Gas Laws and Clinical Application

Issues of Concern

Boyle’s Law

Boyle’s law or Boyle–Mariotte law or Mariotte's law (especially in France) takes the name of Robert Boyle (1627–1691) and is based on the research of Richard Towneley (1629-1707) and Henry Power (1623–1668). It states that at a constant temperature, the pressure is inversely proportional to volume:

• P alpha 1/V or P·V = k, where k is a constant and is dependent on the temperature.

NB: alpha means 'is proportional to.'

For the same gas under different conditions at the same temperature, it can also be expressed as:

• P1·V1 = P2·V2

Charles’ Law

Charles’s law, discovered by Jacques Charles (1746-1823) in 1787 and refined by Joseph Louis Gay-Lussac (1778-1850) in 1808, states that at constant pressure, the volume is directly proportional to absolute temperature, for a fixed mass of a gas:

• V alpha T, which can also be stated as V/T = k, where k is a constant, and similarly, V1/T1 = V2/T2

Gay-Lussac’s Law

Gay-Lussac’s Law or Third Gas Law states that for a constant volume, the pressure is directly proportional to absolute temperature:

• P alpha T; also stated as P/T = K, where K is a constant, and similarly, P1/T1 = P2/T2

 Those three laws can be mathematically combined and expressed as:

• P1V1/T1 = P2V2/T2

In addition to the three fundamental laws, other gas laws must be considered.

Avogadro's Law

Equal volumes of gases at the same temperature and pressure contain the same number of molecules (6.023·10^23, Avogadro’s number). In other words, the volume occupied by an ideal gas is proportional to the number of moles of gas and the molar volume of an ideal gas (the space occupied by 1 mole of the "ideal" gas) is 22.4 liters at standard temperature and pressure.

Ideal Gas Law

The ideal gas law is a combination of Boyle’s law, Charles’s law, Gay-Lussac’s law, and Avogadro’s law:

• P·V = n·R·T

n is the number of moles of the gas (mol), R is the ideal gas constant (8.314 J/(K·mol), or 0.820 (L·atm)/(K·mol)), T is the absolute temperature (K), P is pressure, and V is volume.

Dalton’s Law and Henry’s Law

Dalton’s law of partial pressures states that, for a mixture of non-reacting gases, the sum of the partial pressure of each gas is equal to the total pressure exerted by the mixture, at constant temperature and volume:

• Ptotal= P1 + P2 + … Pz, or Ptotal= (n1·R·T1/V1) + (n2·R·T2/V2) + ... (nz·R·Tz/Vz)

Henry’s law states that for a constant temperature, the amount of dissolved gas in a liquid is directly proportional to the partial pressure of that gas (in contact with its surface). This relationship is no longer linear once a gas mixture is used, due to stabilization and destabilization effects on solubility[2], and deviations are found with increasingly high pressures or concentrations[3]:

• P = K·M, where P is the partial pressure of the gas, K­ is Henry’s constant of proportionality, and M is the molar concentration of the gas.

Graham's Law

The rate of diffusion (or effusion) of a gas is inversely proportional to the square root of the mass of its particles. When a gas had particularly large particles (or is particularly dense), it will mix more slowly with other gases, and oozes more slowly from its containers.

Clinical Significance

Boyle’s Law

Boyle’s law can be used to describe the effects of altitude on gases in closed cavities within the body, and to calculate the total intra-thoracic gas volume by body plethysmography. As altitude increases, ambient pressure decreases, and therefore, by Boyle’s Law, volume expansion occurs in enclosed spaces. This effect can be demonstrated by observing the expansion of a sealed bag of potato chips on an ascending commercial flight. In one artificial pneumothorax model, a 40 mL pneumothorax increased in volume by up to 16% at 1.5 km (approx. 5000 feet) from sea level,[4] an effect which may prompt thoracostomy before helicopter transfer to prevent transition to a tension pneumothorax. It is estimated that an expansion of up to 30% for a closed volume of gas in the human body, e.g., a bulla, can be expected after ascending from sea level to an altitude of 2.5 km[5] (approx. 8200 feet).

Boyles law also explains the use of saline in the cuff of an endotracheal tube during hyperbaric therapy; to prevent an air leak due to the reduction of volume as pressure increases. When ascending from depth, if a diver holds their breath, the gases in their lungs will expand and can cause barotrauma, arterial gas embolism, mediastinal emphysema, or even pneumothorax.[6][7]

Using Boyle’s law, P1V1 = P2V2, we can calculate the change in volume at different altitudes. For example, a patient with a simple pneumothorax being airlifted to their local hospital. They have a pneumothorax with a volume of 1500 mL at sea level (101.3 kPa). At an altitude of 1 km (90 kPa), assuming the patient remains at a constant temperature, we can rearrange the formula to V2= (P1·V1)/P2 ­to calculate that the pneumothorax will now have a volume of 1688 mL, assuming a constant temperature.

Charles’s Law

Charles’s law is apparent in the action of a gas thermometer, where the change in volume of a gas (such as hydrogen) is used to display the change in temperature, or it can be seen more practically by placing a balloon filled with a gas into a freezer, and observing the reduction in volume that occurs. As gases are inspired, we can see from the relationship described in Charles’s law that warming from 20 degrees C (273 degrees K) to 37 degrees C (310K) will cause an increase in the volume of inspired gases. For example, an adult tidal breath of 500 ml of air at room temperature will increase to a volume of 530 ml, when it reaches the site of gas exchange as it warms up to body temperature.

