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Haber's rule

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Contents
  1. 1Statement of the rule
  2. 2Why the rule holds or fails
  3. 3Carbon monoxide
  4. 4Hydrogen cyanide, low oxygen and carbon dioxide
  5. 5Use in fire toxicity assessment
  6. 6Worked example
  7. 7Limits and common misconceptions
  8. 8See also
  9. 9Notes
  10. 10References

Haber's rule is the principle in inhalation toxicology that the toxic effect of a gas depends on the dose inhaled, and that the product of concentration and exposure time is constant for a given effect.[1] If the concentration is doubled, the same effect is expected in half the time. Toxicologists use the rule as an approximation when they need to predict the effect of a higher concentration for a shorter time, or a lower concentration for a longer time.[2] The rule is named after F. Haber, who already noted in 1924 that hydrogen cyanide does not follow it.[3]

In fire safety engineering, the rule underlies the idea that toxic gases act through an inhaled dose, which is the basis of the fractional effective dose (FED).[4] It works well for carbon monoxide during short exposures to high concentrations and for lung irritants that cause damage after a fire. It fails for hydrogen cyanide, low oxygen, high carbon dioxide and the immediate sensory effects of irritant smoke, where the concentration itself matters more than the dose.[2][4][3] For these gases, the toxic hazard models of the SFPE Handbook replace the single constant dose by a dose that depends on the concentration.[5]

Statement of the rule

The two basic quantities of any inhalation exposure are the concentration of the toxic substance in the air and the duration of the exposure.[2] Their product, called the Ct product, gives a first estimate of the dose that has been inhaled. Introduction to Fire Safety Engineering states the same idea in plain words: the dose is the time multiplied by the concentration.[6] Haber's rule adds one assumption: for a given effect, this dose is a constant.[2] In equation form, the dose that produces a particular effect is the concentration multiplied by the exposure time: W=C×t

Symbol Description Unit
W constant dose for a specific effect (the Ct product) ppm·min or mg·min/L
C concentration of the gas in the inhaled air ppm or mg/L
t duration of the exposure min

The constant W is specific to each effect, so the dose for incapacitation differs from the dose for death.[2] For lethal effects in animal tests, the effect is usually the death of half of the animals. The corresponding dose is then the product of the lethal concentration for 50 per cent of the animals, the LC50, and the exposure time, expressed in mg·min/L.[2] In the general safety evaluation of new chemicals, the standard exposure lasts four hours, and the LC50 counts deaths during the exposure or within 14 days after it.[2]

A substance that obeys the rule has a one-hour LC50 four times as high as its four-hour LC50.[2] Carbonyl fluoride, a highly toxic lung irritant released when PTFE decomposes, is an example. Its one-hour LC50 of 0.990 mg/L is exactly four times its four-hour LC50 of 0.248 mg/L.[2] Both exposures correspond to about 59.5 mg·min/L.

Two straight dose lines from zero: the line for four times the concentration reaches the lethal dose after 1 hour, the other after 4 hours.
Under Haber's rule the inhaled dose grows in a straight line with time, so four times the concentration reaches the same lethal dose in a quarter of the time. Based on Purser (2002), p. 2-87.

Why the rule holds or fails

Haber's rule implies that the toxic substance is taken up at a steady rate, in direct proportion to the concentration inhaled.[2] This is the case when the body detoxifies or excretes the substance only slowly. The rule holds for many substances that act mainly on the lungs.[2] In fire toxicology it can be applied to the dose of lung irritants that may cause deaths from inflammation of the lungs some hours after a fire.[2]

Some gases, such as carbon monoxide, are both taken up and given off by the lungs. Their rate of uptake depends on the difference between the concentration in the inhaled air and the concentration already in the body.[2] Uptake is then fast at first and levels off as the body approaches equilibrium. The handbook describes this with an exponential expression, which forms the basis of the Coburn-Forster-Kane (CFK) equation for the uptake of carbon monoxide in humans:[2] W=C(1−e−tk)

Symbol Description Unit
W quantity taken up into the body not given in the source
C concentration in the inhaled air ppm
t exposure time min
k constant determined by several factors, such as breathing rate, body size and blood physiology min−1

