for use in the State Examinations
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Formulae and Tables
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Draft for consultation
Observations are invited on this draft booklet of
Formulae and Tables, which is intended to replace
the Mathematics Tables for use in the state
examinations.
In 2007, the State Examinations Commission
convened a working group to review and update the
Mathematics Tables booklet, which is provided to
candidates for use in the state examinations. The
Department of Education and Science and the
National Council for Curriculum and Assessment are
represented on the working group.
The group has carried out some consultation and now
presents this draft for wider consultation.
This draft is available in English only. The final
booklet will be provided in both Irish and English.
Anyone receiving this document should note its draft
status, and should not assume that content currently
in the draft will be included in the final version. The
final version will be circulated to all schools in
advance of its introduction in the state examinations.
The working group recognises that the Mathematics
Tables booklet is used by other institutions in their
own examinations and by various individuals for
other purposes. Whereas the needs of such users will
be noted, final decisions will be taken in the context
of the booklet’s primary purpose as a reference for
candidates taking examinations conducted by the
State Examinations Commission.
Comments should be forwarded by e-mail to
tables@examinations.ie before 31 May 2008.
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Contents
Length and area ....................................................................................................................................................4
Surface area and volume.......................................................................................................................................5
Area approximations.............................................................................................................................................6
Trigonometry ........................................................................................................................................................7
Co-ordinate geometry .........................................................................................................................................10
Geometry ............................................................................................................................................................11
Algebra ...............................................................................................................................................................12
Sequences and series ..........................................................................................................................................12
Number sets – notation .......................................................................................................................................13
Calculus ..............................................................................................................................................................14
Financial mathematics ........................................................................................................................................16
Statistics and probability.....................................................................................................................................18
Units of measurement .........................................................................................................................................26
Common quantities, their symbols and units of measurement ...........................................................................28
Frequently used constants...................................................................................................................................33
Particle physics ...................................................................................................................................................34
Mechanics...........................................................................................................................................................36
Heat and temperature..........................................................................................................................................41
Waves .................................................................................................................................................................41
Electricity............................................................................................................................................................43
Modern physics...................................................................................................................................................45
Electrical circuit symbols ...................................................................................................................................46
The elements.......................................................................................................................................................51
.
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Length and area
Parallelogram
b
h
C
Arc / Sector
Triangle
c
θ
h
r
a
Perimeter = 2a + 2b
Area = ah
= ab sin C
When θ is in radians:
Length of arc = rθ
Circle / Disc
Area of sector =
r
Length of circle = 2 π r
Area of disc = πr 2
1
2
r 2θ
a
Perimeter = a + b + c
Area =
1
2
ah
=
1
2
ab sin C
When θ is in degrees:
⎛ θ ⎞
Length of arc = 2 π r ⎜
⎟
⎝ 360° ⎠
⎛ θ ⎞
⎟
⎝ 360° ⎠
Area of sector = π r 2 ⎜
b
=
C
s ( s − a)( s − b)( s − c) ,
where s =
a+b+c
.
2
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Surface area and volume
Cylinder
Sphere
Solid of uniform cross-section
(prism)
r
r
h
h
Curved surface area = 2 π rh
Volume = π r 2 h
A
Surface area = 4 π r 2
Volume =
4
3
πr
Volume = Ah , where A is the
area of the base.
2
Pyramid on any base
Cone
Frustum of a cone
l
h
r
1
3
πr 2 h
h
R
Curved surface area = π rl
Volume =
h
r
Volume =
1
3
(
2
π h R + Rr + r
A
2
)
Volume =
1
3
Ah , where A is the
area of the base.
