# Clarke number

Clarke number or clarke is the relative abundance of a chemical element, typically in Earth's crust. The technical definition of "Earth's crust" varies among authors, and the actual numbers also vary significantly.

## History

In the 1930s, USSR geochemist Alexander Fersman defined the relative abundance of chemical elements in geological objects, denoted in percents, as Russian: кларки, lit. 'the clarkes'.$^{:141}$ This was in honor to the American geochemist Frank Wigglesworth Clarke, who pioneered in estimating the chemical composition of Earth's crust, based on Clarke and colleague's extensive chemical analysis of numerous rock samples, throughout 1889 to 1924.
Examples based on Fersman's definition:
* Russian: весовой кларк, lit. 'weight clarke': When the whole mass of a planet X is $Mx$ [kg], and the mass of oxygen there is $Mo$ [kg], then the weight clarke of oxygen in planet X is $Km={\frac {Mo}{Mx}}$ (dimensionless)
* Russian: кларк числа атомов, lit. 'clarke of atom count': When the whole count of atoms in a rock Y is $Ay$ [mol], and the atom count of silicon there is $As$ [mol], then silicon's clarke of atom count in rock Y is $Ka={\frac {As}{Ay}}$ (dimensionless)
* Fersman's "clarke of Earth's crust" is the Earth's surface including 16 km-thick lithosphere, hydrosphere and atmosphere.$^{:141}$

### In Russian

Russian: кларки is synonymous to "the relative abundance of elements" in any object, either in weight ratio or in atomic (number of atoms) ratio, regardless of how "Earth's crust" is defined, and denotation is not restricted to percents.

### In English

In the English speaking world, the term "clarke" was not even used in Wells(1937)$^{:4}$ which introduced Fersman's proposal, nor in later USGS articles such as Fleischer(1953). They used the term "relative abundance of the elements". Brian Mason also mentioned the term "clarke" in Mason(1952)$^{:42}$(mistakenly attributing it to Vladimir Vernadsky, later corrected to Fersman in Mason(1958)$^{:47}$), but the definition slightly differed from Fersman's, limiting it only to the average percentage in Earth's crust, but allowed to exclude hydrosphere and atmosphere. Besides for explaining the term, Mason himself did not use the term "clarke".
A variant term "clarke value" is occasionally used (examples:$^{:778}$). However, "clarke value" can have a different meaning, the clarke of concentration (example:$^{:412}$).
Terms "clarke number" and "Clarke number" are found in articles written by Japanese authors (example:$^{:55}$).

### Usage in Japan

In Japan, "clarke" is translated as kurākusū (クラーク数, clarke number). The word sū (数, number) is always added, which happens to make the term appear similar in form with scientific constants such as abogadorosū (アボガドロ数, the Avogadro constant). The term may have a narrower sense than Fersman's. Several of the following constraints may apply:
* Only of Earth's crust
* Lithosphere approximated as a 10 mile-deep layer from sea level
* Must include all of three layers: lithosphere (93.06%), hydrosphere (0.91%) and atmosphere (0.03%)
* Only mass ratio)
* Denote in percents) (not in ppm or ppb)
* (What the quoter believes to be) data from Clarke and Washington(1924)
Another peculiarity in Japan is the existence of a popular version of data, which was tabulated in reference books such as the annual "Chronological Scientific Tables" (RCST1939(1938)$^{:E46}$), the "Dictionary of Physics and Chemistry" (IDPC(1939)$^{:app.VI}$) and other prominent books on geochemistry and chemistry.$^{:(62)}$ This version Kimura(1938) was devised by chemist Kenjiro Kimura [ja].$^{:5}$ It was often quoted as The "Clarke numbers" (unsourced examples:$^{:443}$,$^{:429t2}$). The numbers differed from any versions by Clarke / Clarke&Washington (1889–1924), or anything listed in foreign (non-Japanese) articles such as the USGS compilation$^{:4t2}$, thus unknown outside of Japan. Yet the numbers were sometimes quoted in English articles without citation (example:$^{:55}$).
As geological definition of "Earth's crust" evolved, the "10 mile-deep" approximation were deemed out-of-date, and some people considered the term "clarke number" obsolete too. Yet other people may have meant broader senses, not limiting to Earth's crust, leading to confusion.$^{:355}$ RCST1961(1961) switched their "clarke number" table from Kimura(1938) to Mason(1958) based, and the label "clarke number" on table was removed in RCST1963(1962). IDPC(1971) removed the "clarke number" table which was a Kimura(1938)'s variant. IDPC(1981) said the term is mostly abandoned, and the dictionary entry for "clarke number" itself was removed from IDPC(1998).$^{:431}$ So "clarke numbers" became associated almost solely with Kimura(1938)'s data, but Kimura's name forgotten. Incidentally, in major reference books, there was no data table titled "clarke numbers" which showed Clarke's original tables.
Despite being removed from major reference books, data from Kimura(1938) and phrases such as "the Clarke number of iron is 4.70", unsourced, continue to circulate, even in the 2010s (example:$^{:799}$).

