Periodic Table of the Elements .:. Table ofUniversal …€¦ ·  · 2012-06-20Calculate the equilibrium constant ... (en)3]2+, together with their molar absorptivity values, are

UNIVERSITY OF SWAZILAND

FINAL EXAMINATIONS

ACADEMIC YEAR 20112012

TITLE OF PAPER INORGANIC CHEMISTRY

COURSE NUMBER C30l

TIME ALLOWED THREE (3) HOURS

INSTRUCTIONS THERE ARE SIX (6) QUESTIONS ANSWER ANY FOUR (4) QUESTIONS EACH QUESTION IS WORTH 25 MARKS

THE FOLLOWING HAVE BEEN PROVIDED WITH TillS ExAMINATION PAPER

Periodic Table of the Elements

Table of Universal Constants

Tanabe-Sugano diagrams for octahedral complexes

Character tables for Clv and D3h point groups

Decision Tree ( Flow chart) for point groups

PLEASE DO NOT OPEN THIS PAPER UNTIL AUTHORISED TO DO SO BY THE CIDEF INVIGILATOR

Marks will be awarded for method clearly labeUed diagrams organization and presentation ofthoughts in clear and concise language

1

Question One

a) Give the IUPAC name for each of the following

i) Ks[Mo(CNh] ii) [Cr~3)6][Cr(CN)6]

iii) [Co(OH2)6](CI04)2 [6]

b) Give the formula of each of the following

i) Sodium pentacyanonitrosylferrate(II) dihydrate ii) Potassium pentachloronitridoosmate(IV) iii) Tetraammineaquacobalt(III)-J1-cyanobromotetracyanocobaltate(III)

[6 ]

c) Briefly discuss the observed trends in the stability of oxidation states across the periods from left to right and down the groups Use examples to illustrate your answer

[6 ]

d) What is the chelate effecfl Give two ways ofexplaining how the chelate effect leads to greater stability of complexes

[7]

Question Two

a) Sketch the structures of all possible isomers that may arise from the following complexes

i) [Ni(SCH2CH2NH2)2] square planar ii) [Co(en)3]3+

[6 ]

b) For each of the following complexes give the oxidation state of the metal and its dN configuration

i) [Mn(CN)6t ii) [Cr(acach]

[4 ]

c) What is the hole formalism Give two examples to illustrate your answer [4 ]

d) State the following rules

i) The spin selection rule ii) The La Porte selection rule

[3 ]

2

e) Explain the origins of LMCT and MLCT absorptions in the electronic spectra ofd-block metal complexes Give two examples (one for each type) to illustrate your answer

[8 ]

Question Three

a) For each of the complexes given below determine the oxidation number and electron configuration dN

of the transition metal ion i) Mg[Mo041 ii) [Fe(CO)4]2shy

[6] b) Give two types ofminerals in which titanium Ti is found Your answer

should include the chemical formula of the chemical species present Give a brief outline of one method that is used to extract the metal from its ore (Le from one of the minerals in which it is found)

[8]

c) A student in the year 1895 prepared three chromium compounds all of which corresponded to the same formula of CrCh6H20 When each of the compounds is dissolved in water the number of cr ions released are as indicated below

Complex colour cr ions in solution per Formula Unit

A Violet 3 B Light green 2 C Dark green 1

i)

ii)

Write the formulas of these compounds

Suggest a method for confirming the number of cr ions in the outer coordination sphere per formula unit of each complex

[11]

3

Question Four

a) For each of the following pairs of species indicate which member of the pair is more acidic Explain each of your answers

[6]

b) In order to separate gold from solid impurities the ore is treated with a sodium cyanide (NaCN) solution in the presence of air to dissolve the gold by forming a soluble complex

i) Write a balanced chemical equation that depicts the formation of the complex

ii) Give the geometry of the complex [6]

c) Predict the spin-only magnetic moment ofeach of the following octahedral complexes In each case use a suitable CF (d-orbital) splitting diagram to illustrate how the spin quantum number (S) is obtained

i) [Fe(CN)6t iii) [FeF6]3shy

[7] d) Assume that the trans effect increases in the order NH3 ltcr lt N02- lt PR3shy

Suggest the structures of the products arising from the reactions of [PtC4t with the ligands given below In each case the structures of the products and sequences of reactions should be clearly shown

i) Two equivalents ofNH3 ii) One equivalent ofPEt3 followed by one equivalent ofN02-

[6]

Question Five

a) Consider formation constants for the following reactions (at 298K)

Reaction Reaction Equation

10 xl

3 - +2NH3

No 1

2

2

i)

ii)

Calculate the equilibrium constant (K3) for the third reaction in the table above

Comment on the relative sizes ofthe equilibrium constants

[5]

4

b) Overall stability constants for [Au(CN)2r and [Pd(CN)4]2- are log~2=39 and log~4=623 respectively Write reaction equations that describe the processes to which these constants refer Then write the expressions to define 132 and ~4 (in terms ofconcentrations of appropriate species)

[6 c) Predict whether the equilibrium for the following reactions is expected

to lie more on the right hand side or more on the left hand side Explain each of your answers

i) Cdh + CaF2 z CdF2 + Cah

ii) [Culit + [CuC4t z [Cuc4t + [Culi]3shy

iii) CH3Hg+ + HCN z CH3HgCN +W [6]

d) Electronic spectra of Ni(II) complex ions [Ni(H20)6f+ and [Ni(en)3]2+ together with their molar absorptivity values are given in the table below

