Physical Chemistry 3th Castellan

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STANDARD ATOMIC MASSES 1979(Scaled to the relative atomic mass , A ,.(I2C) = 12)Name Actinium Aluminium Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Caesium Calcium Californium Carbon Cerium Chlorine Chromium Cobalt Copper Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Atomic Symbol number Ac 89 Al 13 Am 95 Sb 51 Ar 18 As 33 At 85 Ba 56 Bk 97 Be 4 Bi 83 B 5 Br 35 Cd 48 55 Cs Ca 20 Cf 98 C 6 Ce 58 Cl 17 Cr 24 Co 27 Cu 29 Cm 96 Dy 66 Es 99 Er 68 Eu 63 Fm 100 F 9 Fr 87 Gd 64 Ga 31 Ge 32 Au 79 Hf 72 He 2 Ho 67 H I In 49 I 53 Ir 77 Fe 26 Kr 36 La 57 Lr 103 Pb 82 Li 3 Lu 71 Mg 12 Mn 25 Md 101 Hg 80 Atomic mass 227.0278 26 .98154 (243) 121.75* 39 .948 74.9216 (210) 137 .33 (247) 9.01218 208.9804 10.81 79 .904 112.41 132 .9054 40 .08 (25 I) 12.011 140. 12 35.453 51.996 58 .9332 63 .546* (247) 162 .50* (252) 167.26* 151.96 (257) 18.998403 (223) 157.25 * 69.72 72 .59* 196.9665 178.49* 4.00260 164.9304 1.0079 114.82 126.9045 192 .22 * 55 .847* 83 .80 138.9055 * (260) 207 .2 6.941 * 174 .967 * 24.305 54.9380 (258) 200 .59* Name Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten (U nnilhexium) (Unnilpentium) (U nnilquadium) Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Atomic Symbol number Mo Nd Ne Np Ni Nb N No Os 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 II 38 16 73 43 52 65 81 90 69 50 22 74 106 105 104 92 23 54 70 39 30 40 Atomic mass 95.94 144.24* 20 . 179 237 .0482 58.69 92.9064 14 .0067 (259) 190.2 15 .9994* 106.42 30.97376 195.08* (244) (209) 39.0983 140.9077 (145) 231 .0359 226 .0254 (222) 186.207 102 .9055 85.4678 * 101.07* 150.36* 44.9559 78 .96* 28.0855 * 107 .868 22 .98977 87. 62 32 .06 180.9479 (98) 127.60* 158.9254 204 .383 232 .0381 168.9342 118 .69* 47 .88 * 183.85* (263) (262) (261) 238.0289 50.9415 131.29* 173 .04 * 88 .9059 65.38 91.22

0Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb TI Th Tm Sn Ti

W(Unh) (Unp) (Unq) U V Xe Yb Y Zn Zr

Source: Pure and Applied Chemistry , 51, 405 (1979 ). By permission . Value s are considered reliable to I in the last digit or 3 when followed by an asterisk(*). Values in parentheses are used for radioactive elements whose atomic weight s cannot be quoted precisel y without knowledge of the origin of the elements; the value given is the atomic mass number of the isotope of th at element of longest known half-life.

FUNDAMENTAL CONSTANTS(approximate values; best values are in Appendix IV) Quantity Gas constant Zero of the Celsius scale Standard atmosphere Standard molar volume of ideal gas A vogadro constant Boltzmann constant Standard acceleration of gravity Elementary charge Faraday constant Speed of light in vacuum Planck constant Rest mass of electron Permittivity of vacuum SymbolR

Value 8.314 J K- 1 mol-I 273.15 K 1.013 x 105 Pa 22.41 x 10- 3 m3 mol-I 6.022 x 1023 mol I 1.381 x 10- 23 J K- 1 9.807 m s -2

ToPo

Vo

=

RTolpo

!

,

e

F c Il IiIn

=

NAe

=

h121T'

en

Bohr radius Hartree energy"-

41T'eo 1/41T'eo ao = 41T'eoIi2/me2Eh = el l41T'e oao

1.602 9.648 2.998 6.626 1.055 9.110 8.854 LIB 8.988 5.292 4.360

'>\.

X

x xX

X'>\.

X

x xx:

10 19 C 104 C mol-I lOR m s I 10 34 J s 10- 34 J s 10- 31 kg 10- 12 C2 N- 1 m 2 10- 10 C 2 N- I m -2 109 N m 2 C- 2 10 II m 10 I~ J

CONVERSION FACTORS1 L = 10- 3 m' (exactly) = 1 dm 3 I atm = 1.01325 Pa (exactly) I atm = 760 Torr (exactly) 1 Torr = 1.000 mmHg 1 cal = 4.184 J (exactly) 1 erg = 1 dyne cm = 10- 7 J (exactly) 1 eV = 96.48456 kJ/mol1 A = 10 10 m = 0.1 nm = 100 pm I inch = 2.54 cm (exactly) 1 pound = 453.6 g I gallon = 3.785 L 1 Btu = 1.055 kJ I hp = 746 W

i

I

MATHEMATICAL DATA1T =

3.14159265 ...

e = 2.7182818 ...(all x)

In x = 2.302585 ... log x

In I(l

+

x)

=x

-

Y~x"

+X~

IA\"3 -

Y4X 4

+(x~

(x"

< 1)

+

x )-1x )-1

- X

(l (I -

+

X

+ +

x" -

X3 X3

+ +4x 3

>

In the kinetic theory of gases we deal with integrals of the general type

(4. 3 5)

(4. 36)

M athematica l I nterl u d e

63

so that

In C[3) =

Loo x2n+ 1 e-Px2

dx =

n !)[3 - (n + 1 ).