Charles’s law can be also be used to calculate the amount of nitrous oxide remaining in a gas cylinder. A nitrous oxide cylinder will contain a mixture of gas and liquid at 20 degrees C room temperature (as its critical temperature is 36.5 degrees C). As nitrous oxide gets removed, the liquid nitrous will boil, and the nitrous oxide gas will then expand, so some (e.g., Bourdon) pressure gauges will indicate a constant pressure until all the liquid nitrous oxide has boiled and there is relatively little nitrous oxide left. Therefore, to calculate the amount of nitrous oxide left, you need to weigh the cylinder. Using Avogadro’s law (1 gram molecular weight of gas will occupy 22.41 L at STP), and knowing the molecular weight of nitrous oxide is 44, we can calculate the amount of nitrous oxide available to us.

If the empty weight of an ‘E’ cylinder is 5.9 kg and the current weight is 8.8 kg, we will have approximately 2900 g of liquid nitrous oxide and therefore (2900 x 22.41)/44 = 1477 liters of nitrous oxide at 273 degrees K. We can then apply Charles’s law; as room temperature is 293 K (273+20), to work out that there are (1477/273)x293 = 1585 liters of nitrous oxide remaining in the cylinder.

Gay-Lussac’s Law

Gay-Lussac’s law describes the relationship between pressure and temperature and applies in the mechanism of pressure relief valves on gas cylinders. As the pressure inside a gas cylinder increases due to increasing temperature, above a certain pressure limit, the pressure relief valve will open to prevent an explosion. Most physiological processes invariably occur at 37 degrees C, so there are few clinical applications of Gay-Lussac’s law.

Ideal Gas Law

One clinical application of the ideal gas law is in calculating the volume of oxygen available from a cylinder. An oxygen ‘E’ cylinder has a physical volume of 4.7 L, at a pressure of 137 bar (13700 kPa or 1987 PSI). Applying the ideal gas law at room temperature, P1·V1=n1·R1·T1 (inside the cylinder) and P2·V2=n2·R2·T2 (outside the cylinder) assuming a negligible reduction in temperature as gas is removed from the cylinder, i.e., T1 = T2 and n are constant, we are left with P1·V1= P2·V.2. Rearranging the equation, we now have V2= (P1·V1)/P2, and substituting in the values of a full ‘E’ cylinder, we get (13700 x 4.7)/101 = 637 liters of oxygen. With a basal oxygen consumption of 250 ml/min for the average-sized adult (BSA 1.8 m), we will have enough oxygen for 42.5 hours. If we increase the administered rate to 15 L/min, we will have just 42 minutes of oxygen from a full 'E' tank. This is a useful calculation when determining the size and number of cylinders needed to transfer a ventilated patient, though care must be taken to account for the oxygen consumed in driving the ventilator.

Dalton’s Law and Henry’s Law

Henry’s law can be used to understand the decompression sickness divers undergo if they surface too quickly, and how volatile anesthetic gases are used clinically. As diving depth increases, the partial pressure of each gas inspired will increase, leading to a higher concentration of nitrogen dissolving into the blood (if they are breathing a mixture of oxygen and nitrogen). At depth, this is not an issue as the high ambient pressure will maintain the dissolved state of nitrogen. However, if regular stops aren’t made during ascent to allow for transport and expiration of the excess nitrogen, as the ambient pressure decreases, the amount of nitrogen dissolved into the blood will decrease, and form bubbles, causing decompression sickness.[8][9] At 25 degrees C, Henry’s constant (atm/(mol/L)) for nitrogen gas is 1600, oxygen is 757, and carbon dioxide is 30. Henry’s law applies only at specific temperatures as we know by Le Chatelier’s principle, at a given partial pressure, the solubility of a gas is generally inversely proportional to the temperature.

Henry’s and Dalton’s laws also describe partial pressures of the volatile anesthetic gases at the alveoli (and therefore anesthetic depth). The partial pressure of anesthetic gas in the blood is proportional to its partial pressure in the alveoli, and this is determined both by its vapor pressure and concentration in the delivered mixture. Vapor pressure changes with temperature (not barometric pressure) and remains generally constant (some heat gets lost during vaporization from its liquid form), so changing the concentration of the anesthetic gas will influence the depth of anesthesia. With low barometric pressure at high altitudes, the delivered concentration will be higher than that at sea level, at the same concentration setting, due to a reduction in the number of molecules of other gases passing through the vaporizer for the same number of anesthetic agent molecules. For example, with a variable bypass vaporizer, a delivered concentration of 3% sevoflurane at 1 atm, the partial pressure of sevoflurane will be 0.03 x 1 = 0.03 atm. If the vaporizer is still set to deliver 3% sevoflurane, at a barometric pressure of 0.5 atm (4.8 km above sea level), the delivered concentration will be 0.03 x (1/0.5) = 6%, but the partial pressure will still be 0.06 x 0.5 = 0.03 atm, according to Dalton’s law.[10] As a consequence, titrating anesthetic depth to concentration by using the minimum alveolar concentration (MAC) parameter may not be very accurate. For each inhaled agent administered, a MAC 1 value describes the concentration required, at 1 atm ambient pressure, to prevent 50% of subjects moving in response to a stimulus. The use of MAC instead of partial pressure (MAPP, minimum alveolar partial pressure) may lead to significant underdosing of the anesthetic agent, and therefore increases the risk of anesthesia awareness at altitude.[11]

Dalton’s law explains the changes in the atmospheric content of specific gases at different altitudes. This is of particular importance to the mountaineer ascending Everest, but it also finds use in the alveolar gas equation which enables us to calculate the partial pressure of oxygen in the alveolus. At sea level, the partial pressure of oxygen is 21% (157 mmHg or 21 kPa). At the summit of Mount Everest with a barometric pressure of 33.7 kPa or 0.3 atm, and using Dalton’s law, the partial pressure of oxygen is only 7 kPa or 52 mmHg, leading to oxygen-hemoglobin saturation of less than 80% without supplementation.