Transitions Complexes and absorption peak positions (em-I) and molar absorbanees (in brackets)

[Ni(OH2)6y-tshy [Ni(en)3]t+

r 2-r1 9000 (e 2) 11000 (e 7)

r 3-rl 14000 (e 2) 18000 (e 5)

r 4-rl 25000 (e 5) 30000 (e 95)

i)

ii)

iii)

Use a d8 Tanabe-Sugano diagram to identify the terms r I r 2

r 3 and r 4 and assign the transitions to the absorption bands

Comment on the differences in peak positions of [Ni(OH2)6f+ versus those of [Ni(en)3]2+

Comment on the differences in peak intensities of [Ni(OH2)6f+ versus those of [Ni(en)3]2+

[81

5

Question Six

a) With the help of the flow~chart which is provided determine the point group for each of the following

[7] b) The carbonate ion COl can serve as a ligand When it does so the

symmetry is lowered from D3h (for the uncoordinated ion) to C2v (for the monodetate ligand or bidentate ligand) Thus infrared spectroscopy makes it possible to distinguish coordinated-tarbonate from uncoordinated carbonate Using internal coordinates II determine the symmetries and number ofc-o stretching IR active and Raman active bands for

i) Uncoordinated carbonate ii) Monodentate carbonate M-O-C02 where M= metal center

Free Carbonate ion coordinated carbonate ion to a metal ion M (D3h point group) (C2v point group)

I

[18]

6

The Periodic Table -

2

3

11 oC -I)

D

4

5

8

7

1 3 Li

8941 11

Na 2299

19 K

3910 31 Rb

8547 55 Cs

1329 87 Fr

2230

1 2 H He

2 1008 13111 14IV 16N lSNI 17NII 4003 ~ 5 8 7 8 9 10

Be B C N 0 F Ne 9012 1081 1201 1401 1600 1900 2018

M~ 13 14 15 18 17 18

AI Si P S CI Ar 2430 3 4 amp e 7 8 9 10 11 12 2698 2809 3097 3207 3545 399amp

20 21 22 23 24 25 2 27 28 29 30 31 32 33 34 35 36Ca 5c TI V Cr Mn Fe Co Ni Cu Zn Ga Ge As 5e Br Kr

4008 4498 4787 amp094 5200 6494 amp58amp 6B93 6869 6355 6539 8972 1281 7492 7896 7990 8380 38 39 40 41 42 43 44 45 middot48 47 48 49 50 51 62 53 64 Sr V Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

8782 8891 9122 9291 9694 98~91 1011 1029 1064 _1079 1124 1148 l1B7 121B 1278 1289 1313 68 Lashy 72 73 74 7amp 78 77 78 79 80 81 82 83 84 85 88 Ba Lu Hf Ta W Re Os Ir Pt Au Hg TI Pb Bi Po At Rn

1373 1785 1809 1838 1882 1902 1922 1951 1970 2006 2044 2072 2090 2100 2100 2220 88 Acshy 104 105 108 107 108 109 Ra Lr - Unq Unp Unh Uns Uno Une

2280 I dblock pblockbull block

lath~ld Actln

Iblock

57 La

68 Ce

59 Pr

80 Nd

81 Pm

82 Sm

63 Eu

64 Gd

65 Tb

66 Dy

67 Ho

68 Er

69 Tm

70 Vb

71 Lu

138 1401 409 1482 49 1504 1520 1572 1589 1825 1849 1673 1689 1730 1750 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

2210 2320 2310 2380 2370 2391 2411 2441 2491 2521 2521 2171 2581 2591 2821

APPENDIX 5 731

5 d with C = 488 7 d with C=47098

G 70

10 20 30

~ 3~

T2 t~e

5y-2 t~ T t 5

f)

3r 2 2

3r

toB

6 d with C = 46338

A

50

40

CQill

~---L--L-----IL-__ 3A2 t12 10 20 30

toB

70

60 ~D

50

10 30

Useful relations

At 29815 K RT = 24790 kJ mol-I and RTF = 25693 mV

I atm = 101325 kPa =760 Torr (exactly)

I bar = lOS Pa

1 eV = 1602 18 x 10-19 J = 96485 IcJ mol-I = 80655 em-

1 em-I = 1986 x 10-23 J = 1196 J mol-I =01240 MeV

1 cal =4184 J (exactly)

I D (debye) = 3335 64x 10-30 em IT=l()G

I A(angstrom) = 100 pm

I M = 1 mol dm-3

General data and fundamental constants

Quantity Symbol Value Speed of light c 2997 925 x lOS m S-I

Elementary charge e 1602 177 x 10-19 C Faraday constant F=eNA 96485 x lQ4 C mol- I

Boltzmann constant k 138066 x 10-23 J K- I

86174 x 1O-~ eV K- I Gas constant R=kNA 831451 J K- I mol- I

820578 x 10-2 dm3 atm K- I mol- I Planck constant h 6626 08 x 10-34 J S

11 = h27C 1054 57 x 10-34 J 5

- Avogadro constant NA 6022 14 x Ion mol- I

Atomic mass unit u 1660 54 x 10-27 kg

)f- Mass of electron me 9109 39 x 10-31 kg

Vacuum permittivity Eo 8854 19 x 10-12 rl C2 m-I

47tEo 111265 x 10-10 J-I C2 m-I

Bohr magneton 8 =elt2me 927402 x 10-24 J T-I

Bohr radius ao =47CEo1t2meel 5291 77 x 10- 11 mRydberg constant RlaquoJ = mee48~c~ 109737 x 1~ em-I =IO~]3)x 107 111-