The constant k depends on factors such as the breathing rate, body size and the physiology of lungs and blood.[7] When the inhaled concentration is high compared with the concentration in the body, the exponential curve is close to a straight line and Haber's rule is a good approximation.[2] A third group of effects depends hardly on dose at all. The pain caused by sensory irritants starts as soon as exposure begins and depends on the concentration, and it may even lessen as the person adapts.[2][4] Low oxygen and high carbon dioxide also act mainly through their concentration once the blood has adjusted, which takes a few minutes.[4] Sensory irritation is therefore assessed with a fractional effective concentration rather than with a dose.[8]

Carbon monoxide

Carbon monoxide is a highly toxic gas without irritant effect. It can incapacitate people if both the concentration and the exposure time are high enough.[9] It binds to haemoglobin in the blood, forming carboxyhaemoglobin (COHb), which can no longer carry oxygen. Without other factors, a COHb level of 50 per cent or more is generally considered lethal for fire victims.[9] Exposure data quoted in the Fire Dynamics Tools report of the U.S. Nuclear Regulatory Commission (NUREG-1805) show the role of both concentration and time. About 400 to 500 ppm can be inhaled for one hour without appreciable effect, 1500 to 2000 ppm for one hour is dangerous, and more than 4000 ppm is fatal in less than an hour.[10]

For short exposures at high concentrations, as in flaming fires, the uptake of carbon monoxide departs only a little from Haber's rule.[7] Experiments with active monkeys over a range of 1000 to 8000 ppm showed this directly. At every concentration, the animals lost consciousness after inhaling roughly the same dose, between 26,086 and 28,097 ppm·min.[4][7] The handbook concludes that linear uptake models can be used for such situations without serious error.[2]

For long exposures at lower concentrations the rule breaks down. According to the CFK equation, a person at rest reaches a lethal 50 per cent COHb after one hour at 2200 ppm. Reaching the same level in four hours requires 840 ppm, which is 0.38 times the one-hour concentration.[4] Haber's rule would predict 550 ppm, or 0.25 times, and so overestimates the hazard of long exposures.[4] A simpler linear equation by Stewart and co-workers gives the COHb level for short exposures:[7] %COHb=(3.317×10−5)(ppm CO)1.036(RMV)(t)

Symbol Description Unit
%COHb carboxyhaemoglobin in the blood per cent
ppm CO carbon monoxide concentration in the inhaled air ppm
RMV respiratory minute volume, the volume of air breathed per minute L/min
t exposure time min

The breathing rate is the most important variable, and it depends strongly on physical activity.[7] The handbook uses 8.5 L/min for a 70 kg person sitting at rest, 25 L/min for light work such as walking at 6.4 km/h, and 50 L/min for heavy work such as slow running.[7] A person escaping from a fire therefore takes up carbon monoxide much faster than a person at rest. Children breathe more air per kilogram of body weight, so they take up carbon monoxide more quickly than adults.[7]

Chart of time to incapacitation against carbon monoxide concentration for rest, light work and heavy work; heavier work gives much shorter times.
Time to incapacitation by carbon monoxide at rest, during light work and during heavy work, calculated with the Stewart equation. Based on Purser (2002), p. 2-102.

Hydrogen cyanide, low oxygen and carbon dioxide

Hydrogen cyanide is the clearest exception to Haber's rule. In primate tests, incapacitation at 100 ppm required a dose of about 2000 ppm·min, while at 200 ppm it took only about 400 ppm·min.[3] Haber himself recognised in 1924 that the dose for hydrogen cyanide depends on the concentration. The exact reason is not known, but it appears to be related to the rate of uptake and the distribution of the gas over the fluids of the body.[3] As a result, concentrations above about 150 ppm are more toxic than tests at lower concentrations would suggest.[3] The handbook gives an exponential expression for the time to incapacitation at a constant concentration of hydrogen cyanide, derived from primate and human data:[5] tICN=exp⁡(5.396−0.023×CHCN)

Symbol Description Unit
tICN time to incapacitation at a constant concentration min
CHCN concentration of hydrogen cyanide ppm