(4. 37)

The higher-order integrals can be obtained from those of lower order by differentiation ; differentiating Eq. with respect to [3 yields

(4. 3 7)

dU[3) d[3

or

(4. 3 8)Two cases commonly arise. In this case we apply Eq.

(4. 3 7) directly and no difficulty ensues. The lowest member is 1p . All other members can be obtained from Eq. (4. 3 7) or by differentiating and using Eq. (4. 3 8).1 0 ([3) = -1 1 0 ([3)Case n.n

Case I.

n

= or a positive integer.

=

In this case we may also use Eq. directly, but unless we know the value ofthe factorial function for half-integral values of the argument we will be in trouble. If = - 1, the function takes the form

- 1, 1, t or n =

m

(4. 3 7)

- 1 where m = or a positive integer.

When

m 0, we have

=

Im - 1 /2 ([3) =

Loo x2me- Px2

n m

dx =

H(m - 1) !][3 - (m+ 1 /2).

(4. 39)( .

where in the second writing, x = [3 - 1 / 2 y, has been used. Comparing this result with the last member of Eq. we find that

L 1 /2 ([3) =

Loo e-px2 d

1 x = [3 - /2

LOO e - y2

dy = [3 - 1 / 2 L 1 /z (1),

4 40

)

(4. 39)

L 1 /2 (1) =

Loo e- y2and

dy = 1(

- 1)! .

(4. 4 1)

The integral, 1 - 1 / 2 (1), cannot be evaluated by elementary methods. We proceed by writing the integral in two ways,1 _ 1/ 2 (1) =

LOO e- x2

dx

then multiply them together to obtain,1 1 /2 (1) =

The integration is over the area of the first quadrant; we change variables to r 2

Loo LXle- (x2 + y2)

dx dy.

= x 2 + y2

64

T h e Struct u re of G ases

0 I1 /2 (1 ) = f/2d IXl e - r2r dr = G) 1''' e - r2 d(r2) = IXle - Y dy. The last integral is equal to O ! = 1; taking the square root of both sides, we have (4.42) I - l li l) = !In. Comparing Eqs. (4. 4 1) and (4.42), it follows that ( !) ! = In ; now from Eqs. (4. 40) and (4.42), L1 /2 (P) = !Jn p - 1 /2 . By differentiation, and by using Eq. (4. 3 8) we obtain dL 1 / n I 1 /2(P) - - --2 - 2yC (1.2 p - 3/2) dP_

and replace dx dy by the element of area in polar coordinates, r d dr. To cover the first quadrant we integrate from zero to and r from to 00 : the integral becomes

nl2

1.

1 I3/2(P) = dIdP/2 = 2yfir (1.. 1.2 p - 5/2). 2 Repetition of this procedure ultimately yields (P) = f oo x2me - PX2 dx = 2yfir 2(2m) .! p - (m + 1 /2) (4.43) Im - 1 /2 2m By comparing this result with Eq. (4. 3 9) we obtain the interesting result for half-integral factorials ( 2) = y n 2(2m) !, . (4.44) 2m Table 4.1 collects the most commonly used formulas._

and

1.

n

o

1.

n

1 m.

.

m

_

1. 1

C

m.

Ta b l e 4 . 1 i nteg ra ls that occ u r i n t h e k i netic t h e o ry of gases

(6) (7) (3) (4) (5)

1''' 1''' foOO

x 2 e - px2 dx = tfi tP - 3 / 2 x4 e - px2 dx = tfiiP - S / 2 (2n) !p - (n + l / 2 ) 2 2 nn !-

(8) (9)

x 2 ne - Px2 dx = 1.fi 2

f oo fooo 1 1-

x 2 n + l e - px2 dx = O xe - px2 dx = tP - 1

00

O

x 3 e - px 2 dx = tP - 2 x S e - px2 dx = p - 3

00

M athematica l I nterl u d e

65

* 4 . 7 . 1 The E r r o r F u n ct i o n

W e frequently have occasion t o use integrals o f the type o f Case I I above in which the upper limit is not extended to infinity but only to some finite value. These integrals are related to the error function (erf). We define erf (x) If the upper limit is extended to x --+=

00,

the integral is t.fi so that=

fe-U2 duo1.

(4.45)

erf (oo)

Thus a s x varies from zero t o infinity, erf (x) varies from zero t o unity. Ifwe add the integral from x to 00 multiplied by 2/.fi to both sides of the equation, we obtain erf (x) + Therefore

.fi foo e- u 2 du "2

=

[f0"e- u2 du foo e-U2 du] .fi "2+

=

2 .fi

fooe- u2 du 0

=

1.

2

.fi

fooe- u2 du"=

=

1

-

erf (x).

We define the co-error function, erfc (x), by erfc (x) Thus 1-

erf (x).

(4.46)

(4.47) Some values of the error function are given in Table 4.2.

Ta b l e 4 . 2 The error fu nct i o n :erf(x)=

In 0 e - ,,2 du2

I

X

x0.00 0. 10 0.20 0.30 0.40 0.50 0.60 0.70

erf(x)0.000 0. 1 12 0.223 0.329 0.428 0.521 0.604 0.678

x0.80 0.90 1.00 1 . 10 1 .20 1.30 1.40 1.50

erf(x)0.742 0.797 0.843 0.880 0.9 10 0.934 0.952 0.966

x1.60 1.70 1