physics in Dimensions and Units

7/28/2019 physics in Dimensions and Units

1/3

Appendix II: Dimensionsand UnitsThe numerical values of most physical quantities are expressed in terms ofunits. The distance between two points, for example, can be specified by thenumber of meters (or feet. Angstroms, etc.). Similarly, time can be expressed inseconds, days or, say, years. However, the number of days per year varies fromone year to another. The quantities, distance (length) and time, as well as mass,are usually chosen to be primary quantities. In terms of them Newton's secondlaw for the force on an object, can be written as force = mass.distance/(time)^.The definition of the primary quantities allows dimensional expressions to bewritten, such as [force] = MLT~^ in the present example. Note, however, thatin everyday life one speaks of the weight of an object (or a person). Of coursethe weight is not the mass, but rather the force acting on the object by theacceleration due to gravity: [acceleration] = LT~^.The dimensional expressions given above are determined by the particularchoice of primary quantities. In the international system (SI) the base units ofmass, length and time have been chosen to be kilogram, meter and second,respectively. Then, the newton (N, a derived unit) is the unit of force. In thecgs system the centimeter, gram and second are considered to be the baseunits, leading to the force expressed in dynes. In this particular example thedimensional equation [force] = MLT~^ applies to either choice of units. Asa force that produces a change in distance involves work or energy, theirdimensions are given by ML^T~^. The unit of energy in SI is the joule (J),while in cgs it is the erg (note that 1 erg = 10"^ J).The primary quantities M , L, T are sufficient to describe most problems inmechanics. In thermodynamics and other thermal applications it is customaryto add an absolute temperature. In this case the dimension of the Boltzmannconstant, for example, is given by [k] = ML^T~^^"\ where the symbol 0 isused here for the dimension of the absolute or thermodynamic temperature.The dimensions of units in electricity and magnetism are the origin of muchconfusion. In the days when mechanical and thermal quantities were expressedin cgs, two different systems were introduced for the electrical and magneticquantities. They are the esu (electrostatic units) and the emu (electromagnetic


2/3

352 MAT HEM ATICS FOR CHEMISTRY AN D PHYSICSunits), respectively. Their addition to the cgs system results in a hybrid thatis usually referred to as the Gaussian system. An apparent advantage of theGaussian system is the disappearance of the factor 4nso which is foreverpresent in SI in problems involving inherent spherical symmetry. On the otherhand, in the Gaussian system a given quantity usually has different values inesu and emu. It then becomes necessary to introduce various powers of thevelocity of light (c) to assure internal consistency.

The so-called atomic units are often employed in quantum mechanical calculations. They are combinations of fundamental constants that are treated as ifthey were units. The base dimensions are chosen to be mass, length, chargeand action. They are respectively the rest mass of the electron (m^), the radiusof the first Bohr orbit (ao = Ans^^fi^/niee^), the elementary charge {e ) and theaction {h = h/ln, where h is Planck's constant). The corresponding energy,given by Eh = fi^/nieal = ruee^/(ATiSoffp-, is expressed in hartree.

The following tables summarize the units used in this book. For moreextensive tabulations, the reader is referred to the "Green Book", Ian Mills,et al. (eds), "Quan tities, Units and Symbols in Physical Chemistry", BlackwellScientific Publications, London (1993).Ta ble 1 The SI base units.

Physical quantity (dimension in SI)length (L)mass (M)time (T)thermodynamic or absolute temperature {0)electric current (A)amount of substance (mol)luminous intensity (cd)

Physical quantityfrequencyforcepressureenergy, workpowerelectric chargeelectric potential (emf)electric resistanceelectric capacitance

Table 2SI unithertznewtonpascaljoulewattcoulombvoltohmfarad

SI unitmeterkilogramsecondkelvinamperemolecandela

Some derived units.SymbolHzNPaJWCVQF

Expression in SIs-^m kg s~^N m - 2N mJ s - iA sj c - ^V A"^c v-

SymbolmkgsKAmolcd

Dimension in SIT - iM L T 2M L -^T -^M L ^ TM L ^ TA TM L ^ TM L ^ TM-^L"^

-2-3

-^A-^-3 A -2IjApl


3/3

II. DIMENSIONS AND UNITS 353Table 2 (Continued).

Physical quantity SI unit Symbol Expression in SI Dimension in SIelectric field strengthCelsius temperature degree Celsius Cdensitymolar volumewavenumber"perm ittivity (vacuu m ) ^(^0)

V m - ^Kkg m~^m^mol"^m-^F m - i

L^T-Â-^0M L - ^L^mor^L - iM-^L-^TÂ^

"Wavenumber is invariably expressed in cmTable 3 Prefixes in SI.

Submultiple Prefix Symbol Multiple Prefix Symbol10-110-210-31 0 - ^1 0 - ^10-1210-1510-18

decicentimillimicronanopicofemtoatto

dcmMnPfa

1010210^10*10"10'210'510'

decahectokilomegagigaterapetaex a

dahkMGTPE

Table 4 Some of the fundamental constants in the SI system.Quantity Symbol Valuepermeability of vacuumpermittivity of vacuumspeed of light in vacuumPlanck constantelementary chargeelectron rest massproton rest massAvogadro constantBoltzmann constantgas constantzero of the Celsius scaleBohr radiusHartree energyRydberg constant

Mo^0 = (moC ^)-!CohentenipNAkRao = ATiEoh^ /mêÊh = fi^lnieal^00 = Eh/2hco

4 7 r l O -^H m- i o r V s A -^m "!(defined)8.854 1 8 7 8 1 6 . . . 1 0 - i 2 F m - io r C V - ^ m - i299 792 458 m s-^ (defined)6.626 075 5(40) lO'^^ J s1.602 177 33(49) 10-^^ C9.109 389 7(54) lO-^^ kg1.672 623 1(10) 10-27 kg6.022 136 7(36) lO^^moP^1.380 658 (12) 10-23J K'^8.314 510 (70) J K-^mor^273.15 K (defined)5.291 772 49(24) 10"^^ m4.359 748 2(26) 10-^^ J1.097 373 153 4(13) lO^m"^

Documents

physics in Dimensions and Units