Gas Laws Part II

Introduction

• The safe and effective use of medical gases and inhaled volatile anesthetics relies upon a precise understanding of the relationships between pressure, solubility, and diffusion.
• The laws detailed in this summary, including Dalton’s law of partial pressures, Henry’s law of solubility, Fick’s law of diffusion, and Graham’s law of diffusion, which are the physicochemical principles that govern gas behavior within the patient’s respiratory system and circulating blood.

Dalton’s Law¹

• Dalton’s law of partial pressures states that in a mixture of non-reacting gases, the total pressure (P_TOTAL) exerted is equal to the sum of the partial pressures (PN) that each individual gas would exert if it occupied the total volume alone.
• Partial pressure, therefore, represents the fraction of the total pressure contributed by a specific gas in the mixture.

• Clinical application I: alveolar gas equation²
  • The alveolar gas equation is a clinical derivation of Dalton’s law of partial pressures, enabling calculation of the ideal alveolar partial pressure of oxygen (PAO₂), a crucial variable that is not directly measurable.
  • Dalton’s law states that the total pressure in the alveolus (P_BAROMETRIC ) must equal the sum of the partial pressures of all gases within it:

• • The alveolar gas equation aims to solve for PAO₂ (partial pressure of alveolar oxygen).
  • The simplified alveolar gas equation is:

• • The variables are defined as:
    i. PAO₂: The calculated partial pressure of oxygen within the alveoli.
    ii. PiO₂: The inspired partial pressure of oxygen, calculated as: FiO₂ × (P_BAROMETRIC – P_H2O).

• • The partial pressure of inspired oxygen (PiO₂) is the pressure exerted by oxygen alone, and it is the true determinant of how much oxygen diffuses into the pulmonary capillaries. It is calculated as:

• • Thus, PiO₂ = FiO₂ x barometric pressure.
  • Since the barometric pressure at sea level is 760 mmHg, PiO₂ = 0.21 × 760, which is approximately 160 mmHg.
  • This is the partial pressure of oxygen in dry ambient air. To calculate the PiO₂, water vapor pressure (47 mmHg at room temperature) must be subtracted.

• • As a mountaineer or a patient is taken to a higher altitude, the following occurs:
    • Decreased P_TOTAL: The surrounding atmospheric pressure drops significantly and linearly. For example, at the summit of Mount Everest, P_TOTAL drops to approximately 253 mmHg from about 760 mmHg at sea level.
    • Constant FiO₂: In ambient air, the fractional concentration of inspired oxygen (FiO₂) remains fixed at 21% (0.21).
    • Critical Drop in PiO₂: Because PiO₂ is a product of the fractional concentration and the total pressure, the lower PTOTAL causes a proportional, steep drop in PiO₂.

• • For patient management at high altitude, merely checking the FiO2 on a gas tank or ventilator is insufficient. The total barometric pressure is the ultimate determinant of the patient’s available PiO₂. To prevent hypoxia, anesthesia providers must compensate for the low P_TOTAL by:
    • Significantly increasing the FiO₂ (giving supplemental oxygen).
    • In extreme cases, increasing P_TOTAL (e.g., using a hyperbaric chamber or sealed, pressurized transport).

Henry’s Law³

• Henry’s law states that at a constant temperature, the amount (concentration) of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with the liquid.
• Mathematically, this relationship is expressed as:

• Where:
  • C is the concentration of the dissolved gas (e.g., mol/L or mL/dL).
  • P is the partial pressure of the gas above the liquid (e.g., kPa or mmHg).
  • k is the Henry’s Law solubility constant, which is unique for every solute/solvent pair at a specific temperature.

• The speed of diffusion depends on two main sets of factors: those that increase the rate and those that decrease the rate.
• The rate of gas movement is directly proportional to these factors:
  • Surface area (A): The larger the area available for exchange (e.g., the vast surface of the lung alveoli), the faster the diffusion.
  • Partial pressure difference (P₁ – P₂): The steeper the concentration gradient (the bigger the difference in gas pressure between the two sides of the tissue), the faster the gas is pushed across.
  • Solubility (Sol): Gases that dissolve more easily in the tissue (like CO₂ in blood) diffuse faster.
• The rate of gas movement is inversely proportional to these factors:
  • Thickness (T): The thicker the tissue barrier, the slower the diffusion rate.
  • Square root of molecular weight (MW): Lighter gases diffuse slightly faster than heavier gases.

• When two gases A and B are compared, it is expressed as:
• Graham’s law is a fundamental component of Fick’s law of diffusion, as the inverse relationship between diffusion rate and the square root of molar mass is incorporated within the diffusion constant.
• Clinical application I: the second gas effect

• • The second gas effect is a concentration phenomenon in which the rapid, high-volume uptake of N₂O (the First Gas) from the alveoli causes a net decrease in alveolar volume.
  • This volume reduction concentrates the remaining gases, including the simultaneously delivered volatile anesthetic (the second gas), thereby increasing its alveolar partial pressure (PA) and accelerating its uptake into the blood.
  • N₂O (molecular weight = 44) is significantly lighter than volatile agents (e.g., Isoflurane, molecular weight = 184.5). Graham’s law predicts that the lower molar mass of N₂O contributes to its faster molecular motion, which drives rapid mass transfer.
  • The reverse process is the rapid outward diffusion of N₂O from the blood back into the alveoli upon discontinuation, which can cause diffusion hypoxia if 100% oxygen is not administered.