# Fundamental gas laws relevant to critical care medicine

Even though this chapter is most relevant to Section W(i) from the 2023 CICM Primary Syllabus ("Describe the laws governing the behavior of gases"), it acts as an effective portal into the Respiratory section, as logically it should (that system being the main governor of the gases' movements and behaviours throughout the human body). As such, the discussion of gas laws here will be somewhat more detailed and skewed towards respiratory mechanisms, rather than treating them with dry abstraction. Elsewhere, the raw untreated mathematics of these laws are massaged with the appropriate level of reverence.

There being no past paper SAQs on this topic, the CICM Part One candidate is at a loss as to what to study, and how much of it. One can only assume that some rote-learned equations might one day be called upon, or that at some stage during a viva station one will be expected to hold forth on the definitions of these laws. Whether one ought to expect a mindless recitation or some level of analysis, that is not clear. In any case, a short memorable blurb is called for:

Boyle's Law:

• For a fixed mass of gas at constant temperature, the pressure (P) and volume (V) are inversely proportional, such that P ×V = k, where k is a constant.

Charles' Law:

• The volume occupied by a fixed mass of gas at constant pressure is directly proportional to its absolute temperature (V/T = k).

Third gas law (Gay-Lussac's Law):

• The pressure of a fixed mass of gas at constant volume is directly proportional to its absolute temperature (P/T = k).

• Equal volumes of gases at the same temperature and pressure contain the same number of molecules (6.023 × 1023, Avogadro’s number).

Universal (Ideal) Gas Law:

• The state of a fixed mass of gas is determined by its pressure, volume and temperature(PV = nRT)

In terms of peer-reviewed resources, the best single offering is probably Physics of Gases by Thomas & Stamatakis (2009). It is short, dense and to-the-point. Unfortunately, at the time of writing, Anaesthesia and Intensive Care are asking US\$39.95 for the pdf copy. If one is severely affected by class inequality, one may resort to reading Gases and Vapours by  Ben Gupta (at that stage, in 2012 he was an Anaesthetics registrar at the Sir Charles Gairdner Hospital in Perth). This article contains virtually the same information but is free.  The definitive resource, if one has infinite time and money, is Basic physics and measurement in anaesthesia by Parbrook (actually, by the Parbrooks- Geoffrey and Evelyn). At the time of writing the 2003 5th edition is current, though the author of this chapter referred to his 4th edition as if the gas laws hadn't changed since 1995.

## Boyle's Law

As volume increases, the pressure of the gas decreases in proportion:
P1V1 = P2V2

Boyle's 1660 publication, "New Experiments Physico-Mechanicall, Touching the Spring of the Air, and its Effects" has been covered brilliantly by JB West (yes, John B. West, of West's Physiology).

In brief summary, the law states that at any given temperature, the product of pressure and volume is constant. Conversely, as you change one, the other will change predictably.

Boyle's apparatus

Boyle wrote an extensive essay (in the form of a letter to his nephew) detailing his experiment- to be precise, his 43 experiments - on the behaviour of vacuum and rarified gas. Of the engravings he commissioned for this publication, some are selected for display above - not for the purposes of education, but to recall a time when experimental apparatus was lovingly rendered by artists, rather than by software.

Boyle's experiment is discussed in great detail in the course of West's article. In short, Boyle evacuated a huge glass bulb using a ratcheted piston, and then observed what happens to the stuff he put in there. The bulb was huge - around 28 litres - and he was apparently able to reduce it to a pressure of 22.8mmHg, according to the recently developed Toricellian mercury barometer. Among the objects which enjoyed a brief exposure to such a vacuum were lamb's bladders, burning candles, coals, loaded pistols, insects, as well as several unlucky birds and mice, who "droop and appear sick, and very soon after [were] taken with as violent and irregular Convulsions as are wont to be observ’d in Poultry, when their heads are wrung off" as air was removed from the chamber. Unfortunately, there being no Priestley for another hundred years or so, Boyle was unaware of oxygen, and piles of dead animals were generated in the course of his experiments. The exceptions to the rule were house snails: apparently, they were entirely unaffected by the extremes of pressure, and continued to function normally in the near-vacuum. This was no accident - a little digging has revealed that in fact, molluscs are the kings of anoxia, and many bivalves live their lives at about 1.0ppm of pO2 (0.00072 mmHg), so at around 5mmHg the snails were probably quite comfortable.

