of 100% oxygen (FiO₂ = 1.0)

• The PAO₂ is calculated as:

• • The application of the alveolar gas equation demonstrates a profound 6.6-fold increase in PAO₂ upon switching from ambient air to 100% oxygen (from 100 mmHg to 663 mmHg).
  • This increase is critical for generating a large alveolar oxygen reservoir, which dictates the duration of the apneic safety period before the arterial PaO₂ falls below 60 mmHg.
  • This provides the physiological foundation for the universally adopted clinical practice of pre-oxygenation prior to procedures associated with apnea, such as rapid sequence intubation.
  • See the OA summary on Preoxygenation for more details. Link