Prefixes

f p n m c d k M G femto pico nano micro milli centi deci kilo mega giga 1O-1~ 10-12 10-9 10-6 10-3 10-2 10-1 103 10 109

APPENDICES

4 The Cp Groups

lD C

Al

Al

Bl B

Cb

AI

A2 E

E

1 1 1 1

Cl a(xz) O~z)

1 1

-1 -1

1 -1

1 -1

1 -1 -1

1

Z

R x R yR

Xl yl Z1

xy xz yz

E 2C1 3aD

1 1 1 z X+yz

1 1 -1 R 2 -1 0 (x y)(R R) (Xl _ y2 xyXxz yi)

ch

Al

A1

Bl Bl E

E

1

1 1 1 2

2C~ C2 20 20d I

1 1

-1 -1

0

1 1 1 1

-2

1 -1

1 -I

0

1 -1 -1

1 0

Z

R

(x y)(R R)

x2 + y2 Zl

X_yl

xy (xzyz)

------~----~---

6 APPENDICES

6 The DAGroups

E Cz(z) C(y) C1(x) o(xy) o(xz)Dn

I I I I IA 1 1 I -1 1B 1bull I -I 1 -I IB2 1 -1 -I 1 1B 1 1 1 1 -IA I 1 -I - 1 -IB Ibull 1 -1 1middot I -1B2bull 1 -1 -I 1 -IBI bull

gt ~ E 2C 3C2 2S 3aDu

1 1 1 I I 1AI 1 I -I 1 I -1A

E 2 -1 0 2 I 0 1 I 1 -1 -1 -1A l 1 I -1 -I -1 IA 2 -I 0 -2 1 0E

I I 1 -I

-I 1 -I -1 -I -I -1 1

1 -1 I 1

R (x y)

z (R R)

o(yz)

I -1 -1

I -1

1 1

-I

R R R

z y x

X2+y1z2

(x_ y2xy)

(xz yz)

D4I E 2C C l 2C2 2C2 2S a_ laD 2ad

I I 1 I 1 1 1 I 1 1AI 1 1 1 -I 1 I 1 1 -1 -IA1

I -I 1 1 -1 1 -1 1 1 -IB 1

I 1 I -1 1 I -1 1 -I I8 1

2 0 -2 0 0 1 0 -2 0 0E 1 1 1 1 1 -1 -1 -I -I -IAI I I 1 -1 -I -I -I -I 1 IA l bull 1 -I 1 1 middot-1 -1 1 -1 1 1B l bull 1 -1 J -I 1 -I 1 -1 1 -IB 2 0 -2 0 0 -2- 0 1 0 0E

R

_T~ _ _shy

x 2y2z xy xz yz

x+ y2 z2

x2_ yZ xy (xzyz)(R R)

Z

(xy)

gtshy

Table A1 Continued 253A P PE N 0 I X 2

z 000 001 002 003 004 005 006 007 008 009 C CStatistical tables gt (I)-19 00287 00294 00301 00307 00314 00322 00329 00336 00344 00351 1

-18 00359 00367 00375 00384 00392 00401 00409 00418 00427 00436 c X-17 00446 00455 00465 00475 00485 00495 00505 00516 00526 00537 N-16 00548 00559 00571 00582 00594 00606 00618 00630 00643 00655

-15 00668 00681 00694 00708 00721 00735 00749 00764 00778 00793

-14 00808 00823 00838 00853 00869 00885 00901 00918 00934 00951 -13 00968 00985 01003 01020 01038 01056 01075 01093 01112 01131 -12 01151 01170 01190 01210 01230 01251 01271 01292 01314 01335 -11 01357 01379 01401 01423 01446 01469 0149201515 01539 01562 -10 01587 0611 01635 01660 01685 01711 01736 01762 01788 01814

-09 01841 01867 01894 01922 01949 01977 02005 02033 02061 02090 -08 02119 02148 02117 02206 02236 02266 02296 0232i 02358 02389 -07 02420 02451 02483 02514 02546 02578 02611 02643 02676 02709The following tables are presented for the convenience of the reader and for use -06 02743 02776 02810 02843 02871 02912 02946 02981 03015 03050with the simple statistical tests examples and exercises mthis book They are -05 03085 03121 03156 03192 03228 03264 03300 03336 03372 03409presented in a fonnat that is compatible with the needs of analytical chemists

the significance level P = 005 has been used in most cases and it has been -04 03446 03483 03520 03557 03594 03632 03669 03707 03745 03783 assumed that the number of measurements available is fairly smalL Most of -03 03821 03859 03897 03936 03974 04013 04052 04090 04129 04168 these abbreviated tables have been taken with permission from Elementary -02 04207 04247 04286 04325 04364 04404 04443 04483 04522 04562 Statistics Tables by Henry R Neave published by Routledge (fables A2-A4 -01 04602 04641 04681 04721 04761 04801 04840 04880 04920 04960 A7 AB All-A14) The reader reqUiring statistical data corresponding to 00 05000 05040 05080 05120 05160 05199 05239 05279 05319 05359 significance levels andor numbers of measurements not covered in the tables

01 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753is referred to these sources 02 05793 05832 05871 05910 05948 05987 06026 06064 06103 06141 03 06179 06217 06255 06293 06331 06368 06406 06443 06480 06517Table A1 Fez) the standard normaL cumulative distribution function 04 06554 06591 06628 06664 06700 06736 06172 06808 06844 06579

z 000 001 002 003 004 005 006 007 008 009 05 06915 06950 06965 07019 07054 07088 07123 07157 07190 07224