Anyway, the key issue is that for any given constant temperature the pressure of any gas can be predicted from its mass and the volume of the chamber it occupies. West finishes his assessment of Boyle's works by saying that "modern students who are interested in high-altitude physiology should be aware of this classic book".

## Charles's Law

The volume of a gas is directly proportional to its absolute temperature.

The law was published by Joseph Louis Gay-Lussac who credited Jaques Charles with its discovery. Apparently, the unpublished material from which this law was derived dates to 1787. A certain team of chemistry teachers has an excellent entry regarding these two chemists, complete with snide remarks regarding Charles' appearance (looks drunk, they say).

Charles's contribution to gas laws derives much of its success from his excited attempts to replicate Joseph and Etienne Montgolfier's hot air balloon experiences. The first hydrogen balloon, of Charles' design, was a rubberised silk bag filled with the hydrogen which was produced when vast quantities of sulfuric acid were poured over approximately 1000kg of iron filings. The resulting gas was hot - so hot, in fact, that it had to be cooled by passing through lead pipes. Observers then remarked that as the gas cooled, so the balloon deflated. The experiment came to a splendid conclusion after the hydrogen balloon came to rest some ten miles away near the village of Gonesse, whose villagers apparently tore it to shreds with a variety of improvised weapons.

Enraged villagers destroying Charle's hydrogen balloon. Apparently, it made weird noises and had a foul odour. (c 1783)

Fortunately, this early balloon was too small to carry Charles himself. Though he himself did not publish anything of scientific merit, J.L.Gay-Lussac published on his behalf. A specific experiment was the inspiration for what we now (mostly) call Charles's Law. Five balloons of similar volumes were filled with different gases, and then heated to about 80C°. The resulting expansion was measured, and was noted to be the same irrespective of the gas involved.

## Gay-Lussac's law

The pressure of a gas is directly proportional to its temperature.

This was the direct extension of Charles's law. It is variably credited to Gay-Lussac and Guillaume Amontons, both of whom had arrived at certain conclusions regarding the behaviour of gases of fixed mass and fixed volume. A translation of Gay-Lussac's original publication can be found online.

The volume occupied by an ideal gas is proportional to the number of moles of gas.

Specifically, the molar volume of an ideal gas ( the space occupied by 1 mole of the "ideal" gas, or any gas for that matter ) is 22.4 litres at standard temperature and pressure.

In other words, for any gas, at any temperature and pressure, there will be the same amount of molecules present in the specified volume.

Avogadro's law is of particular interest because it uncovers a fundamental constant value, which is encountered everywhere: the Universal Gas Constant, R. This constant is also a feature of Boyle's and Charles's laws, the equations of which can be rearranged to demonstrate it. R is the constant of proportionality which relates the energy scale to the temperature scale; it is expressed as work (joules) per degree per mole. Its approximate value is 8.314.

## Graphical representation of the ideal gas laws

In the event that one recalls visual information more easily than equations, one might potentially derive some benefit from being able to represent the gas laws as graphical plots:

Boyle's law relationship is hyperbolic. Pressure is inversely proportional to volume. The other two laws are relationships which are directly proportional, and their graphs are straight diagonal lines. One needs to remember to correctly represent temperature in the case of Charles' law; at zero degrees Celsius most volatile gases occupy a nonzero volume.

## The Ideal Gas Law

This is the mutant offspring of Boyle's, Charles's, Gay-Lussac's and Avogadro's principles.

### PV = nRT

Where

• P is the pressure of the gas
• V is the volume of the gas
• n is the number of moles of the gas
• R is the Universal Gas Constant (8.314)
• T is the absolute temperature of the gas in Kelvin.

## Graham's Law

Graham's law describes the rate of gases mixing together (diffusion) or escaping from their confinement through miniscule pores in the storage medium (effusion).

