Dave Peterson

Much has been written about the political arena’s “fuzzy math.” While politicians debate the concept, in the emergency-response field, fuzzy math will simply not do. Can you imagine a pump operator setting the pump discharge pressure to a handline based on fuzzy math or vague numbers? Precision is required for proper nozzle pressures, flow rates, and for personnel safety. Fuzzy math will also not do for hazmat response. Precise computations concerning some hazmat concepts will enable response personnel to respond safely and to conduct better risk assessments.

One concept in which mathematics can be applied to enhance safety is called “vapor density.” This concept is defined as the relative weight of a gas or vapor compared to air, which has an arbitrary value of one. If a gas has a vapor density of less than one it will generally rise in air. If the vapor density is greater than one, the gas will generally sink in air.

This concept is important for responders because it will indicate where the gas or vapors can generally be expected to be located at hazmat releases. Accordingly, responders can better select their staging and equipment set-up areas, as well as the hazard isolation zones for the incident. Additionally, knowledge of where the gas may be found can better define the monitoring instrument strategy.

The only problem is the concept of vapor density, which has also been termed “specific gravity of vapor,” offers only a vague value of where the gas or vapor may travel. In other words, response guidebooks such as the Department of Transportation Emergency Response Guidebook (ERG) which state that “vapors may be heavier than air and found in low areas” do not indicate how heavy the vapors will be. To be more precise, then, vapor densities can be calculated.

Air is a complex mixture of several gases with nitrogen and oxygen being the most prevalent. The composition of air at sea level by weight is as follows:

Nitrogen		75.53%
Oxygen		23.16%
Others		1.31% (Argon, carbon dioxide, neon, helium, methane, krypton, nitrous oxide, hydrogen, xenon, ozone)

Additionally, air has a molecular weight of 29 atomic mass units (AMU’s) at sea level. In essence, this is the weight of a sample of air and which can be used for comparison purposes with other gases and vapors.

Given a material’s identity, its molecular weight can be calculated by its chemical formula and in conjunction with a periodic table of elements. All atoms have mass, and weight is the attraction of mass by gravity. For our purposes, we refer to the mass of a compound as its weight. Finding the chemical formula for a compound and adding the weight of all of its atoms can calculate molecular weight. For instance, the molecular weight of anhydrous ammonia is 17 because the formula is NH3 where one atom of nitrogen (N) is 14 AMU’s and three atoms of hydrogen (H) is 3 AMU’s.

To compute a compound’s vapor density, simply divide the molecular weight of the compound by the molecular weight of air. This will provide a numerical value that can be compared to air’s value of one.

For example, hydrogen’s molecular weight is 2 AMU’s (hydrogen gas is diatomic) and the molecular weight of air is 29 (28.9 to be exact). The quotient of 2/29 equals 0.068. Since this answer is below the value of one, hydrogen will rise in air (remember the Hindenburg?). Conversely, a product such as hexane will emit vapors that will sink in air. The computation of hexane’s weight is based on its chemical formula of C₆H₁₄ which provides a molecular weight of 84 AMU’s. The quotient of 84/29 is 2.9. Hexane vapors, then, are 2.9 times heavier than air.

To aid in remembering which gases are lighter than air, some mnemonics or acronyms have been devised. A New York City fire officer around the turn of the century developed a well-known mnemonic for vapor densities. To train his fellow firefighters, he used the term “HA HA MICE” to remember the lighter than air gases. The letters stand for

M — Methane
I — Illuminating Gases (old term for natural gas)
C — Carbon Monoxide
E — Ethylene

This acronym was useful for years, but today we now know there are more than eight gases that are lighter than (or the same weight as) air. To remember the 13 gases that are lighter than air, a new acronym may be used as a mnemonic. The term “4H MEDIC ANNA” identifies the lighter-than-air gases, and they are

GAS	MOLECULAR FORMULA	AMU	VAPOR DENSITY
H - Hydrogen	H₂	2	0.07
H - Helium	He	4	0.14
H - Hydrogen Cyanide	HCN	29	1.0
H - Hydrogen Fluoride	HF	10	0.34
M - Methane	CH₄	16	0.55
E - Ethylene	C₂H₄	28	0.96
D - Diborane	B₂H₂	27.7	0.96
I - Illuminating Gases	CH₄/C₂H₆	17.4	0.6
C - Carbon Monoxide	CO	28	0.96
A - Acetylene	C₂H₂	26	0.9
N - Neon	Ne	10	0.34
N - Nitrogen	N₂	28	0.96
A - Ammonia	NH₃	17	0.59

(Note: Illuminating gases is “natural gas” which is a mixture of approximately 90% methane and 10% ethane.)

