Parts-per notation
In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are parts-per-million (ppm, 10^{−6}), parts-per-billion (ppb, 10^{−9}), parts-per-trillion (ppt, 10^{−12}) and parts-per-quadrillion (ppq, 10^{−15}). This notation is not part of the International System of Units (SI) system and its meaning is ambiguous.
Overview
Parts-per notation is often used describing dilute solutions in chemistry, for instance, the relative abundance of dissolved minerals or pollutants in water. The quantity "1 ppm" can be used for a mass fraction if a water-borne pollutant is present at one-millionth of a gram per gram of sample solution. When working with aqueous solutions, it is common to assume that the density of water is 1.00 g/mL. Therefore, it is common to equate 1 kilogram of water with 1 L of water. Consequently, 1 ppm corresponds to 1 mg/L and 1 ppb corresponds to 1 μg/L.
Similarly, parts-per notation is used also in physics and engineering to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2 micrometers per meter of length for every degree Celsius and this would be expressed as "α = 1.2 ppm/°C". Parts-per notation is also employed to denote the change, stability, or uncertainty in measurements. For instance, the accuracy of land-survey distance measurements when using a laser rangefinder might be 1 millimeter per kilometer of distance; this could be expressed as "Accuracy = 1 ppm."^{[1]}
Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter" (2 nm/m = 2 nano = 2 × 10^{−9} = 2 ppb = 2 × 0.000000001), so the quotients are pure-number coefficients with positive values less than or equal to 1. When parts-per notations, including the percent symbol (%), are used in regular prose (as opposed to mathematical expressions), they are still pure-number dimensionless quantities. However, they generally take the literal "parts per" meaning of a comparative ratio (e.g. "2 ppb" would generally be interpreted as "two parts in a billion parts").^{[2]}
Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the coefficient of thermal expansion of a certain brass alloy, α = 18.7 ppm/°C, may be expressed as 18.7 (µm/m)/°C, or as 18.7 (µin/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of length.^{[3]} Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate Q_{p} = 125 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 µL/L, 125 µgal/gal, 125 cm^{3}/m^{3}, etc.
In nuclear magnetic resonance (NMR) spectroscopy
In nuclear magnetic resonance spectroscopy (NMR), chemical shift is usually expressed in ppm. It represents the difference of a measured frequency in parts per million from the reference frequency. The reference frequency depends on the instrument's magnetic field and the element being measured. It is usually expressed in MHz. Typical chemical shifts are rarely more than a few hundred Hz from the reference frequency, so chemical shifts are conveniently expressed in ppm (Hz/MHz). Parts-per notation gives a dimensionless quantity that does not depend on the instrument's field strength.
Parts-per expressions
1 of → = ⭨ of ↓ |
per cent (%) |
per 1 000 (‰) |
per 10 000 (‱) |
per 100 000 (pcm) |
per million (ppm) |
per billion (ppb) |
---|---|---|---|---|---|---|
% | 1 | 0.1 | 0.01 | 0.001 | 0.0001 | 10^{−7} |
‰ | 10 | 1 | 0.1 | 0.01 | 0.001 | 10^{−6} |
‱ | 100 | 10 | 1 | 0.1 | 0.01 | 10^{−5} |
pcm | 1 000 | 100 | 10 | 1 | 0.1 | 0.0001 |
ppm | 10 000 | 1 000 | 100 | 10 | 1 | 0.001 |
ppb | 10^{7} | 10^{6} | 10^{5} | 10 000 | 1 000 | 1 |
- One part per hundred is generally represented by the percent sign (%) and denotes one part per 100 (10^{2}) parts, and a value of 10^{−2}. This is equivalent to about fifteen minutes out of one day.
- One part per thousand should generally be spelled out in full and not as "ppt" (which is usually understood to represent "parts per trillion"). It may also be denoted by the permille sign (‰). Note however, that specific disciplines such as oceanography, as well as educational exercises, do use the "ppt" abbreviation. "One part per thousand" denotes one part per 1000 (10^{3}) parts, and a value of 10^{−3}. This is equivalent to about one and a half minutes out of one day.
- One part per ten thousand is denoted by the permyriad sign (‱). Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10000 (10^{4}) parts, and a value of 10^{−4}. This is equivalent to about nine seconds out of one day.