06 07257 07291 07324 0735i 07389 07422 07454 07486 07517 07549 -34 00003 00003 00004 00004 00004 00004 00004 00004 00005 00005 07 07580 07611 07642 07673 01704 01734 01764 07794 07823 07852 -33 00005 00005 00005 00005 00006 00006 00006 00006 00006 00007 08 07881 07910 01939 07967 07995 08023 08051 08078 08106 08133 -32 00007 00007 00007 00008 OOOOS 00008 00008 00009 00009 00009 09 08159 08186 08212 08238 08264 08289 08315 08340 08365 08389 -31 00010 00010 00010 00011 00011 00011 00012 00012 00013 00013 10 08413 08438 08461 08485 08508 08531 ~~8554 08577 08599 08621 -30 00013 00014 00014 00015 00015 00016 00016 00017 00018 00018

11 08643 08665 08686 08708 08729 08749 08770 08790 08810 08830-29 00019 00019 00020 00021 00021 00022 00023 00023 00024 00025

12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015-28 00026 00026 00027 00028 00029 00030 00031 00032 00033 00034

13 09032 09049 09066 09082 09099 og115 09131 09147 0916~ 09177-27 0003500036 00037 00038 00039 00040 00041 00043 00044 00045

14 09192 09207 09222 09236 09251 09265 09279 09292 09306 0931926 00047 00048 00049 00051 00052 00054 00055 00057 00059 00060 15 09332 09345 09357 09370 09382 09394 09406 09418 09429 09441-25 00062 00064 00066 00068 00069 00071 00073 00075 00078 00080 16 -09452 09463 09474 09484 09495 09505 09515 09525 09535 09545-24 00082 0OOS4 00087 00089 00091 00094 00096 00099 00102 00104 17 09554 09564 09573 09582 09591 09599 09608 09616 09625 09633-23 00107 00110 00113 00116 00119 Q0122 00125 00129 00132 00136

-22 00139 00143 0014Q 00150 00154 00158 00162 00166 00170 00174 18 09641 09649 09656 09664 09671 09678 09686 09693 09699 09706 -21 00179 00183 00188 00192 00197 00202 00207 00212 00217 00222 19 09713 09719 09726 09732 09738 09744 09750 09756 09761 09767 -20 00228 00233 00239 00244 00250 00256 00262 00268 00274 00281 20 09772 09718 09783 09788 09793 09798 09803 09808 09812 09817

-A

254 Table AI Continued Table A3 Critical values of F for a one-tailed test (P = 005)

z 000 001 002 003 004 005middot 006 007 008 009 Yf Yl

00 (I) 21 09821 09826 09830 09834 09838 09842 09846 09850 09854 09857I 2 3 4 5 6 7 8 9 10 12 15 Q 22 09861 09864 09868 09871 09875 09878 09881 09884 09887 09890 X 23 09893 09896 09898 09901 09904 09906 09909 09911 09913 09916 N 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2455

24 09918 09920 09922 09925 09927 09929 09931 09932 09934 09936 2 1851 1900 1916 1925 1930 1933 1935 1937 1938 1940 1941 1943

25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952 3 1013 9552 9277 9117 9013 8941 8887 8845 8812 8786 8745 8703

26 09953 09955 09956 09957 09959 09960 Q9961 09962 09963 09964 4 7709 6944 6591 6388 6256 6163 6094 6041 5999 5964 5912 5858

27 09965 09966 09967 09968 09969 09970 09971 09972 09973 09974 5 6608 5786 5409 5192 5050 4950 4876 4818 4772 4735 4618 4619

28 09974 09975 09976 09977 09977 09978 09979 09979 09980 09981 6 5981 5143 4151 4534 4387 4284 4207 4141 4099 4060 4000 3938

29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 7 5591 4737 4347 4120 3972 3866 3787 3726 3677 3637 3575 3511

30 09987 09987 09987 09988 09988 09989 09989 09989 09990 09990 8 5318 4459 4066 3838 3687 3581 3500 3438 3388 3347 3284 3218

31 09990 09991 09991 09991 09992 09992 09992 09992 09993 09993 9 5117 4256 3863 3633 3~2 3374 3293 3230 3119 3137 3073 3006

32 09993 09993 09994 09994 09994 09994 09994 09995 09995 09995 10 4965 4103 3708 3478 3326 3217 3135 3072 3020 2978 2913 2845

33 09995 09995 09995 09996 09996 09996 09996 09996 09996 09997 11 4844 3982 3587 3357 3204 3095 3012 29~ 2896 2854 2788 2719

34 09997 09997 09997 09997 09997 09997 09997 09997 09997 09998 12 4741 3885 3490 3259 3106 2996 2913 2849 2796 2753 2681 2617

13 4667 3806 3411 3179 3025 2915 2832 2167 2714 2611 2604 2533

14 4600 3739 3344 3112 2958 2~ 2764 2699 2646 2602 2534 2463Table A2 The t-distribution 15 4543 3682 3287 3056 2901 2790 2707 2641 2588 2544 2475 2403

Value of t for a confidence interval of 90 95 98 99 Critical value ofItIfor Pvalues ofnumber ofdegrees offreedom 010 005 002 001 16 4494 3634 3239 3007 2852 2741 2657 2591 2538 2494 2425 2352

