Ideal gas law - Revision history

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2026-11-02T17:12:26Z

Add Intermediate SVG figures with Fire Safety Academy watermark at planned locations; preserve article text

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	A worked example shows the size of the effect. Air at 20 °C, or 293 K, has a density of 353/293, which is about 1.20 kg/m³.<ref name="VR-3.2" /> In a hot gas layer at 600 °C, or 873 K, the density falls to 353/873, about 0.40 kg/m³. One kilogram of gas then fills about 2.47 m³ instead of 0.83 m³, so it has expanded by a factor of almost three. The volume of a gas at constant pressure is proportional to its absolute temperature.<ref name="VR-3.2" /> In a smoke and heat exhaust example in the book, smoke at 463 °C has a density of about 0.48 kg/m³.<ref name="VR-5.3" />		A worked example shows the size of the effect. Air at 20 °C, or 293 K, has a density of 353/293, which is about 1.20 kg/m³.<ref name="VR-3.2" /> In a hot gas layer at 600 °C, or 873 K, the density falls to 353/873, about 0.40 kg/m³. One kilogram of gas then fills about 2.47 m³ instead of 0.83 m³, so it has expanded by a factor of almost three. The volume of a gas at constant pressure is proportional to its absolute temperature.<ref name="VR-3.2" /> In a smoke and heat exhaust example in the book, smoke at 463 °C has a density of about 0.48 kg/m³.<ref name="VR-5.3" />

			[[File:fse-ideal-gas-law-v1.svg\|thumb\|upright=1.5\|alt=Falling curve of gas density against temperature from 0 to 1000 °C: 1.20 kg/m³ at 20 °C, 0.75 at 200 °C, 0.40 at 600 °C.\|Density of air and smoke at atmospheric pressure, ρ ≈ 353/T with T in kelvin. Based on Van Rossum (2027), §3.2.]]

	The difference between mass flow and volume flow follows directly from this. The mass flow through an opening equals the density times the volume flow.<ref name="VR-3.6">Van Rossum (2027), §3.6.</ref> Air flowing at 2 m/s through an opening of 10 m² gives a volume flow of 20 m³/s. At a density of 0.7 kg/m³ this is a mass flow of 14 kg/s.<ref name="VR-3.6" /> Because hot smoke has a lower density, the same mass of smoke takes up a larger volume. Supply and exhaust flows must therefore be compared with their temperatures and densities taken into account.<ref name="VR-3.6" />		The difference between mass flow and volume flow follows directly from this. The mass flow through an opening equals the density times the volume flow.<ref name="VR-3.6">Van Rossum (2027), §3.6.</ref> Air flowing at 2 m/s through an opening of 10 m² gives a volume flow of 20 m³/s. At a density of 0.7 kg/m³ this is a mass flow of 14 kg/s.<ref name="VR-3.6" /> Because hot smoke has a lower density, the same mass of smoke takes up a larger volume. Supply and exhaust flows must therefore be compared with their temperatures and densities taken into account.<ref name="VR-3.6" />
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	== Expansion and pressure in a fire compartment ==		== Expansion and pressure in a fire compartment ==
	What happens to a heated gas depends on whether it can expand. ''Introduction to Fire Safety Engineering'' distinguishes two cases. If the pressure stays the same, the volume increases and the density falls. If the amount of gas and the volume stay the same, the density stays the same, but the pressure rises.<ref name="VR-3.2" /> A real fire compartment lies between these two cases, depending on how leaky it is.		What happens to a heated gas depends on whether it can expand. ''Introduction to Fire Safety Engineering'' distinguishes two cases. If the pressure stays the same, the volume increases and the density falls. If the amount of gas and the volume stay the same, the density stays the same, but the pressure rises.<ref name="VR-3.2" /> A real fire compartment lies between these two cases, depending on how leaky it is.