In line with this expression, doubling the concentration from 100 to 200 ppm shortens the time to incapacitation from about 20 minutes to about 2 minutes.[5] Below about 80 ppm, only minor effects are expected over periods of up to an hour.[11] Exposure data quoted in NUREG-1805 describe hydrogen cyanide as about 20 times more toxic than carbon monoxide, with 3000 ppm or more rapidly fatal.[12]

Two charts of dose to incapacitation against concentration: flat for carbon monoxide, falling steeply for hydrogen cyanide.
Dose needed for incapacitation against concentration: nearly constant for carbon monoxide, as Haber's rule predicts, but falling steeply for hydrogen cyanide. Based on Purser (2002), pp. 2-88 and 2-105.

Low oxygen and high carbon dioxide do not follow the rule either. Short exposures to severe oxygen lack cause incapacitation very quickly, while long exposures to a moderate lack have little effect.[13] At an oxygen deficit of 17 percentage points the dose to incapacitation is about 5.6 %·min, but at 11.3 percentage points it is about 87 %·min.[13] For carbon dioxide, the dose is about 20 %·min at 10 per cent but about 175 %·min at 5 per cent.[14] In both cases a weaker exposure needs a much larger dose, so a single constant cannot describe the effect.

Use in fire toxicity assessment

Haber's rule is used throughout fire toxicology to put exposure data on a common basis. Lethal concentrations measured over different times are often normalised to a 30-minute exposure by assuming that lethality depends on dose.[15][16] For hydrogen chloride, for example, a 30-minute lethal concentration for rats of 3800 ppm corresponds to an exposure dose of 114,000 ppm·min.[17] Inflammation of the lungs some hours after a fire is basically dose-related and approximately follows the rule. A 30-minute exposure to about 10 mg of decomposed material per litre of air, a dose of about 300 mg·min/L, is expected to cause dangerous lung damage.[18]

The FED method builds directly on the rule. The dose received in each short period of the fire is divided by the dose that causes the effect, and the fractions are added until they reach 1.[4] For substances that obey Haber's rule, the dose in the denominator is a constant. For substances that deviate, it is the dose that would cause the effect at the concentration of that period.[4]

Introduction to Fire Safety Engineering describes the FED in the same way, as a dose built up by adding the exposure to each gas over time.[19] In New Zealand, C/VM2 (second edition, 2025) uses an FED for carbon monoxide as one of its tenability criteria for fire toxicity, including the effects of carbon monoxide, carbon dioxide and oxygen.[20] In Australia, the handbook to the Fire Safety Verification Method (2020) adds a carbon monoxide criterion for smouldering fire scenarios, to be agreed between the stakeholders.[21]

Simpler hazard assessments also rely on the rule. Lethal toxic potency data for burning materials, obtained from small-scale tests with rats, can be used for a quick estimate of the hazard in a fire if they are assumed to follow Haber's rule.[22] The handbook considers the error introduced by this assumption unimportant in such a rough analysis.[22] It also warns that such potency data do not allow for the deviations from the rule that matter for asphyxiant gases.[23] The Australian verification method handbook notes that models of human exposure range from simple tenability limits to more complex models that include the FED and the probit function. For buildings, either simple limits or FED models are generally adopted.[21]

Worked example

The first case shows the rule working. With the constant dose of about 27,000 ppm·min found for active monkeys, Haber's rule predicts loss of consciousness after 27,000 / 1500 = 18 minutes at 1500 ppm of carbon monoxide.[3] At 3000 ppm, the predicted time halves to 9 minutes. The measured doses in the four primate tests lay within 5 per cent of their average of about 26,900 ppm·min, which confirms the rule for these short, high exposures.[4]

The second case shows the rule failing. In an example from the handbook, a person breathes 90 ppm of hydrogen cyanide for 15 minutes and then 180 ppm for 2 minutes.[5] The total dose is 90 × 15 + 180 × 2 = 1710 ppm·min. The expression for the time to incapacitation gives 27.83 minutes at 90 ppm, which corresponds to a dose of about 2500 ppm·min. If Haber's rule held, the person would have received only 1710 / 2500 = 0.68 of an incapacitating dose. The handbook instead adds the fraction for each period, each with the time to incapacitation at its own concentration: FICN=1527.83+23.51=0.54+0.57=1.11