## Example data

This section lists only historical data. For recent data, see Abundance of elements in Earth's crust.
Technical definition of "clarke", "Earth's crust" and "lithosphere" differ among authors, and the actual numbers vary accordingly, sometimes by several times. Even the same author presents multiple versions, with various estimation parameters or knowledge refinements. Yet they are often quoted without source, rendering the data unverifiable. Clarke & Washington$^{:114}$$^{:34t17}$ presented estimations of the average composition of the outer part of Earth with four variants:
* 10-mile crust, hydrosphere and atmosphere.
* 20-mile crust, hydrosphere and atmosphere.
* 10-mile crust, only igneous rocks and sedimentary rocks. (i.e. exclude hydrosphere and atmosphere)
* 10-mile crust, only igneous rocks. (i.e. exclude hydrosphere and atmosphere)
"The earth's crust" in Clarke and Washington works can mean two different things: (a) The whole outer part of Earth, i.e. lithosphere, hydrosphere and atmosphere; (b) Only the lithosphere, which in their works just meant "the rocky crust of the earth". "Crust" here means (b).
* Following tables do not cover all elements. Some elements not on the table may have larger abundance. Some minor elements are listed here to aid identifying the origin of unsourced documents.
* Some entries contain data for the disputed element 43 masurium.
* Precision (number of digits) may be adjusted to improve legibility.

