267 E The SI (mks) Unit System The magnitude of the physical quantity is expressed in comparison with a certain unit of the size of the standard. The physical quantity appearing in mechanics and dynamics is determined by giving length, mass, and time. A unit system that adopts meter (m), kilogram (kg), second (s) as their standard is called the mks unit system. As a metric unit system concluded in 1875, there were several problems. Apart from the mks system, there were others such as the cgs unit system based on centimeter (cm), gram (g), second (s), and the gravity unit system based on a unit of gravity instead of mass. However, in order to avoid confusion, the SI International System of Units (Système International d Unitès) has been adopted. E.1 Three Basic Units E.1.1 Unit of Length: Meter At the end of the eighteenth century, the French Academy of Sciences determined 1 meter (m) as one millionth of the length of the meridian measured from the Arctic to the equator. In 1799, a plate-shaped meter prototype (Mètre des Archives) with 1 m length made of platinum was made. After that, an international metric prototype which reproduced Mètre des Archives was made, and in 1889, meters were defined at the first international conference on measurement. Thereafter, in order to solve the problem of the change in the artificial prototype, it was suggested that the wavelength of the atomic spectrum be used as a reference. In 1960, the 86 Kr atom had a length equal to 1 650 763.73 times the wavelength of the light emitted in the transition between 2p 10 and 5d 5 levels in the vacuum. Subsequently, with the stabilization of the laser and the progress of the frequency measurement technique, by defining the speed of light in vacuum as c = 299 792 458 m/s in 1983, one meter was defined by the distance traveling through the vacuum during 1/299 792 458 of 1 s. Friction at the Atomic Level: Atomistic Approaches in Tribology, First Edition. Motohisa Hirano. 2018 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2018 by Wiley-VCH Verlag GmbH & Co. KGaA.
268 E The SI (mks) Unit System E.1.2 Unit of Time: Second Until 1960, one second was decided 1/86 460 of the day (mean solar day). The number 86 460 is calculated from 24 times multiplied by 3600 times. It is one mean solar day until the sun is seen south, which is called southing, until the next south, and the average sun day was the average of a sun day. Later, in 1967, it was decided to set the oscillation period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of 133 Cs atom to 9 192 631 770 times 1 s. This number was determined to be close to 86 460 times one averaged solar day (a mean solar day) defined as the rotation cycle of the Earth. The time defined in this way is called international atomic time. At atomic hour, the time difference between itself and the world time 1 is controlled not to exceed ±0.9 s, considering the general relativistic effect of gravity potential. Today, the accuracy of 10 13 10 14 s has been achieved. E.1.3 Unit of Mass: Kilogram After 1 m was first determined, 1 kg was determined as the mass of water of 1000 cm 3, a cube with a side of 10 cm, at the beginning of the nineteenth century. Since the density of water depends on temperature and pressure, it was decided to refer to the properties of water at 4 C water, having the maximum density under 1 atm. In 1799, platinum cylinder standard equipment, called Kilogramme des Archives, was made. After that, in an effort to make a prototype having a long-lasting nature, a new platinum, platinum 90-iridium 10 alloy with a diameter of around 39 mm, was made, and the definition of 1 kg by this prototype is still used. E.1.3.1 Atomic Mass Unit There are two important units when discussing the mass of an atom: mole and atomic mass unit (a.m.u., unit symbol is u). Mole is a basic unit of the SI unit system; a.m.u is not included in the SI unit, but is conventionally allowed to be used. The definition of 1 mol is as follows. It is an amount composed of the number of elements of the substance equal to the number of atoms contained in 0.012 kg (12 g) of the carbon isotope 12 C having a mass number of 12. For example, when considering an oxygen atom, the abovementioned definition is interpreted as follows. It means that there are Avogadro s number of an oxygen atom in one mole of oxygen atom. Also, it means that there are Avogadro s number of an oxygen molecule in one mole of oxygen molecule. Avogadro s constant N A is N A = 6.0221367(±0.0000036) 10 23 mol. 1 Time added at 12 h to the sun at 0 latitude (the time when the sun came south at 0 o clock) was added.