			[[File:fse-ideal-gas-law-v2.svg\|thumb\|upright=1.5\|alt=Left: a cylinder with a free piston that rises when the gas is heated. Right: a closed box whose pressure gauge rises when the gas is heated.\|Heating a gas. Left: at constant pressure the volume grows and the density falls. Right: at constant volume the density stays the same and the pressure rises. Based on Van Rossum (2027), §3.2.]]

	In a compartment with only one opening, cool air flows in and hot smoke flows out. Klote shows that, neglecting the mass of the fuel, the ratio of the volume flows out and in equals the ratio of the absolute temperatures.<ref name="KL-4-277">Klote (2002), p. 4-277.</ref> For smoke at 700 °C the ratio is 3.32. If 1.5 m³/s of air flows into the compartment, about 4.98 m³/s of smoke flows out.<ref name="KL-4-277" /> With open doors or windows, the pressure difference caused by this expansion is negligible. In a tightly sealed compartment it may be important.<ref name="KL-4-277" />		In a compartment with only one opening, cool air flows in and hot smoke flows out. Klote shows that, neglecting the mass of the fuel, the ratio of the volume flows out and in equals the ratio of the absolute temperatures.<ref name="KL-4-277">Klote (2002), p. 4-277.</ref> For smoke at 700 °C the ratio is 3.32. If 1.5 m³/s of air flows into the compartment, about 4.98 m³/s of smoke flows out.<ref name="KL-4-277" /> With open doors or windows, the pressure difference caused by this expansion is negligible. In a tightly sealed compartment it may be important.<ref name="KL-4-277" />
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	For a fire compartment, the same type of equation gives the buoyancy pressure of the hot smoke. At a compartment temperature of 800 °C, the pressure difference 1.52 m above the [[neutral plane]] is about 13 Pa.<ref name="KL-4-276">Klote (2002), p. 4-276.</ref> In full-scale tests by Fang, the largest pressure difference across the wall of the burn room was 16 Pa at the ceiling. For a compartment at 700 °C and a height of 10.7 m above the neutral plane, the difference reaches 88 Pa.<ref name="KL-4-276" /> The neutral plane is the level where the pressures inside and outside are equal. Below it, air flows into the compartment, and above it, smoke flows out.<ref name="VR-3.2" />		For a fire compartment, the same type of equation gives the buoyancy pressure of the hot smoke. At a compartment temperature of 800 °C, the pressure difference 1.52 m above the [[neutral plane]] is about 13 Pa.<ref name="KL-4-276">Klote (2002), p. 4-276.</ref> In full-scale tests by Fang, the largest pressure difference across the wall of the burn room was 16 Pa at the ceiling. For a compartment at 700 °C and a height of 10.7 m above the neutral plane, the difference reaches 88 Pa.<ref name="KL-4-276" /> The neutral plane is the level where the pressures inside and outside are equal. Below it, air flows into the compartment, and above it, smoke flows out.<ref name="VR-3.2" />

			[[File:fse-ideal-gas-law-v3.svg\|thumb\|upright=1.5\|alt=Fire room with a door: smoke out above the neutral plane, air in below; a graph shows the pressure difference growing with height.\|Buoyancy in a fire room with one opening: the pressure difference between inside and outside grows with height above the neutral plane, where it is zero. Based on Van Rossum (2027), §3.2, and Klote (2002), p. 4-276.]]

	These pressure differences are small, a few tens of pascals, yet they drive most [[smoke movement]] in buildings. Klote notes that in hot smoke the buoyancy forces can be so great that smoke flows upwards even during reverse stack effect.<ref name="KL-4-276" /> Together with expansion, wind and ventilation systems, they are the driving forces behind smoke spread.<ref name="VR-6.6" /><ref name="KL-4-274" />		These pressure differences are small, a few tens of pascals, yet they drive most [[smoke movement]] in buildings. Klote notes that in hot smoke the buoyancy forces can be so great that smoke flows upwards even during reverse stack effect.<ref name="KL-4-276" /> Together with expansion, wind and ventilation systems, they are the driving forces behind smoke spread.<ref name="VR-6.6" /><ref name="KL-4-274" />