### Of the mass of 10 mile-thick lithosphere plus hydrosphere and atmosphere

Tables of historical data for some elements of their relative abundance in Earth's crust.
| cited as | cited as | Clarke (1889) | Clarke (1891) | Clarke (1908) | Clarke (1911) | Clarke (1916) | Clarke (1920) | Clarke & Washington (1922) | Clarke & Washington (1924) | Clarke (1924) | Berg (1929) | Berg (1932) | Fersman (1923) | Fersman (1934) | RCST1937 (1936) | RCST1939 (1938) | Kimura (1939) |
| cited in | cited in | $^{:138c.3}$ | $^{:39c.3}$ | $^{:32c.3}$ | $^{:34-35c.3}$ | $^{:34c.3}$ | $^{:35c.3}$ | $^{:114c.1}$ | $^{:34t.17c.1}$ | $^{:36c.3}$ | $^{:11}$ | $^{:113t.15}$ | $^{:18t.VIc.5}$ | $^{:148t.XVc.X}$$^{:174t.15c.X}$ | $^{:316}$ | $^{:E46}$ | $^{:5t.4}$ |
| | titled "clarke"? | - | - | - | - | - | - | - | - | - | | | - | Yes | Yes | Yes | Yes |
| | elements | 19 | 19 | 20 | 20 | 20 | 20 | 31 | 27 | 20 | | | 28 | 87 | 89 | 89 | 89 |
| lithosphere | lithosphere | 93% | 93% | 93% | 93% | 93% | 93% | 93% | 93% | 93% | 13.5/14.5 | | Yes | Yes | Yes | Yes | Yes |
| | definition | 10 miles | 10 miles | 10 miles | 10 miles | 10 miles | 10 miles | 10 miles | 10 miles | 10 miles | 16 km | 16 km | | 16 km | 16 km | 16 km | 10 miles |
| | rock types | All | All | All | All | All | All | All | All | All | All | | All | All | All | All | All |
| hydrosphere | hydrosphere | ocean 7% | ocean 7% | ocean 7% | ocean 7% | ocean 7% | ocean 7% | 7% | 7% | 7% | 1/14.5 | | Yes | Yes | Yes | Yes | Yes |
| atmosphere | atmosphere | N 0.02% | N 0.02% | N 0.02% | N 0.02% | N 0.02% | N 0.02% | 0.03% | 0.03% | Yes | | | Yes | Yes | Yes | Yes | Yes |
| remarks | remarks | often mentioned | | | | | | | often cited. >100% | | | >100% | >100% | | | popular in Japan | |
| Z | element | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (weight%) | (weight%) |
| 8 O | oxygen | 49.98 | 49.98 | 49.78 | 49.85 | 50.02 | 50.02 | 49.190 | 49.520 | 49.20 | 49.5000 | 49.500 | 49.70 | 49.130 | 49.500 | 49.500 | 49.500 |
| 14 Si | silicon | 25.30 | 25.30 | 26.08 | 26.03 | 25.80 | 25.80 | 25.710 | 25.750 | 25.67 | 25.7000 | 25.300 | 26.00 | 26.000 | 25.300 | 25.800 | 25.800 |
| 13 Al | aluminium | 7.26 | 7.26 | 7.34 | 7.28 | 7.30 | 7.30 | 7.500 | 7.510 | 7.50 | 7.5000 | 7.500 | 7.45 | 7.450 | 7.500 | 7.560 | 7.560 |
| 26 Fe | iron | 5.08 | 5.08 | 4.11 | 4.12 | 4.18 | 4.18 | 4.680 | 4.700 | 4.71 | 4.7000 | 5.080 | 4.20 | 4.200 | 5.040 | 4.700 | 4.700 |
| 20 Ca | calcium | 3.51 | 3.51 | 3.19 | 3.18 | 3.22 | 3.22 | 3.370 | 3.390 | 3.39 | 3.3900 | 3.390 | 3.25 | 3.250 | 3.390 | 3.390 | 3.390 |
| 11 Na | sodium | 2.28 | 2.28 | 2.33 | 2.33 | 2.36 | 2.36 | 2.610 | 2.640 | 2.63 | 2.6300 | 2.630 | 2.40 | 2.400 | 2.630 | 2.630 | 2.630 |
| 19 K | potassium | 2.23 | 2.23 | 2.28 | 2.33 | 2.28 | 2.28 | 2.380 | 2.400 | 2.40 | 2.4000 | 2.400 | 2.35 | 2.350 | 2.400 | 2.400 | 2.400 |
| 12 Mg | magnesium | 2.50 | 2.50 | 2.24 | 2.11 | 2.08 | 2.08 | 1.940 | 1.940 | 1.93 | 1.9300 | 1.930 | 2.35 | 2.350 | 1.930 | 1.930 | 1.930 |
