3 CONVERSION OF UNITS
Units of the SI are recommended for use throughout science and technology. However, the published literature of science makes widespread use of non-SI units, e.g. Torr, bar, atm, kcal. It is thus often necessary to convert the values of physical quantities between SI units and other units. This chapter is concerned with facilitating this process.
3.1 THE USE OF QUANTITY CALCULUS
Quantity calculus is a system of algebra in which symbols are consistently used to represent physical quantities as the product of a numerical value and a unit and in which we manipulate the symbols for physical quantities, numerical values, and units by the ordinary rules of algebra. Quantity calculus has particular advantages in facilitating the problems of converting between different units and different systems of units. This is demonstrated in the following examples using a limited number of significant digits.
Example 1. The vapor pressure of water at 20 C is recorded to be p(H2O, 20 C) 17.5 Torr The torr, the bar, and the atmosphere are given by the equations 1 Torr 133.3 Pa, 1 bar = 105 Pa, and 1 atm = 101 325 Pa Thus p(H2O, 20 C) 17.5×133.3 Pa 2.33 kPa = (2.33×103/105) bar = 23.3 mbar = (2.33×103/101 325) atm 2.30×10−2 atm
Example 2. Spectroscopic measurements show that for the methylene radical, CH2, the a 1A1 excited state lies at a repetency (wavenumber) 3156 cm−1 above the X 3B1 ground state ( a − X) = T0( a) − T0( X) 3156 cm−1
The excitation energy from the ground triplet state to the excited singlet state is thus E = hc0 (6.626×10−34 J s) (2.998×108 m s−1) (3156 cm−1) 6.269×10−22 J m cm−1 = 6.269×10−20 J = 6.269×10−2 aJ
where the values of h and c0 are taken from the fundamental physical constants (see cover page) and we have used the relation 1 m = 100 cm, or 1 m 1 cm−1 = 100. Since the electronvolt is given by the equation 1 eV 1.6022×10−19 J, or 1 aJ (1/0.160 22) eV,
E 6.269×10−2/0.160 22 eV 0.3913 eV
Similarly the hartree is given by Eh = 2/mea02 4.3597 aJ, or 1 aJ (1/4.3597)Eh, and thus the excitation energy is given in atomic units by E 6.269×10−2/4.3597 Eh 1.438×10−2Eh
Finally the molar excitation energy is given by Em = NA E (6.022×1023 mol−1)(6.269×10−2 aJ) 37.75 kJ mol−1
Also, since 1 kcal = 4.184 kJ, or 1 kJ = (1/4.184) kcal, Em (37.75/4.184) kcal mol−1 9.023 kcal mol−1
In the transformation from E to Em the coefficient NA (Avogadro constant), is not a number, but has the unit mol−1.
3.2 CONVERSION TABLES FOR UNITS
The table below gives conversion factors from a variety of units to the corresponding SI unit [17]. For each physical quantity the name is given, followed by the recommended symbol(s), the SI unit name and its symbol. Entries give other units in common use, with their conversion factors to SI and other units. Systems of units other than the SI are referred to by the following acronyms: au (atomic units), cgs (‘centimetre, gram, second’ system of units), esu (electrostatic system of units), emu (electromagnetic system of units), and Gaussian (Gaussian system of units). The constant which occurs in some of the electromagnetic conversion factors is the exact number 29 979 245 800 and equals c0/(cm s−1).
The inclusion of non-SI units in this table should not be taken to imply that their use is to be encouraged. With some exceptions, SI units are always to be preferred to non-SI units. However, since many of the units below are to be found in the scientific literature, it is convenient to tabulate their relation to the SI.
The table must be read in the following way, for example for the ångström, as a unit of length: Å = 10−10 m means that the symbol Å of a length 13.1 Å may be replaced by 10−10 m, saying that the length has the value 13.1×10−10 m. Conversion factors are either given exactly or rounded to four digits (when the sign is used) for convenience only. An entry in the column named “Symbol” may refer to unit or quantity symbol (see Sections 1.3.1 and 1.3.2, p. 2).
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(1) This number refers to the tropical year for the epoch 1900.0. For more details, see [19] and the 4th edition of the Green Book [20].(2) The Rankine temperature, the Celsius temperature t, and the Fahrenheit temperature tF are related to the thermodynamic temperature T as follows: T/ R = (9/5)T/K (Rankine), t/ C = T/K−273.15 (Celsius), tF/ F = (9/5)(t/ C)+32 (Fahrenheit). (3) See also footnote 2, p. 4. (4) is the exact number = c0/(cm s−1) = 29 979 245 800. (5) The unit in quotation marks for polarizability (‘polarizability volume’) may be found in the literature, although it is formally incorrect. (6) In practice the oersted, Oe, is only used as a unit for H(ir) = 4 H, thus when H(ir) = 1 Oe, H = (103/4 ) A m−1 (see Section 7.3, ref. [20]). In the Gaussian or emu system, gauss and oersted are equivalent units. (7) In practice susceptibilities quoted in the context of emu or Gaussian units are always values for (ir) = /4 ; thus when (ir) = 10−6, = 4 ×10−6 (see Section 7.3, ref. [20]). (8) In practice the units cm3 mol−1 usually imply that the non-rationalized molar susceptibility is being quoted (ir) m = m/4 . For example if (ir) m = −15×10−6 cm3 mol−1, then m = −1.88×10−10 m3 mol−1 (see Section 7.3, ref. [20]).
3.3 TRANSFORMATION OF EQUATIONS OF ELECTROMAGNETIC THEORY BETWEEN THE ISQ (SI) AND GAUSSIAN FORMS
Transformation of other important equations of electromagnetic theory are given in the Green Book, 3rd Edition (2011) [2.c] and 4th Edition [20]. There, the reader will also find general forms of these equations that allow writing them using other systems of units, such as the system of atomic units.
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(1) H(ir) = 4 H, D(ir) = 4 D, (ir) e = e/4 , (ir) = /4 .