UnitsAndDimensions
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UnitsAndDimensions

Course Number: CHE 3051, Spring 2008

College/University: National Central...

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ts pressure and its density. (a) (b) Use dimensional analysis to determine this dependence. If the gas is ideal, how will the speed of sound depend on the molecular weight at a given temperature? The speed of sound in air at room temperature is 330 ms-1. Calculate the speed of sound in gaseous helium at the same temperature. (c) 12. When an oil droplet is released, it falls under the influence of gravity until...

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pressure ts and its density. (a) (b) Use dimensional analysis to determine this dependence. If the gas is ideal, how will the speed of sound depend on the molecular weight at a given temperature? The speed of sound in air at room temperature is 330 ms-1. Calculate the speed of sound in gaseous helium at the same temperature. (c) 12. When an oil droplet is released, it falls under the influence of gravity until it reaches its terminal velocity, at which the gravitational force exactly balances the frictional force exerted by the air through which it passes. The terminal velocity depends on the weight mg of the drop (g is the acceleration due to gravity and m is the mass of the droplet), the viscosity of the medium, and the radius a of the droplet. (a) Use dimensional analysis to work out how the terminal velocity should depend on all of these factors. [The SI units of viscosity are kg m-1 s-1.] What will be the effect of the following changes on the terminal velocity? (i) (ii) Using a gas with twice the viscosity of air. Using an oil drop with double the radius. (b) 13. The rotational energy of a diatomic molecule is a function of its bond length, r, its reduced mass, , and Planck's constant, h. Use dimensional analysis to find out how the energy depends on these quantities. 14. The wind chill factor is the reduction in temperature due to the wind speed. It arises from the conversion of random motion (temperature) into organised motion (wind). The wind chill factor T depends on the wind speed, v, the molecular mass of the gas, m, and Boltzmann's constant k, which has the value 1.38 x 10-23 J K-1. Find the dimensions of each of these quantities and use dimensional analysis to discover how T depends on them. The molecular collision frequency per unit concentration in a gas, Z, has units m3 s-1 and depends on the Boltzmann constant, kB, the temperature, T, the molecular mass, m, and the molecular diameter, d. Use dimensional analysis to determine how Z depends on these quantities. 15. 1 Physical quantities and units 1.1 PHYSICAL QUANTITIES AND QUANTITY CALCULUS The value of a physical quantity can be expressed as the product of a numerical value and a unit: physical quantity = numerical value x unit Neither the name of the physical quantity, nor the symbol used to denote it, should imply a particular choice of unit. Physical quantities, numerical values, and units, may all be manipulated by the ordinary rules of algebra. Thus we may write, for example, for the wavelength A. of one of the yellow sodium lines: A. = 5.896 X 10- 7 m = 589.6 nm (1) where m is the symbol for the unit of length called the metre (see chapter 3), nm is the symbol for the nanometre, and the units m and nm are related by nm = 1O- 9 m (2) The equivalence of the two expressions for A. in equation (1) follows at once when we treat the units by the rules of algebra and recognize the identity of nm and 10- 9 m in equation (2). The wavelength may equally well be expressed in the form A./m = 5.896 x 10 - 7 or A./nm = 589.6 (3) (4) In tabulating the numerical values of physical quantities, or labelling the axes of graphs, it is particularly convenient to use the quotient of a physical quantity and a unit in such a form that the values to be tabulated are pure numbers, as in equations (3) and (4). Examples T/K 216.55 273.15 304.19 2.4 ~ p/MPa 4.6179 3.6610 3.2874 0.5180 3.4853 7.3815 In (p/MPa) -0.6578 1.2486 1.9990 1.6 50.8 c E Po. o -0.8 '-------'--------'------'-------"----'3.6 4.0 4.8 3.2 4.4 10 3 KIT Algebraically equivalent forms may be used in place of 10 3 K/T, such as kK/T or 103(T/K)-1. The method described here for handling physical quantities and their units is known as quantity calculus. It is recommended for use throughout science and technology. The use of quantity calculus does not imply any particular choice of units; indeed one of the advantages of quantity calculus is that it makes changes between units particularly easy to follow. Further examples of the use of quantity calculus are given in chapter 7, which is concerned with the problems oftransforming from one set of units to another. 3 1.2 BASE PHYSICAL QUANTITIES AND DERIVED PHYSICAL QUANTITIES By convention physical quantities are organized in a dimensional system built upon seven base quantities, each of which is regarded as having its own dimension. These base quantities and the symbols used to denote them are as follows: Physical quantity Symbol for quantity length mass time electric current thermodynamic temperature amount of substance luminous intensity m t I T n Iv All other physical quantities are called derived quantities and are regarded as having dimensions derived algebraically from the seven base quantities by multiplication and division. Example dimension of (energy) = dimension of (mass x length Z x time- Z ) The physical quantity amount of substance or chemical amount is of special importance to chemists. Amount of substance is proportional to the number of specified elementary entities of that substance, the proportionality factor being the same for all substances; its reciprocal is the Avogadro constant (see sections 2.10, p.46, and 3.2, p.70, and chapter 5). The SI unit of amount of substance is the mole, defined in chapter 3 below. The physical quantity 'amount of substance' should no longer be called 'number of moles', just as the physical quantity 'mass' should not be called 'number of kilograms'. The name 'amount of substance' and 'chemical amount' may often be usefully abbreviated to the single word 'amount', particularly in such phrases as 'amount concentration' (p.42)1, and 'amount of N z' (see examples on p.46). (1) The Clinical Chemistry Division of IUPAC recommends that 'amount-of-substance concentration' be abbreviated 'substance concentration'. 