17 4451 3592 3197 2965 2810 2699 2614 2548 2494 2450 2381 2308

18 4414 3555 3160 2928 2773 2661 2577 2510 2456 2412 2342 22691 631 1271 3182 6366 19 4381 3522 3127 2895 2140 2628 2544 2477 2423 2318 2308 22342 292 430 696 992 20 4351 3493 3098 2866 2711 2599 2514 2447 2393 2348 2278 22033 235 318 454 584

4 213 278 375 460 Y1 = number of degrees of freedom of the numerator and 2 = number of degrees of freedom 015 202 257 336 403 denominator6 194 245 314 371

7 189 236 300 350 8 186 231 290 336 9 183 226 282 325 10 181 223 276 317 12 178 218 268 305 14 176 214- 262 298 16 175 212 258 292 18 i73 iip 2~55 288~

20 172 209 253 285 30 170 204 246 275 50 168 201 240 268 00 164 196 233 258 bull

l

The critieal values of ItIare appropriate for a two-tailed test For a one-tailed test the value l

is taken from the column for twice the desired P-value eg for a onemiddottailed test P=005 5 degrees of freedom the cotieal value is read from the P= 010 column and is equal to 202

t

257Table 16 Critical values of G(P =005) for able 14 Critical values of F for a two-tailed test (P = 005) I two-sided test

gt-Vt II ~Critical valueSamplB size ~

(I) l

2 3 4 5 6 7 8 9 10 12 15 20 I Q11553 )(

4 N1481 6478 7995 8642 8996 9218 9371 9482 9567 9633 9686 9767 9849 9931

5 1115 3851 3900 3917 3925 3930 3lU 3936 3937 3939 3UO 3941 3U3 3945

6 1887 1744 1604 1544 1510 1488 1473 1462 1454 1447 1442 1434 1425 1417

7 2020 1222 1065 9979 9605 9364 9197 9074 8980 8905 8844 8751 8657 8560

8 2126 1001 8434 7764 7388 7146 6978 6853 6757 6681 6619 6525 6428 6329

9 r 2215 8813 7260 6599 6227 5988 5820 5695 5600 5523 5461 5366 5269 5168

2290108073 6542 5890 5523 5285 5119 4995 4899 4823 4761 4666 4568 4467

7571 6OS9 5416 5053 4817 4652 4529 4433 4357 4295 4200 4101 3999 Taken from Outliers in Statistical Data Vic

7209 5715 5078 4718 4484 4320 4197 4102 4026 3964 3868 3769 3667 Barnett androby Lewis 2nd Edition 1984 John

6937 5456 4826 4468 4236 4072 3950 3855 3779 3717 3621 3522 3419 Wiley amp Sons limited6724 5256 4630 4275 4044 3881 3759 3664 3588 3526 3430 3330 3226 I ~6554 5096 4474 4121 3891 3728 3607 3512 3436 3374 3277 3177 3073

6414 4965 4347 3996 3767 3604 3483 3388 3312 3250 3153 3053 2948

6298 4857 4242 3892 3663 3501 3380 3285 3209 3147 3OS0 2949 2844

6200 4765 4153 3804 3576 3415 3293 3199 3123 3060 2963 2862 2756 Table 17 Critical values of X2 (P =005) 6115 4687 4077 3729 3502 3341 3219 3125 3049 2986 2889 2788 2681

Critical valueNumberofdeg~esof~edom6042 4619 4011 3665 3438 3277 3156 3061 2985 2922 2825 2723 2616 5978 4560 3954 3608 3382 3221 3100 30OS 2929 2866 2769 2667 2559

3845922 4508 3903 3559 3333 3172 3OS1 2956 2880 2817 2720 2617 2509 1 5995871 4461 3859 3515 3289 3128 3007 2913 2837 2774 2676 2573 2464 2 7813

I 4 949number of degrees of freedom of the numerator and 112 = number of degrees of freedom of the 1107minator 5 12596 14077 15518 16929 183110

Table A5 Critical values of Q(P =005) for a two-sided test

Sample size Criticaillalue

4 0831 5 0717 6 0621 7 0570

Taken from King E P 1958 J Am Statist Assoc 48 531

I 262 Table A13 The Spearman rank correlation coefficient Table A15 Critical vaLues for C 263

Critical vaLues for p at P = 005 (P = 005) for n = 2

gt-0 -0 rIgt I Q x N

n One-tailed test Two-tailed test

5 0900 1000 6 0829 0886 7 0714 0786 8 0643 0738 9 0600 0700

10 0564 0649 11 0536 0618 12 0504 0587 13 0483 0560 14 0464 0538 15 0446 0521 16 0429 0503 17 0414 0488 18 0401 0472 19 0391 0460 20 0380 0447

Table A14 The Kolmogorov test Critical two-tailed values for a specified distribution and for unspecified normal istributions at P= 005

n Spedjied distributions Unspedjied normal distributions

3 0708 0376 4 0624 0375 5 0563 0343 6 0519 0323 7 0483 0304 8 0454 0288 9 0430 0274

10 0409 0262 11 0391 0251 12 0375 0242 13 0361 0234 14 0349 0226 15 0338 0219 16 0327 0213 17 0318 0207 18 middot0309 0202 19 0301 0197 20 _ 0294 0192