If you can remember this mnemonic for the lighter-than-air gases, everything else is heavier, including the vapors from flammable liquids. In general terms, the heavier the vapor, the lower it will accumulate when released.

Vapor density is merely a general concept to have an idea where vapors may be found when released. However, this physical parameter is not absolute, and it can be affected by

One other precaution with the vapor-density concept is the presumption that the ratios found between a gas or vapor and air are absolute. In reality, this is hardly ever the case.

Vapor-density values are frequently misinterpreted in cases where substances are released at an ambient temperature that prevents them from existing as a pure gas or vapor at normal atmospheric pressures. An inaccurate conclusion may mislead responders on the actions of the released gas or vapor and could compromise responder safety.

Since many substances (liquids) have boiling points well above ambient temperatures, they will not evaporate or evolve into pure vapors when released. (Pure vapors being defined as 100 percent concentration above the spill). This point is important, because vapor densities are computed by using molecular weight ratios which assumes pure vapors. A more accurate method to determine the vapor density of a substance would be to compare the mixture of the substance’s vapor in air with that of pure air. This ratio will provide a more accurate determination of the substances vapor density. The procedure to do so is as follows:

Step 1: Compute the approximate density of the pure chemical at a specific temperature.
pV = 1.3691 x molecular weight of substance divided by the temperature (in degrees Farenheit) + 460

Step 2: Compute the approximate density of air at ambient temperature.
pA = 39.566 divided by the temperature ( in degrees Farenheit) + 460

Step 3: Compute the relative vapor density of the chemical-air mixture.
Relative vapor density = (C x pV) + [(100 – C) X pA] divided by 100 X pA

(C stands for the “saturated concentration” of the chemical vapor in air in percent by volume.)

* Multiplying 100 by the vapor pressure of a substance and dividing that product by 760 calculate saturated concentration.

A good example of this concept is with benzene. Benzene has a molecular weight, based on the formula of C₆H₆, of 78.1 AMU’s. The vapor density ratio becomes 2.69 when benzene’s molecular weight of 78.1 is divided by the molecular weight of air (29). This ratio would indicate that the vapors of benzene would accumulate near the surface of the spill and the terrain when released. In actuality, the vapors would be only slightly heavier than air by computing with the above formulas.

The vapor pressure of benzene is 100 mm/Hg at 79° F, so multiplying 100 mm/Hg by 100 and dividing that product by 760 can calculate the saturated concentration. The answer is 13.16%, which indicates the maximum concentration of benzene vapors above a release at 79° F. Use of these values in the equations above, with the assumption of an air temperature of 79° F, provides a more accurate vapor density value of 1.22. This value means that the benzene-air mixture directly above the spill of benzene at 79° F is only 1.22 times heavier than air and not the 2.7 ratio that is frequently reported as the vapor density for benzene.

These equations and concepts are used in vapor dispersion software applications to better predict where and how far vapor clouds will travel when released. A mixture with vapor density values close to those of air will quickly mix with air as it drifts away from the spill. It will not take long for this mixture to approach the density of air and behave as a neutrally buoyant vapor-air mixture. Negatively buoyant mixtures will behave as heavier-than-air mixtures for a large distance from a spill. Also, positively buoyant mixtures will behave as lighter-than-air mixtures.

Precision with information such as this will aid in our risk assessments and could alter our tactics. With an eye toward increased personnel safety, consider using these concepts and formulas at your future responses. As with all concepts and theories, there are exceptions; be sure to remain vigilant and be careful!

The next issue will discuss another hazmat math concept that can assist in determining product concentrations in containers.

Resource: Handbook of Chemical Hazard Analysis Procedures by Federal Emergency Management Agency (FEMA), U.S. Department of Transportation (DOT), U.S. Environmental Protection Agency (EPA)

Nitrogen		78.00%
Oxygen		20.95%
Argon		0.93%
CO₂		0.033%
Others		<0.003%