In contrast, in finance, the basis point is typically used to denote changes in or differences between percentage interest rates (although it can also be used in other cases where it is desirable to express quantities in hundredths of a percent). For instance, a change in an interest rate from 5.15% per annum to 5.35% per annum could be denoted as a change of 20 basis points (per annum). As with interest rates, the words "per annum" (or "per year") are often omitted. In that case, the basis point is a quantity with a dimension of (time^{−1}).^{[4]} - One part per hundred thousand, per cent mille (pcm) or milli-percent denotes one part per 100000 (10^{5}) parts, and a value of 10^{−5}. It is commonly used in epidemiology for mortality, crime and disease prevalence rates, and nuclear reactor engineering as a unit of reactivity. In time measurement it is equivalent to about 5 minutes out of a year; in distance measurement, it is equivalent to 1 cm of error per km of distance traversed.
- One part per million (ppm) denotes one part per 1000000 (10^{6}) parts, and a value of 10^{−6}. It is equivalent to about 32 seconds out of a year or 1 mm of error per km of distance traversed.
- One part per billion (ppb) denotes one part per 1000000000 (10^{9}) parts, and a value of 10^{−9}. This is equivalent to about three seconds out of a century.
- One part per trillion (ppt) denotes one part per 1000000000000 (10^{12}) parts, and a value of 10^{−12}. This is equivalent to about thirty seconds out of every million years.
- One part per quadrillion (ppq) denotes one part per 1000000000000000 (10^{15}) parts, and a value of 10^{−15}. This is equivalent to about two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed.^{[5]}
Criticism
Although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI).^{[2]} Note that although "percent" (%) is not formally part of the SI, both the BIPM and the International Organization for Standardization (ISO) take the position that "in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01" for dimensionless quantities.^{[2]}^{[6]} According to IUPAP, "a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt".^{[7]} Although SI-compliant expressions should be used as an alternative, the parts-per notation remains nevertheless widely used in technical disciplines. The main problems with the parts-per notation are set out below.
Long and short scales
Because the named numbers starting with a "billion" have different values in different countries, the BIPM suggests avoiding the use of "ppb" and "ppt" to prevent misunderstanding. The U.S. National Institute of Standards and Technology (NIST) takes the stringent position, stating that "the language-dependent terms [...] are not acceptable for use with the SI to express the values of quantities".^{[8]}
Thousand vs. trillion
Although "ppt" usually means "parts per trillion", it occasionally means "parts per thousand". Unless the meaning of "ppt" is defined explicitly, it has to be determined from the context.^{[citation needed]}
Mass fraction vs. mole fraction vs. volume fraction
Another problem of the parts-per notation is that it may refer to mass fraction, mole fraction or volume fraction. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, mol/mol or m^{3}/m^{3} (even though they are all dimensionless).^{[9]} The difference is quite significant when dealing with gases, and it is very important to specify which quantity is being used. For example, the conversion factor between a mass fraction of 1 ppb and a mole fraction of 1 ppb is about 4.7 for the greenhouse gas CFC-11 in air. For volume fraction, the suffix "V" or "v" is sometimes appended to the parts-per notation (e.g. ppmV, ppbv, pptv).^{[10]}^{[11]} Unfortunately, ppbv and pptv are also often used for mole fractions (which is identical to volume fraction only for ideal gases).
To distinguish the mass fraction from volume fraction or mole fraction, the letter "w" (standing for "weight") is sometimes added to the abbreviation (e.g. ppmw, ppbw).^{[12]}
The usage of the parts-per notation is generally quite fixed inside most specific branches of science, leading some researchers to draw the conclusion that their own usage (mass/mass, mol/mol, volume/volume, or others) is the only correct one. This, in turn, leads them to not specify their usage in their publications, and others may therefore misinterpret their results. For example, electrochemists often use volume/volume, while chemical engineers may use mass/mass as well as volume/volume. Many academic publications of otherwise excellent level fail to specify their usage of the parts-per notation.
SI-compliant expressions
SI-compliant units that can be used as alternatives are shown in the chart below. Expressions that the BIPM explicitly does not recognize as being suitable for denoting dimensionless quantities with the SI are shown in red text.