| 1 H | hydrogen | 0.94 | 0.94 | 0.95 | 0.97 | 0.95 | 0.95 | 0.872 | 0.880 | 0.87 | 0.8700 | 0.870 | 1.00 | 1.000 | 0.870 | 0.870 | 0.870 |
| 22 Ti | titanium | 0.30 | 0.30 | 0.37 | 0.41 | 0.43 | 0.43 | 0.648 | 0.580 | 0.58 | 0.5800 | 0.630 | 0.50 | 0.610 | 0.630 | 0.460 | 0.460 |
| 17 Cl | chlorine | 0.15 (Cl+Br) | 0.15 (Cl+Br) | 0.21 | 0.20 | 0.20 | 0.20 | 0.228 | 0.188 | 0.19 | 0.1900 | 0.190 | 0.20 | 0.200 | 0.190 | 0.190 | 0.190 |
| 25 Mn | manganese | 0.07 | 0.07 | 0.07 | 0.08 | 0.08 | 0.08 | 0.108 | 0.080 | 0.09 | 0.0900 | 0.090 | 0.09 | 0.100 | 0.090 | 0.090 | 0.090 |
| 15 P | phosphorus | 0.09 | 0.09 | 0.11 | 0.10 | 0.11 | 0.11 | 0.142 | 0.120 | 0.11 | 0.1200 | 0.120 | 0.10 | 0.120 | 0.120 | 0.080 | 0.080 |
| 6 C | carbon | 0.21 | 0.21 | 0.19 | 0.19 | 0.18 | 0.18 | 0.139 | 0.087 | 0.08 | 0.0800 | 0.080 | 0.35 | 0.350 | 0.080 | 0.080 | 0.080 |
| 16 S | sulfur | 0.04 | 0.04 | 0.11 | 0.10 | 0.11 | 0.11 | 0.093 | 0.048 | 0.06 | 0.0600 | 0.060 | 0.10 | 0.100 | 0.060 | 0.060 | 0.060 |
| 7 N | nitrogen | 0.02 | 0.02 | 0.02 | 0.03 | 0.03 | 0.03 | 0.030 | 0.030 | 0.03 | 0.0300 | 0.030 | 0.04 | 0.040 | 0.030 | 0.030 | 0.030 |
| 9 F | fluorine | - | - | 0.02 | 0.10 | 0.10 | 0.10 | 0.030 | 0.027 | 0.03 | 0.0270 | 0.026 | 0.08 | 0.080 | 0.026 | 0.030 | 0.030 |
| 37 Rb | rubidium | - | - | - | - | - | - | - | - | - | 0.0033 | 3.50E-03 | - | 0.008 | 3.50E-03 | 0.030 | 0.030 |
| 56 Ba | barium | 0.03 | 0.03 | 0.09 | 0.09 | 0.08 | 0.08 | 0.075 | 0.047 | 0.04 | 0.0400 | 0.040 | 0.04 | 0.050 | 0.040 | 0.023 | 0.023 |
| 40 Zr | zirconium | - | - | - | - | - | - | 0.048 | 0.023 | - | 0.0230 | 0.023 | - | 0.025 | 0.023 | 0.020 | 0.020 |
| 24 Cr | chromium | 0.01 | 0.01 | - | - | - | - | 0.062 | 0.033 | - | 0.0330 | 0.038 | 0.02 | 0.030 | 0.033 | 0.020 | 0.020 |
| 38 Sr | strontium | - | - | 0.03 | 0.03 | 0.02 | 0.02 | 0.032 | 0.017 | 0.02 | 0.0200 | 0.020 | 0.02 | 0.040 | 0.020 | 0.020 | 0.020 |
| 23 V | vanadium | - | - | - | - | - | - | 0.038 | 0.016 | - | 0.0160 | 0.018 | 0.02 | 0.020 | 0.013 | 0.015 | 0.015 |
| 28 Ni | nickel | - | - | - | - | - | - | 0.030 | 0.018 | - | 0.0180 | 0.018 | 0.02 | 0.020 | 0.018 | 0.010 | 0.010 |
| 29 Cu | copper | - | - | - | - | - | - | 0.010 | 0.010 | - | 0.0100 | 0.010 | 0.02 | 0.010 | 0.010 | 0.010 | 0.010 |
| 58 Ce | cerium | - | - | - | - | - | - | 0.019 (Ce+Y) | 0.014 (Ce+Y) | - | 0.0022 | 2.00E-03 | - | 2.90E-03 | 2.00E-03 | 4.50E-03 | 4.50E-03 |
| 30 Zn | zinc | - | - | - | - | - | - | 0.004 | - | - | 0.0045 | 0.017 | - | 0.020 | 0.017 | 4.00E-03 | 4.00E-03 |
| 32 Ge | germanium | - | - | - | - | - | - | - | - | - | 2.00E-08 | 1.00E-04 | - | 4.00E-04 | 1.00E-04 | 6.50E-04 | 6.50E-04 |
| 43 Ma | masurium | - | - | - | - | - | - | - | - | - | - | - | - | 1.00E-07 | 1.00E-07 | 1.00E-07 | 1.00E-07 |
| others | | 0.00 | 0.00 | 0.48 | 0.47 | 0.47 | 0.47 | 0.00 | 0.032 | 0.47 | | | | | | | |