The appropriate value is compared with the maximum difference between the experimental and theoreticaL cumulative frequency middotcurves as described in the text

gt k Critical value -0

-0

3 4 5 6 7 8 9

10

0967 0906 0841 0781 0727 0680 0638 0602

rIgt I Q x N

7

TABLEB Critical values of Students t-distribution

jI a 09 05 04 02 01 005 002 001 0001 a 11

1 1S8 1000 1376 3078 6314 12706 31821 63657 636619 I 2 142 816 1061 1886 2920 4303 6965 9925 31598 2 3 137 765 978 1638 2353 3182 4541 5841 12924 3 4 134 741 941 1533 2132 2776 3747 4604 8610 4 5 132 727 920 1476 2015 2571 3365 4032 6869 5

6 131 718 906 1440 1943 2447 3143 3707 5959 6 7 130 711 896 1415 1895 2365 2998 3499 5408 7 8 130 706 889 1397 1860 2306 2896 3355 5041 8 9 129 703 883 1383 1833 2262 2821 3250 4781 9

10 129 700 879 1372 1812 2228 2764 3169 4587 10

II 129 697 876 1363 1796 2201 2718 3106 4437 11 12 128 695 873 1356 1782 2179 2681 3055 4318 12 13 128 694 870 1350 1771 2160 2650 3012 4221 13 1 128 692 868 1345 1761 2145 2624 2977 4140 14 15 128 691 866 1341 1753 2131 2602 2947 4073 15

16 128 690 865 1337 1746 2120 2583 2921 4015 16 17 128 689 863 1333 1740 2110 2567 2898 3965 17 18 127 688 862 1330 1734 2101 2552 2878 3922 18 19 12 688 861 1328 1729 2093 2539 2861 3883 19 20 127 687 860 1315 1725 2086 2528 2845 3850 20

21 127 686 859 1323 1721 2080 2518 2831 3819 21 22 127 686 858 1321 1717 2074 2508 2819 3792 ~2 23 127 685 858 1319 1714 2069 2500 2807 3767 23 24 127 685 857 1318 1711 2064 2492 2797 3745 24 15 127 684 856 1316 1708 2060 2485 2787 3715 25

26 127 684 856 1315 1706 2056 2479 2779 3707 26 27 127 684 855 1314 1703 2052 2473 2771 3690 27 28 127 683 855 1313 1701 2048 2467 2763 3674 28 29 127 683 854 1311 1699 2045 2462 2756 3659 29 30 127 683 854 1310 1697 2042 2457 2750 3646 30

40 126 681 851 1303 1684 2021 2423 2704 355i 40 60 126 679 848 i296 1671 2000 2390 2660 3460 60

120 126 677 845 1289 1658 1980 2358 2617 3373 120 00 126 674 842 1282 1645 1960 2326middot 2576 3291 QO

Area a corresponding to percentage points comprises two tails of a2 each

3

f 2

1

I

24

TABLED Critical values of the chi-square distribution

995 975 9 5 1 05 025 01 005 001 a a

1 0000 0000 0016 0455 2706 3841 5024 6635 2 0010 0051 0211 1386 4605 5991 7378 9210 3 0072 0216 0584 2366 6251 7815 9348 1t345 4 0207 0484 1064 3357 7179 9488 11143 13277 5 0412 0831 1610 4351 9236 11070 12832 15086

6 0676 1237 2204 5348 10645 12592 14449 16812 7 0989 1690 2833 6346 12017 14067 16013 18475 8 1344 2180 3490 7344 13362 IS507 17535 20090 9 1735 2700 4168 8343 14684 16919 19023 21666

10 2156 3247 4865 9342 15987 18307 20483 23209

11 2603 3816 5578 10341 17275 19675 21920 24725 12 3074 4404 6304 11340 18549 21026 23337 26217 13 3565 5009 7042 12340 19812 22362 24736 27688 14 4075 5629 7790 13339 21064 23685 26119 29141 15 4601 6262 8547 14339 22307 24996 27488 30578

16 5142 6908 9312 15338 23542 26296 28845 32000 17 5697 7564 1008S 16338 24769 27587 30191 33409 18 6265 8231 10865 17338 25989 28869 3lS26 34805 19 6844 8907 11651 18338 27204 30144 32852 36191 20 7434 9591 12443 19337 28412 31410 34170 37566

21 8034 10283 13240 20337 2M15 32670 35479 38932 22 8643 10982 14042 21337 30813 33924 36781 40289 23 9260 11688 14848 22337 32007 35172 38076 41638 24 9886 12401 15659 23337 33196 36415 39364 42980 25 10520 13120 16473 24337 34382 37652 40646 44314

26 11160 13844 17292 25336 35563 38885 41923 45642 27 11808 14573 18114 26336 36741 40113 43194 46963 28 12461 15308 18939 273j6 37916 41337 44461 48278 29 13121 16047 19768 28336 39088 42557 45722 49588 30 13787 16791 20599 29336 40256 43173 46979 50892

31 14458 17539 21434 30336 41422 44985 48232 52191 32 15134 18291 22271 31336 42585 46194 49480 53486 33 15815 19047 23110 32336 43745 47400 50725 54776 34 16501 19806 23952 33336 44903 48602 51966 56061 35 17192 20569 24797 34336 46059 49802 53203 57342

36 17887 21336 25643 35336 47212 5099~ 54437 58619 37 18586 22106 26492 36335 48363 52192 55668 59892 38 19289 22878 27343 37335 49513 53384 56896 61162 39 19996 23654 28196 38335 50660 54572 58120 62428 40 20707 24433 29051 39335 51805 55758 59342 63691

