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.
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." [lower-alpha 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 n m / 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"). [1]
Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the expansion coefficient of some brass alloy, α = 18.7 ppm/°C, may be expressed as 18.7 (μm/m)/°C, or as 18.7 (μ in/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of length. [lower-alpha 2] Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate Q p = 12 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 μL/L,125 μ gal / gal, 125 cm3/m3, etc.
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.
1 of → = ⭨ of ↓ | per cent (%) | per mille (‰) | 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 | 107 | 106 | 105 | 10,000 | 1,000 | 1 |
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). [1] 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. [1] [4] According to IUPAP, "a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt". [5] 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.
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". [6]
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 ]
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 units out, such as kg/kg, mol/mol or m3/m3, even though they are all dimensionless. [7] 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). [8] [9] However, 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). [10]
The usage of the parts-per notation is generally quite fixed within each specific branch of science, but often in a way that is inconsistent with its usage in other branches, leading some researchers to assume that their own usage (mass/mass, mol/mol, volume/volume, mass/volume, or others) is correct and that other usages are incorrect. This assumption sometimes leads them to not specify the details of their own 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, while chemists, the field of occupational safety and the field of permissible exposure limit (e.g. permitted gas exposure limit in air) may use mass/volume. Unfortunatelly, many academic publications of otherwise excellent level fail to specify their use of the parts-per notation, which irritates some readers, especially those who are not experts in the particular fields in those publications, because parts-per-notation, without specifying what it stands for, can mean anything.[ citation needed ]
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 marked with !.
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% [11] | 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.
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. [5] In 2004, a report to the International Committee for Weights and Measures (CIPM) stated that the response to the proposal of the uno "had been almost entirely negative", and the principal proponent "recommended dropping the idea". [12] To date, the uno has not been adopted by any standards organization.
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In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.
A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is the algebraic multiplication of a numerical value and a unit of measurement. For example, the physical quantity mass, symbol m, can be quantified as m=n kg, where n is the numerical value and kg is the unit symbol.
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.
The International System of Units, internationally known by the abbreviation SI, is the modern form of the metric system and the world's most widely used system of measurement. Established and maintained by the General Conference on Weights and Measures (CGPM), it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.
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The Avogadro constant, commonly denoted NA or L, is an SI defining constant with an exact value of 6.02214076×1023 reciprocal moles. It is used as a normalization factor in the amount of substance in a sample (in units of moles), defined as the number of constituent particles (usually molecules, atoms, or ions) divided by NA.
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A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement.
A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.