E.2 The SI (mks) Unit System 269 This value historically was introduced on the basis of Avogadro s hypothesis stating that gas under the same temperature, same pressure, and the same volume contains the same number of molecules. Avogadro s constant N A has been measured by various methods such as observation of Brownian motion, observation of monomolecular layer, measurement of Faraday constant and elementary ratio, measurement of α particle from radioactive substance, and measurement of crystal density by X-ray diffraction crystallography. The a.m.u (u) corresponds to 1/12 of the mass of one atom of the isotope 12 Cas 12 1u = 6.0221367 10 23 12 = 1.6605402 10 24 g. Since the mass of one atom is thus extremely small, it is determined on the basis of the atomic weight of the atom, which is also called relative atomic mass. Although the mass of an atom differs depending on the nuclide, the existence ratio of the isotope for most elements is constant for each element, and then the atomic weight can be determined for each element. The atomic weight of an element is defined by the ratio of the average atomic weight of the element with the natural isotopic ratio to 1/12 of the atomic mass of 12 C. 2 E.2 The SI (mks) Unit System The SI system is an mks system that uses the meter, kilogram, and second as base units. All other units are derived from the base units, which are completely listed in Table E.1. The SI system is fully consistent, and there is only one recognized unit for each physical quantity. Three types of units are used: base units, supplementary units, and derived units. The base units listed in Table E.1 are dependent only on accepted standards or reproducible phenomena. The supplementary units listed in Table E.2 have not been classified as being base units or derived units. The derived units listed in Tables E.3 and E.4 are made up of combinations of base and supplementary units. In addition, there is a set of non-si units 2 The atomic number of carbon 12 is 6 and the mass number is 12; it is written by 12 C. The nucleus 6 of a carbon atom contains six protons and six neutrons. There are as many electrons as there are protons, so carbon atoms are electrically neutral. If neutrons are the same number as protons, the mass number of 12 Cisobtainedby12= 6protons+ 6 neutrons. Naturally even with the same element, the number of neutrons differs, so that isotopes with different mass numbers exist at a certain ratio. For example, 1 1 H with mass number 1 exists 99.9885%, and 2 H with mass number 2 1 exists 0.0115%. Most of the naturally occurring carbon atoms (98.9%) are 12 C, and only a small amount of 13 C (seven neutrons) has a small contribution to the average atomic weight. Carbon 14 is a radionuclide with a half-life of 5730 years. It occurs naturally in the nuclear reaction of neutrons and nitrogen, which are made by cosmic rays in the atmosphere upper layer as n + 12 6 N 14 6 C + p, and is widely distributed in the carbon on the earth such as atmospheric carbon dioxide in the atmosphere and inorganic and organic compounds in surface sea water. It was discovered in 1947 1950 that 14 C existed in nature, and radiocarbon dating method has been developed to contribute to archeology.
270 E The SI (mks) Unit System Table E.1 SI base units. Quantity Name Symbol Length Meter m Mass Kilogram kg Time Second s Electric current Ampere A Temperature Kelvin K Amount of substance Mole mol Luminous intensity Candela cd Table E.2 SI supplementary units. Quantity Name Symbol Plane angle Radian rad Solid angle Steradian sr Table E.3 Some SI derived units with special names. Quantity Name Symbol Expressed in terms of other units Frequency Hertz Hz /s Force Newton N kg m/s 2 Pressure, stress Pascal Pa N/m 2 Energy, work, quantity of heat Joule J Nm Power, radiant flux Watt W J/s Quantity of electricity, electric charge Coulomb C Electric potential Volt V W/A Electric capacitance Farad F C/V Electric resistance Ohm Ω V/A Electric conductance Siemen S A/V Magnetic flux weber Wb Vs Magnetic flux density Tesla s Wb/m 2 Inductance Henry H Wb/A Luminous flux Lumen lm Illuminance Lux lx lm/m 2
E.2 The SI (mks) Unit System 271 Table E.4 Some SI derived units. Quantity Description Expressed in terms of other units Area Square meter m 2 Volume Cubic meter m 3 Speed linear Meter per second m/s Speed angular Radian per second rad/s Acceleration linear Meter per second squared m/s 2 Acceleration angular Radian per second squared rad/s 2 Density, mass density Kilogram per cubic meter kg/m 3 Concentration (of amount of substances) Mole per cubic meter mol/m 3 Specific volume Cubic meter per kilogram m 3 /kg Luminance Candera per square meter cd/m 2 Absolute viscosity Pascal second Pa s Kinematic viscosity Square meters per second m 2 /s Moment of force Newton meter N m Surface tension Newton per meter N/m Heat flux density, irradiance Watt per square meter W/m 2 Heat capacity, entropy Joule per kelvin J/K Specific heat capacity, specific entropy Joule per kilogram kelvin J/(kg K) Specific energy Joule per kilogram J/kg J Thermal conductivity Watt per meter kelvin W/(m K) Energy density Joule per cubic meter J/m 3 Electric field strength Volt per meter V/m Electric charge density Coulomb per cubic meter C/m 3 Surface density of charge, flux density Coulomb per cubic meter C/m 2 Permittivity Farad per meter F/m Current density Ampere per square meter A/m 2 Magnetic field strength Ampere per meter A/m Permeability Henry per meter H/m Molar energy Joule per mole J/mol Molar entropy, molar heat capacity Joule per mole kelvin J/(mol K) Radiant intensity Watt per steradian W/sr that may be used. This concession is primarily due to the significance and widespread acceptance of these units. Use of non-si units listed in Table E.5 will usually create an inconsistent expression requiring conversion factors. Prefixes listed in Table E.6 are printed without spacing between the prefix and the unit symbol.