41 21421 25215 29907 40335 52949 56942 60561 64950 42 22138 25999 30765 41335 54090 58124 61177 66206 43 22859 26785 31625 42335 55230 59304 62990 67459 44 23584 27575 32487 43335 56369 60481 64202 68710 45 24311 28366 33350 44335 57505 61656 65410 69957

46 25042 29160 34215 45335 58641 62830 66617 71201 47 25775 29956 35081 46335 59774 64001 67821 72443 48 26511 30755 35949 47335 60907 65171 69023 73683 49 27249 31555 36818 48335 62038 66339 70222 74919 50 27991 32357 37689 49335 63167 67505 71420 76154

7879 10828 10597 13816 12838 16266 14860 18467 16750 20515

18548 22458 20278 24322 21955 26124 23589 27817 25188 29588

26757 31264 28300 32910 29819 34528 31319 36123 32801 37697

34267 39252 35718 40790 37tS6 42312 38582 43820 39997 45315

41401 46797 42796middot 48268 4418t 49728 45558 51179 46928 52620

48290 54052 49645 55476 50993 56892 52336 58301 53672 59703

55003 61098 56329 62487 57649 63870 58964 65247 60275 66619

61582 67985 62884 69346 64182 70703 65476 72055 66766 73402

68053 69336 70616 71893 73166

74437 75704 76969 78231 79490

TABLED

a 995 1 2 3 4 5

6 7 8 9

10

11 12 13 14 15

16 17 18 19 20

51 52 53 54 55

56 57 58 59 60

61 62 63 64 65

66 67 68 69 70

Criti

975

28735 33162 29481 33968 30230 34176 30981 35586 31735 36398

32490 37212 33248 38027 34008 38844 34170 39662 35534 40482

36300 41303 37068 42126 37838 42950 38610 43776 39383 44603

40158 45431 40935 46261 41713 47092 42494 47924 43275

44058 44843 45629 46417 47206

47997 48788 49582 50376 51172

51969 52767 53567 54368 55170

48758

49592 50428 51265 52103 52942

53782 54623 55466 56309 57153

57998 58845 59692 60540 61389 bull 62239 l 63089 63941 64793 65647

66501 67356 68211 69068 69925

70783 71642 72501 73361 74222

25

TABLED Critical values of the chi-square distribution (continued))ution

975 9 5 1 OS 025 01 005 001 a1 005 001 a a995

35 7879 10828 10 10597 13816 45 12838 16266 77 14860 18467 86 16750 20515

12 18548 22458 75 20278 24322 90 21955 26124 66 23589 27877 09 25188 29588

25 26757 31264 17 28300 32910 88 29819 34528 41 31319 36123 78 32801 37697

)() 34267 39252 )9 35718 40790 )5 37156 42312 H 38582 43820 56 39997 45315

J2 41401 46797 ~9 42796 48268 18 44181 49728 JO 45558 51179 14 46928 52620

12 48290 54052 3 49645 55476 8 50993 56892 18 52336 58301 12 53672 59703

II 55003 61098 i6 56329 62487 6 57649 63870 )J 58964 65247 12 60275 66619

19 61582 67985 12 62884 69346 2 64182 70703 8 65476 72055 H 66766 73402

0 68053 74745 )6 69336 76084 9 70616 77419 10 71893 78750 7 73166 80077

)1 74437 81400 13 75704 82720 13 76969 84037 9 78231 85351 4 79490 86661

I 2 3 4 5

6 7 8 9

10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

31 32 33 34 35

36 37 38 39 40

41 42 43 44 45

46 47 48 49 50

51SI 28735 33162 38560 50335 64295 68669 72616 77386 80747 87968 5252 29481 33968 39433 51335 65422 69832 73810 78616 82001 89272 5330230 34776 40308 52335 66548 70993 75002 79843 83253 9057353 5430981 35586 41183 53335 67673 72153 76192 81069 84502 9187254

31735 36398 42060 54335 68796 73311 77380 82292 85749 93168 55S5

32490 37212 42937 55335 69918 74468 78567 83513 86994 94460 5656 S7 33248 38027 43816 56335 71040 75624 79752 84733 88237 95751 57 58 34008 38844 44696 57335 72160 76778 80936 85950 89477 97039 58 59 34770 39662 45577 58335 73279 77931 82117 87166 90715 98324 59 60 35534 40482 46459 59335 74397 79082 83298 88379 91952 99607 60

61 36300 41303 47342 60335 75514 80232 84476 89591 93186 100888 61 62 37068 42126 48226 61335 76630 81381 85654 90802 94419 102166 62 63 37838 42950 49111 62335 77745 82529 86830 92010 95649 103442 63 64 38610 43776 49996 63335 78860 83675 88004 93217 96878 104716 64

39383 44603 50883 64335 79973 84821 89177 94422 98105 105988 6565

66 40158 45431 51770 65335 81085 85965 90349 95626 99331 107258 66 67 40935 46261 52659 66335 82197 87108 9lS19 96828 10055 108526 67 68 41713 47092 53548 67334 83308 88250 92689 98028 10178 109791 68 69 42494 47924 54438 68334 84418 89391 93856 99228 10300 111055 69

7070 43275 48758 55329 69334 85527 90531 95023 10043 10421 112317

71 44058 49592 56221 70334 86635 91670 96189 10162 10543 113577 71 72 44843 50428 57113 71334 87743 92808 97353 10282 10665 114835 72