Daan: New FSE wiki article: Ideal gas law (written by Claude Opus 5.5)

2026-11-01T22:38:17Z

New FSE wiki article: Ideal gas law (written by Claude Opus 5.5)

New page

The '''ideal gas law''' is the relation between the pressure, the temperature, the volume and the amount of a gas. It states that, for a given gas, the pressure divided by the density is proportional to the absolute temperature.<ref name="KA-1-2">Kandola (2002), p. 1-2.</ref><ref name="VR-3.2">Van Rossum (2027), §3.2.</ref> A gas that follows this law exactly is called a perfect gas or ideal gas. Real gases follow it closely at temperatures and pressures well away from liquefaction or dissociation.<ref name="KA-1-2" /> For air at atmospheric pressure the error is no more than a fraction of a percent.<ref name="VR-3.2" />

In fire safety engineering the ideal gas law explains why hot smoke rises and spreads. When air and combustion gases are heated at constant pressure they expand, so the same mass occupies more space and the density falls.<ref name="VR-3.2" /> The lighter smoke is pushed upwards by the colder, heavier air around it. This [[buoyancy]] drives fire plumes, the [[stack effect]] and natural smoke venting.<ref name="VR-3.2" /> If a heated gas cannot expand, its pressure rises instead.<ref name="VR-3.2" />

At atmospheric pressure the law reduces to a simple rule: the density of air or smoke in kg/m³ is about 353 divided by the absolute temperature in kelvin.<ref name="VR-3.2" /> Air at 20 °C therefore has a density of about 1.2 kg/m³, and smoke at 200 °C about 0.75 kg/m³.<ref name="VR-3.2" /> Zone models such as CFAST and computational fluid dynamics models such as the Fire Dynamics Simulator use the ideal gas law as the equation of state that links temperature, density and pressure.<ref name="CF-7">Peacock et al. (2026), p. 7.</ref><ref name="FDS-10">McGrattan et al. (2026), p. 10.</ref>

== Statement of the law ==
For a gas of uniform properties and fixed composition, experiments show that the density depends only on the pressure and the temperature. If two of these three quantities are known, the third follows.<ref name="KA-1-2" /> The ideal gas law expresses this dependence in its simplest form. ''Introduction to Fire Safety Engineering'' describes it as the relation between temperature, volume, pressure and the amount of gas.<ref name="VR-3.2" /> In terms of density, the law reads:<ref name="KA-1-2" /><ref name="CF-8">Peacock et al. (2026), p. 8.</ref>

<math display="block">p = \rho \, R \, T</math>

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>p</math> || absolute pressure of the gas || Pa
|-
| <math>\rho</math> || density of the gas || kg/m³
|-
| <math>R</math> || specific gas constant: the universal gas constant divided by the molar mass of the gas; for air 287 || J/(kg·K)
|-
| <math>T</math> || absolute temperature || K
|}

The specific gas constant depends on the gas. It equals the universal gas constant divided by the molar mass of the gas or gas mixture.<ref name="CF-7" /> NUREG-1805 gives a value of 287.045 J/(kg·K) for air and 461.52 J/(kg·K) for water vapour.<ref name="NU-I-17">Iqbal and Salley (2004), p. I-17.</ref> The zone model CFAST uses a constant value of about 289 J/(kg·K), because it assumes that the gas mixture in a fire has the molar mass of air.<ref name="CF-7" /> Air is in reality a mixture of gases, but the law is accepted for air flows with an acceptable degree of accuracy. It is also assumed to hold for a gas in motion as well as at rest.<ref name="KA-1-2" />

Because the gas constant contains the molar mass, gases at the same pressure and temperature have densities in proportion to their molar masses. Kandola lists molar masses of 2.0 for hydrogen, 16.0 for methane, 28.0 for carbon monoxide, 44.0 for propane and 28.9 for air.<ref name="KA-1-2" /> Propane gas is therefore about 1.5 times as dense as air at the same temperature, while methane has little more than half the density of air. For a perfect gas, the coefficient of thermal expansion at constant pressure equals one divided by the absolute temperature.<ref name="KA-1-2" /> At 20 °C, a gas thus expands by about 1/293 of its volume for every kelvin of heating. CFAST also uses the link between the gas constant and the [[specific heat capacity]] of the gas: the gas constant equals the difference between the specific heats at constant pressure and at constant volume.<ref name="CF-7" />