The two-minute peak at 180 ppm alone contributes more than half of an incapacitating dose. The sum passes 1, and incapacitation is predicted between 17 and 18 minutes.[5] Averaging the concentration over the 17 minutes would give about 101 ppm and a predicted time of about 22 minutes. This illustrates the handbook's warning that averaging tends to give longer times to incapacitation than expected.[5]

Top: hydrogen cyanide at 90 ppm for 15 minutes then 180 ppm for 2 minutes. Bottom: fraction rising past 1, Haber line staying near 0.7.
Worked example: fractional incapacitating dose for 15 minutes at 90 ppm followed by 2 minutes at 180 ppm of hydrogen cyanide, with and without Haber's rule. Based on Purser (2002), p. 2-105.

Limits and common misconceptions

Haber's rule is an approximation. It assumes steady uptake, so it describes substances that are retained in the body much better than substances that are exhaled again or act through their concentration.[2] The dose constant W also belongs to one effect only, so a dose that causes incapacitation cannot be used as the dose for death.[2] For the smoke of burning materials, incapacitation is predicted at around a third of the lethal exposure dose.[23] Doses measured in animals depend on their breathing rate and body size, and small laboratory animals take up carbon monoxide faster than humans.[7]

A common misconception is that a dose limit always gives a safe prediction. For hydrogen cyanide the opposite is true, because a dose limit derived from long, low exposures underestimates the hazard of short, high peaks.[3][5] Averaging the concentration over the exposure hides these peaks in the same way.[5] For carbon monoxide during long exposures, the error goes the other way, and the rule overestimates the hazard.[4]

A second misconception is that every toxic effect of smoke can be expressed as a dose. The pain caused by irritant gases depends on the concentration at each moment, and the effect may lessen even while the dose keeps growing.[4] Conversely, the lung damage that the same irritants can cause hours after a fire does follow the dose.[18] Near the fire, visibility is usually the decisive factor for safe escape, and with a visibility above 5 to 10 m the dose of toxic gases is often of secondary importance. Further away, carbon monoxide concentrations can be very high while visibility exceeds 20 m, because soot settles and spreads less far than the gases.[24]

See also

Notes

  1. ↑ Purser (2002), p. 2-166.
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 Purser (2002), p. 2-87.
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 Purser (2002), p. 2-104.
  4. ↑ 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 Purser (2002), pp. 2-88–2-89.
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Purser (2002), p. 2-105.
  6. ↑ Van Rossum (2027), §6.1.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Purser (2002), p. 2-102.
  8. ↑ Purser (2002), p. 2-89.
  9. ↑ 9.0 9.1 Iqbal and Salley (2004), p. B-67.
  10. ↑ Sumi and Tsuchiya (1971), as cited in Iqbal and Salley (2004), p. B-67.
  11. ↑ Purser (2002), pp. 2-104–2-105.
  12. ↑ Sumi and Tsuchiya (1971), as cited in Iqbal and Salley (2004), p. B-68.
  13. ↑ 13.0 13.1 Purser (2002), p. 2-107.
  14. ↑ Purser (2002), p. 2-108.
  15. ↑ Purser (2002), p. 2-92.
  16. ↑ Purser (2002), p. 2-116.
  17. ↑ Purser (2002), p. 2-119.
  18. ↑ 18.0 18.1 Purser (2002), p. 2-117.
  19. ↑ Van Rossum (2027), §7.7.
  20. ↑ Ministry of Business, Innovation and Employment (2025b), p. 17.
  21. ↑ 21.0 21.1 Australian Building Codes Board (2020b), p. 110.
  22. ↑ 22.0 22.1 Purser (2002), p. 2-97.
  23. ↑ 23.0 23.1 Purser (2002), p. 2-93.
  24. ↑ Brandweeracademie (2020), as cited in Van Rossum (2027), §6.1.

References

  • Australian Building Codes Board. (2020b). Handbook: Fire Safety Verification Method (Print version 1.1).
  • Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
  • Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
  • Purser, D. A. (2002). Toxicity assessment of combustion products. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-83–2-171). National Fire Protection Association; Society of Fire Protection Engineers.
  • Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
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