272 E The SI (mks) Unit System Table E.5 Acceptable non-si units. Quantity Name Symbol Relationship to SI unit Area Hectare ha 1 ha = 10 000 m 2 Energy Kilowatt-hour kwh 1 kwh = 3.6 MJ Mass Metric ton a) t 1t= 1000 kg Plane angle Degree (of arc) 1 =1 = 0.017453 rad Speed of rotation Revolution per minute r/min 1 r/min = 2π/60 rad/s Temperature interval Degree celcius C 1 r/min = 2π/60 rad/s Time Minute min 1 min = 60 s Hour h 1 h = 3600 s Day (mean solar) d 1 d = 86 400 s Year (calendar) a 1 a = 31 536 000 s Velocity Kilometer per hour km/h 1 km/h = 0.278 m/s Volume Liter b) l 1 l = 0.001 m 3 a) The international name for metric ton is tonne.the metric ton is equal to the megagram (Mg). b) The international symbol for liter is the lower case l, which can be easily confused with the numeral 1. Several English-speaking countries have adopted the script l and uppercase L as a symbol for liter in order to avoid any misinterpretation. Table E.6 SI prefixes. Prefix Symbol Values exa E 10 18 peta P 10 15 tera T 10 12 giga G 10 9 mega M 10 6 kilo k 10 3 hecto h 10 2 deca da 10 1 deci d 10 1 centi c 10 2 milli m 10 3 micro μ 10 6 nano n 10 9 pico p 10 12 femto f 10 15 atto a 10 18
E.3 The cgs System 273 E.3 The cgs System The cgs system is used widely by chemists and physicists. It is named for three primary units used to construct its derived variables such as centimeter, gram, and second. Table E.7 shows cgs units with special names. When Newton s second law is written in the cgs system, the following combination of units is obtained by Units of force = g cm s 2. This combination of units for force is known as a dyne. Energy variables in the cgs system have units of dyne cm or, equivalently, g cm 2 /s 2. This combination is known as an erg. There is no uniformly accepted unit of power in the cgs system, although calories per second is used. The fundamental volume unit in the cgs system is the cubic centimeter (cc). Since this is the same volume as one thousandth of a liter, units of milliliters (ml) are also used. Some other units are listed in Table E.8 Table E.7 CGS units with special names. Quantity Name Symbol Equivalent in SI units Energy erg Erg 1 erg = 10 7 J Force dyne Dyne 1 dyne = 10 5 N Viscosity Poise P 1 P = 1dyns/cm 2 =0.1 Pa s Kinematic viscosity Stokes St 1 St = 1cm 2 /2 =10 4 m 2 /s Magnetic flux density Gauss G 1 G = 10 4 T Magnetic flux Maxwell Mx 1 Mx = 10 8 Wb Magnetic field Oersted Oe 1 Oe = (1000/4π)A/m Acceleration Gal Gal 1 Gal = 1cm/s 2 = 10 2 m/s 2 Wavenumber Kayser cm 1 1cm 1 = 100/m Table E.8 Other units. Quantity Name Symbol Expressed in other units Pressure Bar bar 1 bar = 0.1 MPa = 10 5 Pa = 10 6 dyn/cm 2 mb 1 mb = 1hPa torr 1 torr = 1 mmhg atm 1 atm = 760 torr = 101 325 Pa (Standard atmospheric pressure) Heat quantity Calorie cal 1 cal = 4.1868 J Length Nautical mile a) 1nauticalmile= 1852 m ångström Å 1 Å= 0.1 nm = 10 10 m a) Length on the ground surface for the central angle of the average meridian of 1 min.