Parts-per notation

thumb|upright|[[Fluorescein aqueous solutions, diluted from to 1part per million in intervals of ten-fold dilution.

At the solution is a deep red colour. As the concentration decreases, the colour becomes greenish-orange, then a vibrant yellowish-green, with the final 1ppm sample a very pale yellowish-green.]]

In science and engineering, parts-per notation is a set of pseudo-units to describe the 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,
parts-per-billion – ppb,
parts-per-trillion – ppt,
parts-per-quadrillion – ppq,

This notation is not part of the International System of Units (the SI system) and its meaning is ambiguous. In chemistry, ambiguity arises because parts-per notation is able to be used to refer to a mole fraction and a mass fraction, which are unequal depending on the molar mass of the substance.

Applications

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 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."

Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter" 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").

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the expansion coefficient of some brass alloy, 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.

Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate is doing so at a rate that may be expressed in a variety of volumetric units, including 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.

==Parts-per expressions==

{| class="wikitable" style="font-size:85%;line-height:0.9;text-align:center;margin-left:1ex;float:right;"

!style="padding:0 1px;"|1 of → = ⭨ of ↓

!per cent (%)

!per mille (‰)

!per myriad (‱)

!per cent mille (pcm)

!per million (ppm)

!per billion (ppb)

|1||0.1||0.01||0.001||0.0001||

!‰

|10||1||0.1||0.01||0.001||

!‱

|100||10||1||0.1||0.01||

!pcm

|1,000||100||10||1||0.1||0.0001

!ppm

|10,000||1,000||100||10||1||0.001

!ppb

|||||||10,000||1,000||1

One part per hundred is generally represented by the percent sign (%) and denotes one part per 100 () parts, and a value of . This is equivalent to about fourteen 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 1,000 () parts, and a value of . This is equivalent to about ninety seconds 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 10,000 () parts, and a value of . 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).<!--NOTE TO EDITORS:

Unlike the % symbol, which constitutes a formatting exception to the rule of the SI, a space goes between the value and both the permille and permyriad symbols (‰ and ). Besides adhering to the rule of the SI, this is also good technical writing practice as it helps to distinguish these two less-common symbols from the percent symbol.

--> 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).

One part per hundred thousand, per cent mille (pcm) or milli-percent denotes one part per 100,000 () parts, and a value of . 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 1,000,000 () parts, and a value of . It is equivalent to about 32 seconds out of a year or 1 mm of error per km of distance traversed. In mining, it is also equivalent to one gram per metric ton, expressed as g/t.

One part per billion (ppb) denotes one part per 1,000,000,000 () parts, and a value of . This is equivalent to about three seconds out of a century.

One part per trillion (ppt) denotes one part per 1,000,000,000,000 () parts, and a value of . This is equivalent to about thirty seconds out of every million years.

One part per quadrillion (ppq) denotes one part per 1,000,000,000,000,000 () parts, and a value of . 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. <!-- EDITORS NOTE REGARDING TRIPS AROUND THE WORLD:

The analogy of paper thickness vs. 146k trips around the world may be so wildly counterintuitive that the statement might seemingly be in error, but please do not change the statement without first understanding the below math (shown here with excess precision):

Paper thickness = 0.0030 inch = 0.0000762 m (0.003 × 25.4 mm/inch ÷ 1,000 mm/m)

Earth's mean circumference = 40,041,470 m

In then follows that...

thumb|One part per [[trillion (short scale)|trillion (1 ppt) is a proportion equivalent to a drop of water diluted into 400 Olympic-size swimming pools.]]

In science and engineering, the parts-per notation is a set of pseudo-units to describe

146,386.7 trips × 40,041,470 m × 0.013E-15 = 0.0000762 m = 0.0030 inch = paper thickness

-->

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). According to IUPAP, "a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt". 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".

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. To distinguish the mass fraction from volume fraction or mole fraction, the letter m (for mass, but is ambiguous with mole fraction) or w (standing for weight) is sometimes added to the abbreviation (e.g. ppmw, ppbw).

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