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Area approximations
Trapezoidal rule:
Area
≈
h
[ y1 + y n + 2( y 2 + y3 + y 4 + L + y n−1 )]
2
or
≈
h
[first + last + twice the rest ]
2
y1
y2
y3
y4
…
yn
h
Simpson’s rule:
Area
≈
h
[ y1 + y n + 2( y3 + y5 + L + y n−2 ) + 4( y 2 + y 4 + L + y n−1 )] , (where n is odd)
3
or
≈
h
[first + last + twice the odds + four times the evens]
3
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Trigonometry
1
Definitions
sin A
tan A =
cos A
1
cot A =
tan A
1
sec A =
cos A
1
cosec A =
sin A
–1
(cos A, sin A)
A
1
–1
Trigonometric ratios of certain angles
A (degrees)
0°
90°
180°
270°
30°
45°
60°
A (radians)
0
π
2
π
3π
2
π
6
cos A
1
0
–1
0
π
4
1
π
3
1
2
sin A
0
1
0
–1
tan A
0
not
defined
0
not
defined
3
2
1
2
1
3
2
1
2
3
2
1
3
Basic identities
cos 2 A + sin 2 A = 1
cos(− A) = cos A
sin( − A) = − sin A
tan(− A) = − tan A
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Compound angle formulae
Double angle formulae
cos( A + B ) = cos A cos B − sin A sin B
cos 2 A = cos 2 A − sin 2 A
cos( A − B ) = cos A cos B + sin A sin B
sin 2 A = 2 sin A cos A
sin( A + B ) = sin A cos B + cos A sin B
tan 2 A =
2 tan A
1 − tan 2 A
cos 2 A =
1 − tan 2 A
1 + tan 2 A
sin 2 A =
2 tan A
1 + tan 2 A
cos 2 A =
1
2
(1 + cos 2 A)
sin 2 A =
1
2
(1 − cos 2 A)
sin( A − B ) = sin A cos B − cos A sin B
tan( A + B) =
tan A + tan B
1 − tan A tan B
tan( A − B) =
tan A − tan B
1 + tan A tan B
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Products to sums and differences
Trigonometry of the triangle
2 cos A cos B = cos( A + B) + cos( A − B )
2 sin A cos B = sin( A + B ) + sin( A − B )
2 sin A sin B = cos( A − B) − cos( A + B)
cos A + cos B = 2 cos
A+ B
A− B
cos
2
2
cos A − cos B = −2 sin
A+ B
A− B
sin
2
2
b
Area =
B
2 cos A sin B = sin( A + B ) − sin( A − B )
Sums and differences to products
A
c
ab sin C
C
a
Sine rule:
1
2
a
b
c
=
=
sin A sin B sin C
Cosine rule: a 2 = b 2 + c 2 − 2bc cos A
In a right-angled triangle,
A+ B
A− B
sin A + sin B = 2 sin
cos
2
2
sin =
opposite
hypotenuse
A+ B
A− B
sin A − sin B = 2 cos
sin
2
2
cos =
adjacent
hypotenuse
tan =
opposite
adjacent
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Co-ordinate geometry
Line
Slope: m =
Distance =
y 2 − y1
x 2 − x1
( x1 , y1 )
( x 2 − x1 ) 2 + ( y 2 − y1 ) 2
( x2 , y 2 )
⎛ x + x 2 y1 + y 2 ⎞
,
Midpoint = ⎜ 1
⎟
2 ⎠
⎝ 2
Equation of a line: y − y1 = m( x − x1 ) or y = mx + c
Area of a triangle with one vertex at the origin =
1
2
x1 y 2 − x 2 y1
⎛ bx + ax 2 by1 + ay 2 ⎞
,
Point dividing a line segment in the ratio a : b = ⎜ 1
⎟
a+b ⎠
⎝ a+b