### Other variants

Some authors call these "clarkes" too, some do not.
| cited as | cited as | Clarke (1889) | Clarke & Washington (1924) | Clarke & Washington (1924) | Clarke & Washington (1924) | Clarke (1924) | Goldschmidt (1937) | Goldschmidt (1937) | Mason (1952) | Mason (1958) | RCST1961 (1961) | Mason (1966) | Mason & Moore (1982) |
| cited in | cited in | $^{:138c.1}$ | $^{:34t.17c.2}$ | $^{:34t.17c.3}$ | $^{:20t.11c.1}$ | $^{:36c.1}$ | $^{:99–100}$ | $^{:4t.2}$ | $^{:41t.8}$ | $^{:44t.9}$ | | $^{:45t.3.3}$ | $^{:46t.3.5}$ |
| | titled "clarke"? | - | - | - | - | - | | - | - | - | Yes | - | - |
| | elements | 19 | 27 | 27 | 35 | 20 | | | 80 | 80 | 85 | 78 | 78 |
| lithosphere | lithosphere | 100% | Yes | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% |
| | definition | 10 miles | 20miles | 10 miles | 10 miles | 10 miles | | | 10 miles | 10 miles | 10 miles | | |
| | rock types | All | All | igneous + sedimentary | igneous | All | igneous | igneous | All | All | All | All | All |
| hydrosphere | hydrosphere | (no) | Yes | (no) | (no) | (no) | (no) | (no) | (no) | (no) | Yes | (no) | (no) |
| atmosphere | atmosphere | (no) | Yes | (no) | (no) | (no) | (no) | (no) | (no) | (no) | Yes | (no) | (no) |
| remarks | remarks | | | | | | | | | | wrong composition | | |
| Z | element | (mass%) | (mass%) | (mass%) | (mass%) | (mass%) | (mass ppm) | (weight%) | (mass ppm) | (mass ppm) | (weight%) | (mass ppm) | (mass ppm) |
| 8 O | oxygen | 47.29 | 48.080 | 46.710 | 46.590 | 46.46 | | 46.600 | 466,000 | 466,000 | 46.600 | 466,000 | 466,000 |
| 14 Si | silicon | 27.21 | 26.720 | 27.690 | 27.720 | 27.61 | 277,200 | 27.720 | 277,200 | 277,200 | 27.720 | 277,200 | 277,200 |
| 15 Al | aluminium | 7.81 | 7.790 | 8.070 | 8.130 | 8.07 | 81,300 | 8.130 | 81,300 | 81,300 | 8.130 | 81,300 | 81,300 |
| 26 Fe | iron | 5.46 | 4.870 | 5.050 | 5.010 | 5.06 | 50,000 | 5.000 | 50,000 | 50,000 | 5.000 | 50,000 | 50,000 |
| 20 Ca | calcium | 3.77 | 3.520 | 3.650 | 3.630 | 3.64 | 36,300 | 3.630 | 36,300 | 36,300 | 3.630 | 36,300 | 36,300 |
| 11 Na | sodium | 2.36 | 2.690 | 2.750 | 2.850 | 2.75 | 28,300 | 2.830 | 28,300 | 28,300 | 2.830 | 28,300 | 28,300 |
| 19 K | potassium | 2.40 | 2.490 | 2.580 | 2.600 | 2.58 | 25,900 | 2.590 | 25,900 | 25,900 | 2.590 | 25,900 | 25,900 |
| 12 Mg | magnesium | 2.68 | 2.010 | 2.080 | 2.090 | 2.07 | 20,900 | 2.090 | 20,900 | 20,900 | 2.090 | 20,900 | 20,900 |
| 1 H | hydrogen | 0.21 | 0.510 | 0.140 | 0.130 | 0.14 | nicht ber. | - | 1,400 | 1,400 | 0.140 | 1,400 | 1,400 |
| 22 Ti | titanium | 0.33 | 0.600 | 0.620 | 0.630 | 0.62 | 4,400 | 0.440 | 4,400 | 4,400 | 0.440 | 4,400 | 4,400 |