The temperature in the law is always the absolute temperature in kelvin. ''Introduction to Fire Safety Engineering'' gives the conversion as the temperature in degrees Celsius plus 273.15.<ref name="VR-3.2" /> For a temperature difference it does not matter whether degrees Celsius or kelvin are used, because a rise of 100 °C equals a rise of 100 K. For the density of smoke at 200 °C, however, 473 K must be entered, not 200.<ref name="VR-3.2" />

== Density and temperature ==
The most common use of the law in fire safety engineering is to find the density of hot air or smoke. At a pressure of about one atmosphere, as at sea level, the law gives the density of air and smoke as approximately 353 divided by the absolute temperature.<ref name="VR-3.2" /> The book derives this rule from the ideal gas law and uses it throughout, for instance for smoke and heat exhaust and in zone models.<ref name="VR-5.3">Van Rossum (2027), §5.3.</ref><ref name="VR-9.3a">Van Rossum (2027), §9.3 (How a zone model works).</ref>

A worked example shows the size of the effect. Air at 20 °C, or 293 K, has a density of 353/293, which is about 1.20 kg/m³.<ref name="VR-3.2" /> In a hot gas layer at 600 °C, or 873 K, the density falls to 353/873, about 0.40 kg/m³. One kilogram of gas then fills about 2.47 m³ instead of 0.83 m³, so it has expanded by a factor of almost three. The volume of a gas at constant pressure is proportional to its absolute temperature.<ref name="VR-3.2" /> In a smoke and heat exhaust example in the book, smoke at 463 °C has a density of about 0.48 kg/m³.<ref name="VR-5.3" />

The difference between mass flow and volume flow follows directly from this. The mass flow through an opening equals the density times the volume flow.<ref name="VR-3.6">Van Rossum (2027), §3.6.</ref> Air flowing at 2 m/s through an opening of 10 m² gives a volume flow of 20 m³/s. At a density of 0.7 kg/m³ this is a mass flow of 14 kg/s.<ref name="VR-3.6" /> Because hot smoke has a lower density, the same mass of smoke takes up a larger volume. Supply and exhaust flows must therefore be compared with their temperatures and densities taken into account.<ref name="VR-3.6" />

In a two-zone model, the [[hot gas layer]] is described by its mass and its temperature. The mass follows from the plume model, and the density follows from the temperature with the same rule. The volume of the layer, and with it the layer thickness, is then the mass divided by the density.<ref name="VR-9.3a" />

== Expansion and pressure in a fire compartment ==
What happens to a heated gas depends on whether it can expand. ''Introduction to Fire Safety Engineering'' distinguishes two cases. If the pressure stays the same, the volume increases and the density falls. If the amount of gas and the volume stay the same, the density stays the same, but the pressure rises.<ref name="VR-3.2" /> A real fire compartment lies between these two cases, depending on how leaky it is.

In a compartment with only one opening, cool air flows in and hot smoke flows out. Klote shows that, neglecting the mass of the fuel, the ratio of the volume flows out and in equals the ratio of the absolute temperatures.<ref name="KL-4-277">Klote (2002), p. 4-277.</ref> For smoke at 700 °C the ratio is 3.32. If 1.5 m³/s of air flows into the compartment, about 4.98 m³/s of smoke flows out.<ref name="KL-4-277" /> With open doors or windows, the pressure difference caused by this expansion is negligible. In a tightly sealed compartment it may be important.<ref name="KL-4-277" />