7373 45629 51265 58006 72334 88850 93945 98516 10401 10786 116092 74 46417 52103 58900 73334 89956 95081 99678 10520 109Q7 117346 74 15 47206 52942 59795 74334 91061 96217 10084 10639 11029 118599 75

76 47997 53782 60690 75334 92166 97351 10200 10758 11 150 119850 76 17 48788 54623 61586 76334 93270 98484 10316 10877 11270 121100 77 78 49582 55466 62483 77334 94373 99617 10432 10996 11391 122348 78

50376 56309 63380 78334 95476 10075 10547 11114 11512 123594 7919 80 51172 57153 64278 79334 96578 10188 10663 11233 11632 124839 80

51969 57998 65176 80334 97680 10301 10778 11351 11752 126082 8181 82 52767 58845 66076 81334 98780 10414 10894 11469 11873 127324 82 83 53567 59692 66976 82334 99880 10527 11009 11588 11993 128565 83 amp4 54368 60540 67876 83334 10098 10639 11124 11706 12113 129804 84 85 55170 61389 68777 84334 10208 10752 11239 11824 12232 131041 85

86 55973 62239 69679 85334 10318 10865 11354 11941 12352 132277 86 87 56777 63089 70581 86334 10428 10977 11469 12059 12472 133512 87 88 57582 63941 71484 87334 10537 11090 11584 12177 12591 134745 88 89 58389 64793 72387 88334 10647 11202 11699 12294 12711 135978 89 90 59196 65647 73291 89334 10756 11315 11814 12412 12830 137208 90

91 60005 66501 74196 90334 10866 11427 11928 12529 12949 138438 91 92 608lS 67356 75101 91334 10976 11539 12043 12646 13068 139666 92 93 61625 68211 76006 92334 11085 11651 12157 12763 13187 140893 93 94 62437 69068 76912 93334 11194 11763 12272 12880 13306 142119 94 95 63250 69925 77818 94334 11304 11875 12386 12997 13425 143344 95

96 64063 70783 78725 95334 11413 11987 12500 13114 13543 144567 96 97 64878 71642 79633 96334 11522 12099 12614 13231 13662 145789 97 98 9865694 72501 80541 97334 11632 12211 12728 13348 13780 147010 99 66510 73361 81449 98334 11741 12323 12842 13464 13899 148230 99

100 10067328 74222 82358 99334 11850 12434 12956 13581 14017 149449

148

TABLE X

n 8 3 00

05 10

4 00 05 10

5 00 05 10

6 00 05 10

7 00 05 10

8 00 05 10

9 00 05 10

10 00 05 10

11 00 05 10

12 00 05 10

13 00 05 10

14 00 05 10

15 00 05 10

16 00 05 10

17 00 05 10

18 00 05 10

19 00 05 10

20 00 05 10

21 00 05 10

Critical values of the ~corrected one- TABLE Xsample Kolmogorov-Smirnov statistic

u

02 01 005 002 001 35477 41811 46702 53456 57900 n 8 39814 46938 54093 61789 66234 22 0053584 63160 70760 78456 82900 0533435 39075 44641 50495 54210 1036765 44022 49894 56387 60924 23 0046154 53829 60468 68377 73409 0531556 37359 42174 47692 51576 10 34698 40945 46328 52718 56853 24 0041172 48153 54273 61133 65692 05 130244 35522 40045 4S440 48988 10 J32704 38466 43593 49407 53327 25 00 J37706 44074 49569 55969 60287 05 128991 33905 38294 43337 46761 10 131005 36464 41200 46701 50438 26 00 135066 40892 46010 51968 55970 05 127828 32538 36697 41522 44819 10 129581 34712 39177 44404 47929 27 00 132925 38365 43160 48732 52519 05 126794 31325 35277 39922 43071 10 128355 33191 37446 42404 45776 28 00 131157 36287 40794 46067 49652 05 125884 30221 34022 38481 41517 10 1127260 31866 35925 40662 43893 29 00 1~29668 34525 38798 43809 47220 05 t25071 29227 32894 37187 40122 10 J26284 30697 34577 39125 42225 30 00 li28388 33008 37084 41864 45127 05 1124325 28330 31869 36019 38856 10 1725410 29648 33376 37751 40738 31 00 1627269 31686 35588 40167 43298 05 1723639 27515 30935 34954 37703 10 1724624 28703 32297 36516 39401 32 00 1626279 30520 34265 38668 41680 05 16123010 26767 30081 33980 36649 10 1723909 27846 31319 35398 38190 33 00 1625395 29478 33086 37331 40238 05 16122430 26077 29296 33083 35679 10 17(23255 27064 30426 34379 37087 34 00 16124600 28541 32026 36128 38940 05 16421895 25439 28570 32256 34784 10 16822653 26347 29608 33446 36076 35 00 15923879 27692 31065 35039 37764 05 16221397 24847 27897 31489 33953 10 16622098 25686 28855 32586 35145 36 00 15723221 26918 30189 34045 36691 05 16020933 24296 27270 30775 33181 10 16421582 25073 28158 31792 34284 37 00 15522617 26208 29386 33134 35707 05 15820498 23781 26685 301OS 32459 10 16221103 24504 27511 31054 33485 38 00 15422060 25553 28646 32295 34801 05 15620089 23298 26137 29484 31784 10 16020656 23973 26908 30366 32741 39 00 15221544 24947 27961 31518 33962 05 15419705 22844 25622 28898 31149 10 15820236 23477 26343 29723 32045 40 00 ISO21064 24384 27325 30796 33182 05 152

10 156