Measure | SI units |
Named parts-per ratio (short scale) |
Parts-per abbreviation or symbol |
Value in scientific notation |
---|---|---|---|---|
A strain of... | 2 cm/m | 2 parts per hundred | 2%^{[13]} | 2 × 10^{−2} |
A sensitivity of... | 2 mV/V | 2 parts per thousand | 2 ‰ | 2 × 10^{−3} |
A sensitivity of... | 0.2 mV/V | 2 parts per ten thousand | 2 ‱ | 2 × 10^{−4} |
A sensitivity of... | 2 µV/V | 2 parts per million | 2 ppm | 2 × 10^{−6} |
A sensitivity of... | 2 nV/V | 2 parts per billion | 2 ppb | 2 × 10^{−9} |
A sensitivity of... | 2 pV/V | 2 parts per trillion | 2 ppt | 2 × 10^{−12} |
A mass fraction of... | 2 mg/kg | 2 parts per million | 2 ppm | 2 × 10^{−6} |
A mass fraction of... | 2 µg/kg | 2 parts per billion | 2 ppb | 2 × 10^{−9} |
A mass fraction of... | 2 ng/kg | 2 parts per trillion | 2 ppt | 2 × 10^{−12} |
A mass fraction of... | 2 pg/kg | 2 parts per quadrillion | 2 ppq | 2 × 10^{−15} |
A volume fraction of... | 5.2 µL/L | 5.2 parts per million | 5.2 ppm | 5.2 × 10^{−6} |
A mole fraction of... | 5.24 µmol/mol | 5.24 parts per million | 5.24 ppm | 5.24 × 10^{−6} |
A mole fraction of... | 5.24 nmol/mol | 5.24 parts per billion | 5.24 ppb | 5.24 × 10^{−9} |
A mole fraction of... | 5.24 pmol/mol | 5.24 parts per trillion | 5.24 ppt | 5.24 × 10^{−12} |
A stability of... | 1 (µA/A)/min | 1 part per million per minute | 1 ppm/min | 1 × 10^{−6}/min |
A change of... | 5 nΩ/Ω | 5 parts per billion | 5 ppb | 5 × 10^{−9} |
An uncertainty of... | 9 µg/kg | 9 parts per billion | 9 ppb | 9 × 10^{−9} |
A shift of... | 1 nm/m | 1 part per billion | 1 ppb | 1 × 10^{−9} |
A strain of... | 1 µm/m | 1 part per million | 1 ppm | 1 × 10^{−6} |
A temperature coefficient of... | 0.3 (µHz/Hz)/°C | 0.3 part per million per °C | 0.3 ppm/°C | 0.3 × 10^{−6}/°C |
A frequency change of... | 0.35 × 10^{−9} ƒ | 0.35 part per billion | 0.35 ppb | 0.35 × 10^{−9} |
Note that the notations in the "SI units" column above are all dimensionless quantities; that is, the units of measurement factor out in expressions like "1 nm/m" (1 nm/m = 1 nano = 1 × 10^{−9}) so the quotients are pure-number coefficients with values less than 1.
Uno
Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, the International Union of Pure and Applied Physics (IUPAP) in 1999 proposed the adoption of the special name "uno" (symbol: U) to represent the number 1 in dimensionless quantities.^{[7]} This symbol is not to be confused with the always-italicized symbol for the variable "uncertainty" (symbol: U). This unit name "uno" and its symbol could be used in combination with the SI prefixes to express the values of dimensionless quantities that are much less—or even greater—than one.^{[14]}
Common parts-per notations in terms of the uno are given in the table below.
Coefficient | Name | Parts-per example |
Uno equivalent |
Symbol form |
Value of quantity |
---|---|---|---|---|---|
10^{−2} | Percent | 2% | 2 centiuno | 2 cU | 2 × 10^{−2} |
10^{−3} | Permille | 2 ‰ | 2 milliuno | 2 mU | 2 × 10^{−3} |
10^{−4} | Per ten thousand | 2 ‱ | 0.2 milliuno | 0.2 mU | 2 × 10^{−4} |
10^{−6} | Parts per million | 2 ppm | 2 microuno | 2 µU | 2 × 10^{−6} |
10^{−9} | Parts per billion | 2 ppb | 2 nanouno | 2 nU | 2 × 10^{−9} |
10^{−12} | Parts per trillion | 2 ppt | 2 picouno | 2 pU | 2 × 10^{−12} |
10^{−15} | Parts per quadrillion | 2 ppq | 2 femtouno | 2 fU | 2 × 10^{−15} |
In 2004, a report to the International Committee for Weights and Measures (CIPM) stated that response to the proposal of the uno "had been almost entirely negative", and the principal proponent "recommended dropping the idea".^{[15]} To date, the uno has not been adopted by any standards organization, and it appears unlikely that it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities. The proposal was instructive, however, as to the perceived shortcomings of the current options for denoting dimensionless quantities.