4 1.3 SYMBOLS FOR PHYSICAL QUANTITIES AND UNITS [5.a] A clear distinction should be drawn between the names and symbols for physical quantities, and the names and symbols for units. Names and symbols for many physical quantities are given in chapter 2; the symbols given there are If recommendations. other symbols are used they should be clearly defined. Names and symbols for units are given in chapter 3; the symbols for units listed there are mandatory. General rules for symbols for physical quantities The symbol for a physical quantity should generally be a single letter of the Latin or Greek alphabet (see p.14W. Capital and lower case letters may both be used. The letter should be printed in italic (sloping) type. When no italic font is available the distinction may be made by underlining symbols for physical quantities in accord with standard printers' practice. When necessary the symbol may be modified by subscripts and/or superscripts of specified meaning. Subscripts and superscripts that are themselves symbols for physical quantities or numbers should be printed in italic type; other subscripts and superscripts should be printed in roman (upright) type. Examples but for for for for for for for heat capacity at constant pressure mole fraction of the ith species heat capacity of substance B kinetic energy relative permeability standard reaction enthalpy molar volume The meaning of symbols for physical quantities may be further qualified by the use of one or more subscripts, or by information contained in round brackets. Examples AfS" (HgCl z , cr, 25 0c) J1.i = -154.3 J K -1 mol- 1 = (aG/an;)T,p,ftj.i Vectors and matrices may be printed in bold face italic type, e.g. A, a. Matrices and tensors are sometimes printed in bold face sans-serif type, e.g. S, T. Vectors may alternatively be characterized ~ ~ by an arrow, A,11 and second rank tensors by a double arrow, S, T. - General rules for symbols for units Symbols for units should be printed in roman (upright) type. They should remain unaltered in the plural, and should not be followed by a full stop except at the end of a sentence. Example r = 10 cm, not cm. or cms. Symbols for units should be printed in lower case letters, unless they are derived from a personal name when they should begin with a capital letter. (An exception is the symbol for the litre which may be either L or I, i.e. either capital or lower case.) (1) An exception is made for certain dimensionless quantities used in the study of transport processes for which the internationally agreed symbols consist of two letters (see section 2.15). Example Reynolds number, Re When such symbols appear as factors in a product, they should be separated from other symbols by a space, multiplication sign, or brackets. 5 Examples m (metre), s (second), but J (joule), Hz (hertz) Decimal multiples and submultiples of units may be indicated by the use of prefixes as defined in section 3.6 below. Examples nm (nanometre), kHz (kilohertz), Mg (megagram) 6 3 Definitions and symbols for units 3.1 THE INTERNATIONAL SYSTEM OF UNITS (SI) The International System of units (SI) was adopted by the 11 th General Conference on Weights and Measures (CGPM) in 1960 [3]. It is a coherent system of units built from seven 81 base units, one for each of the seven dimensionally independent base quantities (see section 1.2): they are the metre, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given in section 3.2. The 81 derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity [3]. In the International System there is only one SI unit for each physical quantity. This is either the appropriate SI base unit itself (see table 3.3) or the appropriate SI derived unit (see tables 3.4 and 3.5). However, any of the approved decimal prefixes, called 81 prefixes, may be used to construct decimal multiples or submultiples of SI units (see table 3.6). It is recommended that only SI units be used in science and technology (with SI prefixes where appropriate). Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units. 69 3.2 DEFINITIONS OF THE SI BASE UNITS [3] metre: The metre is the length of path travelled by light in vacuum during a time interval of 1/299 792 458 of a second (17th CGPM, 1983). kilogram: The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (3rd CGPM, 1901). second: The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom (13th CGPM, 1967). ampere: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed I metre apart in vacuum, would produce between these conductors a force equal to 2 x 10- 7 newton per metre oflength (9th CGPM, 1948). kelvin: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (13th CGPM, 1967). mole: The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles (14th CGPM, 1971). Examples of the use of the mole 1 mol 1 mol 1 mol 1 mol 1 mol 1 mol 1 mol of H 2 contains about 6.022 x 10 23 H 2 molecules, or 12.044 x 10 23 H atoms of HgCl has a mass of 236.04 g of Hg 2 Cl 2 has a mass of 472.08 g of Hg 2 2+ has a mass of 401.18 g and a charge of 192.97 kC of FeO.9IS has a mass of 82.88 g of e - has a mass of 548.60 Ilg and a charge of - 96.49 kC of photons whose frequency is 5 x 10 14 Hz has energy of about 199.5 kJ See also section 2.10, p.46. candela: The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 10 12 hertz

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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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National Central University - CHE - 3051
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