The rate of diffusion or effusion of a gas is inversely proportional to the square root of the mass of its particles

In most circumstances, the mass of the particles and the density of the gas are sufficiently correlated to one another that an approximation can be made:

Thus, if a gas had particularly large particles, or is particularly dense, it will mix more slowly with other gases, and ooze more slowly out of containers.

## References

Parbrook, Geoffrey Donald, Paul D. Davis, and Evelyn O. Parbrook. Basic physics and measurement in anaesthesia. Oxford, UK: Butterworth-Heinemann, 1990.

Thomas, Gary, and Stephen Stamatakis. "Physics of gases." Anaesthesia & Intensive Care Medicine 10.1 (2009): 48-51.

Gupta, Ben. "Gases and Vapours" Anaesthesia 15.2 (2012).

Boyle, Robert. A continuation of new experiments physico-mechanical, touching the spring and weight of the air and their effects. The I. part whereto is annext a short discourse of the atmospheres of consistent bodies / written by way of letter to the right honourable the Lord Clifford and Dungarvan by the honourable Robert Boyle... H. Hall. 1660

West, John B. "Robert Boyle’s landmark book of 1660 with the first experiments on rarified air." Journal of Applied Physiology 98.1 (2005): 31-39.

De Zwaan, A., et al. "Differential sensitivities to hypoxia by two anoxia-tolerant marine molluscs: a biochemical analysis." Marine biology 111.3 (1991): 343-351.

Gillespie, Richard. "Ballooning in France and Britain, 1783-1786: aerostation and adventurism." Isis (1984): 249-268.

Satyapal, Sunita, John Petrovic, and George Thomas. "Gassing up with hydrogen." Scientific American 296.4 (2007): 80-87.

Remington, Preston. "A Monument Honoring the Invention of the Balloon." The Metropolitan Museum of Art Bulletin (1944): 241-248.

Dalton, Jean-Baptiste Biot, Alexander von Humboldt, and Louis Jacques Thenard. "GAY-LUSSAC’S LAW OF COMBINING GAS VOLUMES."SELECTED BOOKS BY: 190.

Gay-Lussac, Joseph Louis. "Recherches sur la dilatation des gaz et des vapeurs." Annales de Chimie. Vol. 43. 1802.

Mary Ellen Bowden. "1. Forerunners." Chemical Achievers: The Human Face of the Chemical Sciences (1997): 3.

Baker, Francis Joseph, and Reginald Edward Silverton. Introduction to medical laboratory technology. Butterworth-Heinemann, 1976.

Severinghaus, John W., Poul Astrup, and John F. Murray. "Blood gas analysis and critical care medicine." American journal of respiratory and critical care medicine 157.4 (1998): S114-S122.

Henry, William (January 1, 1803). "Experiments on the Quantity of Gases Absorbed by Water, at Different Temperatures, and under Different Pressures". Philosophical Transactions of the Royal Society (London) 93: 29–274.

"Textbook errors: III. The solubility of gases in liquids." Journal of Chemical Education 32, no. 8 (1955): 399.

Ashwood, E. R., G. Kost, and M. Kenny. "Temperature correction of blood-gas and pH measurements." Clinical chemistry 29.11 (1983): 1877-1885.

Sendroy, Julius, Robert T. Dillon, and Donald D. Van Slyke. "STUDIES OF GAS AND ELECTROLYTE EQUILIBRIA IN BLOOD XIX. THE SOLUBILITY AND PHYSICAL STATE OF UNCOMBINED OXYGEN IN BLOOD." Journal of Biological Chemistry 105.3 (1934): 597-632.

Yeh, Shu-Yuan, and Richard E. Peterson. "Solubility of krypton and xenon in blood, protein solutions, and tissue homogenates." J Appl Physiol 20.5 (1965): 1041-1047.

Weathersby, P. K., and L. D. Homer. "Solubility of inert gases in biological fluids and tissues: a review." Undersea biomedical research 7.4 (1980): 277-296.

Parbrook, Geoffrey Donald, Paul D. Davis, and Evelyn O. Parbrook. Basic physics and measurement in anaesthesia. Oxford, UK: Butterworth-Heinemann, 1990.