In a closed volume, the ideal gas law requires the pressure to rise with the temperature. In building fires, gas leaks through cracks around windows and doors usually keep this [[fire-induced pressure]] and its rate of rise very small. Some enclosures, such as certain compartments on ships, can be well sealed.<ref name="NU-14-3">Iqbal and Salley (2004), p. 14-3.</ref> ''Introduction to Fire Safety Engineering'' notes that expansion is strongest around the moment of flashover, when smoke is forced through seams, cracks and holes.<ref name="VR-6.6">Van Rossum (2027), §6.6.</ref> For a confined gas explosion, NUREG-1805 estimates the maximum pressure as the initial pressure multiplied by the ratio of the adiabatic flame temperature to the initial temperature. Both temperatures must be absolute.<ref name="NU-15-14">Iqbal and Salley (2004), p. 15-14.</ref>

== Buoyancy, stack effect and the neutral plane ==
Because hot gas is less dense than cold air, a column of hot smoke weighs less than a column of cold air of the same height. The colder air pushes the smoke upwards.<ref name="VR-3.2" /> In a gas at rest, the pressure difference over a height equals approximately the density times the acceleration of gravity times the height. Between a column of cold air and a column of hot smoke, the pressure difference is therefore the acceleration of gravity times the height times the difference in density.<ref name="VR-3.2" /> The larger the density difference and the height, the larger the pressure difference.

The same principle drives the stack effect in buildings. When it is cold outside, the warmer, less dense air inside rises in stairwells, lift shafts and other shafts.<ref name="KL-4-274">Klote (2002), p. 4-274.</ref> Klote gives an example of a building 60 m tall with the neutral plane at mid-height, an outside temperature of −18 °C and an inside temperature of 21 °C. The maximum pressure difference due to stack effect is then 55 Pa.<ref name="KL-4-275">Klote (2002), p. 4-275.</ref> This normal stack effect is stronger for low outside temperatures and tall shafts. When the outside air is warmer than the inside air, the flow in shafts reverses.<ref name="KL-4-274" />

For a fire compartment, the same type of equation gives the buoyancy pressure of the hot smoke. At a compartment temperature of 800 °C, the pressure difference 1.52 m above the [[neutral plane]] is about 13 Pa.<ref name="KL-4-276">Klote (2002), p. 4-276.</ref> In full-scale tests by Fang, the largest pressure difference across the wall of the burn room was 16 Pa at the ceiling. For a compartment at 700 °C and a height of 10.7 m above the neutral plane, the difference reaches 88 Pa.<ref name="KL-4-276" /> The neutral plane is the level where the pressures inside and outside are equal. Below it, air flows into the compartment, and above it, smoke flows out.<ref name="VR-3.2" />

These pressure differences are small, a few tens of pascals, yet they drive most [[smoke movement]] in buildings. Klote notes that in hot smoke the buoyancy forces can be so great that smoke flows upwards even during reverse stack effect.<ref name="KL-4-276" /> Together with expansion, wind and ventilation systems, they are the driving forces behind smoke spread.<ref name="VR-6.6" /><ref name="KL-4-274" />

== Use in fire models ==
Zone models are built on conservation of mass and energy together with the ideal gas law. CFAST derives its equations from these principles, using the ideal gas law as the equation of state that relates temperature and density.<ref name="CF-7" /> The model applies the law to each gas layer, linking the compartment pressure, the layer volume, the layer mass and the layer temperature.<ref name="CF-8" /> The zone model MAGIC also calculates the gas densities with the ideal gas law.<ref name="JO-3-1">Joglar et al. (2007), p. 3-1.</ref>

The Fire Dynamics Simulator solves the conservation equations for an expandable mixture of ideal gases in the low Mach number limit.<ref name="MG-3-2">McGrattan (2007), p. 3-2.</ref> In this approach, the pressure is split into a background pressure and a small perturbation. Only the background pressure is kept in the equation of state.<ref name="FDS-10" /> The temperature is then found from the density through the equation of state.<ref name="MG-3-2" /> Both types of model therefore rest on the same simple relation between pressure, density and temperature.