See also
- International Electrotechnical Commission (IEC)
- Milligram per cent
- Percentage (%) 1 part in 100
- Per mille (‰) 1 part in 1,000
- Permyriad (‱) 1 part in 10,000
- Per cent mille (pcm) 1 part in 100,000
- Per-unit system
References
- ^ This is a simplified explanation. Laser rangefinders typically have a measurement granularity of one to ten millimeters; thus, the complete specification for distance measurement accuracy might read as follows: Accuracy: ±(1 mm + 1 ppm). Consequently, a distance measurement of only a few meters would still have an accuracy of ±1 mm in this example.
- ^ ^{a} ^{b} ^{c} BIPM: 5.3.7 Stating values of dimensionless quantities, or quantities of dimension one].
- ^ In the particular case of coefficient of thermal expansion, the change to inches (one of the U.S. customary units) is typically also accompanied by a change to degrees Fahrenheit. Since a Fahrenheit-sized interval of temperature is only 5/9 that of a Celsius-sized interval, the value is typically expressed as 10.4 (µin/in)/°F rather than 18.7 (µin/in)/°C.
- ^ "What are Basis Points (BPS)?".
- ^ Measurements of dioxin are routinely made at the sub-ppq level. The U.S. Environmental Protection Agency (EPA) currently sets a hard limit of 30 ppq for dioxin in drinking water but once recommended a voluntary limit of 0.013 ppq. Also, radioactive contaminants in drinking water, which are quantified by measuring their radiation, are often reported in terms of ppq; 0.013 ppq is equivalent to the thickness of a sheet of paper versus a journey of 146000 trips around the world._{ }
- ^ Quantities and units. Part 0: General principles, ISO 31-0:1992.
- ^ ^{a} ^{b} Petley, Brian W. (September 1998). "Report on Recent Committee Activities on Behalf of IUPAP to the 1999 IUPAP General Assembly". Archived from the original on 2017-08-15. Retrieved 2017-08-15.
- ^ NIST: Rules and Style Conventions for Expressing Values of Quantities: 7.10.3 ppm, ppb, and ppt.
- ^ Schwartz and Warneck (1995). "Units for use in atmospheric chemistry" (PDF). Pure Appl. Chem. 67: 1377–1406. doi:10.1351/pac199567081377. Retrieved March 9, 2011.
- ^ "EPA On-line Tools for Site Assessment Calculation: Indoor Air Unit Conversion". Environmental Protection Agency.
- ^ Milton R. Beychok (2005). "Air Dispersion Modeling Conversions and Formulas". Fundamentals of Stack Gas Dispersion (4th ed.). Milton R. Beychok. ISBN 0964458802.
- ^ "Introduction to Green Engineering".
- ^ Compliance with the SI regarding the percent symbol (%) is limited in this chart. According to the BIPM’s SI brochure: Subsection 5.3.3, Formatting the value of a quantity, a space is always used to separate the unit symbol from the numeric value. Notable exceptions are the unit symbols for degree, minute, and second for plane angle, °, ′, and ″ (e.g. a latitude of 47° 38′ 8.8″). However, according to 5.3.7 Stating values of dimensionless quantities, or quantities of dimension one, the exception does not apply to the "%" symbol; it states as follows: "When it [the percent symbol] is used, a space separates the number and the symbol %." This practice has not been well adopted with regard to the % symbol, is contrary to Wikipedia’s Manual of Style, and is not observed here._{ }
- ^ Certain mathematical functions can produce proportional quantities with values greater than 1._{ }
- ^ Consultative Committee for Units (13–14 May 2004). "Report of the 16th meeting (13–14 May 2004) to the International Committee for Weights and Measures, of the International Bureau of Weights and Measures" (PDF). Archived from the original (PDF) on 2014-03-10.