| 17 Cl | chlorine | 0.01 | 0.101 | 0.045 | 0.048 | 0.05 | 480 | 0.048 | 314 | 200 | 0.020 | 130 | 130 |
| 25 Mn | manganese | 0.08 | 0.090 | 0.090 | 0.100 | 0.09 | 1,000 | 0.100 | 1,000 | 1,000 | 0.100 | 950 | 950 |
| 15 P | phosphorus | 0.10 | 0.130 | 0.130 | 0.130 | 0.12 | 800 | 0.118 | 1,180 | 1,180 | 0.118 | 1,050 | 1,050 |
| 6 C | carbon | 0.22 | 0.091 | 0.094 | 0.032 | 0.09 | | 0.032 | 320 | 320 | 0.032 | 200 | 200 |
| 16 S | sulfur | 0.03 | 0.050 | 0.052 | 0.052 | 0.06 | 520 | 0.052 | 520 | 520 | 0.052 | 260 | 260 |
| 7 N | nitrogen | - | 0.016 | - | - | - | | - | 46 | 46 | 0.0046 | 20 | 20 |
| 9 F | fluorine | - | 0.028 | 0.029 | 0.030 | 0.03 | | 0.030 | 300 | 700 | 0.070 | 625 | 625 |
| 37 Rb | rubidium | - | - | - | - | - | 310 | 0.031 | 310 | 120 | 0.012 | 90 | 90 |
| 56 Ba | barium | 0.03 | 0.048 | 0.050 | 0.050 | 0.04 | 250 | 0.025 | 250 | 400 | 0.040 | 425 | 425 |
| 40 Zr | zirconium | - | 0.024 | 0.025 | 0.026 | - | 220 | 0.022 | 220 | 160 | 0.016 | 165 | 165 |
| 24 Cr | chromium | 0.01 | 0.034 | 0.035 | 0.037 | - | 200 | 0.020 | 200 | 200 | 0.020 | 100 | 100 |
| 38 Sr | strontium | - | 0.017 | 0.018 | 0.019 | 0.02 | 150 | 0.015 | 300 | 450 | 0.045 | 375 | 375 |
| 23 V | vanadium | - | 0.016 | 0.016 | 0.017 | - | 150 | 0.015 | 150 | 110 | 0.011 | 135 | 135 |
| 28 Ni | nickel | - | 0.018 | 0.019 | 0.020 | - | 100 | 0.010 | 80 | 80 | 0.008 | 75 | 75 |
| 29 Cu | copper | - | 0.010 | 0.010 | 0.010 | - | 100 | 0.010 | 70 | 45 | 0.0045 | 55 | 55 |
| 58 Ce | cerium | - | 0.014 (Ce+Y) | 0.014 (Ce+Y) | 0.015 (Ce+Y) | - | 46.1 | 0.0046 | 46 | 46 | 0.0046 | 60 | 60 |
| 30 Zn | zinc | - | - | - | 0.004 | - | 40 | 0.004 | 132 | 65 | 0.0065 | 70 | 70 |
| 32 Ge | germanium | - | - | - | n*E-11 | - | 7 | 0.0007 | 7 | 2 | 2.00E-04 | 1.5 | 1.5 |
| 43 Ma | masurium | - | - | - | - | - | - | - | - | - | - | - | - |
| others | | | 0.033 | 0.033 | | 0.500 | | | | | | | |

## Clarke of concentration

A related term "clarke of concentration" or "concentration clarke", synonym: "concentration factor (mineralogy)", is a measure to see how rich a particular ore is. That is, the ratio between the concentrations of a chemical element in the ore, and its concentration in the whole Earth's crust (i.e. "clarke")$^{:42}$$^{:43}$.
If the concentration of a commodity in an ore X is $Kx$ [ppm], and the "clarke" of that commodity is $Ke$ [ppm], then "the clarke of concentration" of that commodity X is $Kk={\frac {Kx}{Ke}}$ (dimensionless).
The value represents the degree to which the commodity is concentrated from crustal abundances to the ore by natural geochemical processes; a clue for whether the commodity could be mined economically.