The law is also used in fire measurements. In the experiments for the World Trade Center investigation, the molar flow of smoke-laden gas drawn through a sampling system was calculated from the measured pressure, temperature and volume flow with the ideal gas equation.<ref name="HA-72">Hamins, Maranghides, McGrattan, Johnsson, et al. (2005), p. 72.</ref> The uncertainties in these measured values were 0.3 kPa, 1 °C and 1.1 × 10<sup>−3</sup> L/s, respectively.<ref name="HA-72" /> Hand calculations of smoke and heat exhaust, smoke layer filling and flows through openings use the ideal gas law in the same way, through the density of the hot gas.<ref name="VR-3.2" /><ref name="VR-3.6" />

== Limits and common misconceptions ==
The ideal gas law is a theoretical assumption. It is a good approximation for many common gases, although some limitations exist.<ref name="VR-3.2" /> Ideal gases do not exist, but the law is useful at sufficiently low densities.<ref name="VR-3.2" /> The simple rule of 353 divided by the temperature also assumes a pressure of about one atmosphere.<ref name="VR-3.2" /> The book's pressure formulas are simplifications as well. In practice the shape and position of openings, flow resistance, wind and mechanical ventilation also influence pressure differences and flows.<ref name="VR-3.2" />

A frequent error is to enter the temperature in degrees Celsius. Using 200 instead of 473 for smoke at 200 °C would give a density of about 1.8 kg/m³, higher than that of cold air, which is clearly wrong.<ref name="VR-3.2" /> Absolute temperatures are also required in the expansion ratio and in the pressure rise of a confined explosion.<ref name="KL-4-277" /><ref name="NU-15-14" />

A second misconception is that hot smoke rises because it is at a higher pressure. In an open compartment the pressure hardly changes; the gas expands and its density falls.<ref name="VR-3.2" /><ref name="KL-4-277" /> The flow is driven by the small pressure differences that arise from the density difference over a height. A third misconception is that a fire always produces a large overpressure. In ordinary buildings, leakage keeps the pressure rise from expansion small. Only in well-sealed enclosures does the full pressure rise predicted by the ideal gas law become relevant.<ref name="NU-14-3" />

== See also ==
* [[Vent flow]]
* [[Convection]]
* [[Heat transfer]]

== Notes ==
<references />

== References ==
* Hamins, A., Maranghides, A., McGrattan, K. B., Johnsson, E. L., Ohlemiller, T. J., Donnelly, M. K., Yang, J. C., Mulholland, G. W., Prasad, K. R., Kukuck, S. R., Anleitner, R. L., & McAllister, T. P. (2005). ''Experiments and modeling of structural steel elements exposed to fire'' (NIST NCSTAR 1-5B). National Institute of Standards and Technology.
* 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.
* Joglar, F., Gautier, B., Gay, L., & Texeraud, J. (2007). ''Verification and validation of selected fire models for nuclear power plant applications: Vol. 6. MAGIC'' (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
* Kandola, B. S. (2002). Introduction to mechanics of fluids. 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. 1-1–1-26). National Fire Protection Association; Society of Fire Protection Engineers.
* Klote, J. H. (2002). Smoke control. 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. 4-274–4-291). National Fire Protection Association; Society of Fire Protection Engineers.
* McGrattan, K. (2007). ''Verification and validation of selected fire models for nuclear power plant applications: Vol. 7. Fire Dynamics Simulator'' (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
* McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2026). ''Fire Dynamics Simulator technical reference guide: Vol. 1. Mathematical model'' (6th ed., NIST Special Publication 1018-1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.1018
* Peacock, R. D., McGrattan, K. B., Forney, G. P., & Reneke, P. A. (2026). ''CFAST – Consolidated Fire and Smoke Transport: Vol. 1. Technical reference guide'' (Version 7, NIST Technical Note 1889v1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.TN.1889v1
* Van Rossum, E. (2027). ''Introduction to Fire Safety Engineering''. Fire Safety Academy.

[[Category:Physics & Heat Transfer]]