Distance from a point to a line =
ax1 + by1 + c
a2 + b2
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Circle
Equation of a circle, centre (h, k ) , radius r: ( x − h) 2 + ( y − k ) 2 = r 2
Circle x 2 + y 2 + 2 gx + 2 fy + c = 0 has centre (− g ,− f ) and radius
Tangent to circle through given point:
g2 + f 2 −c
( x − h)( x1 − h) + ( y − k )( y1 − k ) = r 2 or
xx1 + yy1 + g ( x + x1 ) + f ( y + y1 ) + c = 0
Geometry
Notation
Line through A and B:
AB
Vector operations
v 1 = x1i + y1 j , v 2 = x 2 i + y 2 j
Line segment from A to B:
AB
Scalar product:
Distance from A to B:
AB
Vector from A to B:
AB
Vector from origin O to A:
OA = a
v 1 ⋅ v 2 = x1 x 2 + y1 y 2
= v 1 v 2 cos θ
Norm:
v = x2 + y2
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Algebra
Roots of quadratic equation: x =
− b ± b 2 − 4ac
2a
n
Binomial theorem: ( x + y ) n =
∑
r =0
⎛ n ⎞ n − r r ⎛ n ⎞ n ⎛ n ⎞ n −1
⎛n⎞
⎛n⎞
⎛n⎞
⎜⎜ ⎟⎟ x y = ⎜⎜ ⎟⎟ x + ⎜⎜ ⎟⎟ x y + ⎜⎜ ⎟⎟ x n − 2 y 2 + L ⎜⎜ ⎟⎟ x n − r y r + L + ⎜⎜ ⎟⎟ y n
⎝r⎠
⎝0⎠
⎝1⎠
⎝ 2⎠
⎝r⎠
⎝n⎠
De Moivre’s theorem: [r (cos θ + i sin θ )] = r n (cos nθ + i sin nθ ) = r n e inθ or (r cis θ ) = r n cis(nθ )
n
⎛a b ⎞
⎟⎟ :
Inverse of matrix A = ⎜⎜
⎝c d ⎠
n
1 ⎛ d − b⎞
⎜
⎟ , where det A = ad − bc
det A ⎜⎝ − c a ⎟⎠
Sequences and series
Arithmetic sequence or series
Tn = a + (n − 1)d
Sn =
n
[2a + (n − 1)d ]
2
Geometric sequence or series
Tn = ar n −1
Sn =
a (1 − r n )
1− r
S∞ =
a
, where r < 1
1− r
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Number sets – notation
Natural numbers:
Whole numbers:
Integers:
Rational numbers:
Real numbers:
Complex numbers:
N = {1, 2, 3, 4, L}
W = {0, 1, 2, 3, 4, L}
Z = {L − 3, − 2, − 1, 0, 1, 2, 3, L}
⎧p
⎫
p ∈ Z, q ∈ Z , q ≠ 0 ⎬
Q=⎨
⎩q
⎭
R
C = a + bi a ∈ R, b ∈ R, i 2 = −1
{
}
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Calculus
Differentiation
f (x)
f ′(x)
xn
ex
nx n −1
1
x
ex
e ax
ae ax
ax
cos x
a x ln a
− sin x
sin x
cos x
tan x
sec 2 x
1
ln x
cos −1
x
a
x
a
x
tan −1
a
sin −1
−
2
a −x
1
a2 − x2
a
2
a + x2
Product rule
y = uv ⇒
Quotient rule
dy
dv
du
=u
+v
dx
dx
dx
u
y=
v
⇒
dy
=
dx
v
du
dv
−u
dx
dx
2
v
Chain rule
y = u (v( x)) ⇒
dy du dv
=
dx dv dx
Newton-Raphson Iteration
f ( xn )
x n +1 = x n −
f ′( x n )
Maclaurin series
2
f ( x) = f (0) + f ′(0) x +
f ′′(0) 2
f ( r ) ( 0) r
x +L +
x +L
r!
2!
Taylor series
f ( x + h) = f ( x) + hf ′( x) +
h2
hr
f ′′( x) + L +
f
2!
r!
(r )
( x) + L
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Integration
Constants of integration omitted.
Integration by parts
f (x)
∫ f ( x)dx
x n , (n ≠ −1)
x n +1
n +1
1
x
ex
ln x
cos x
sin x
− cos x
tan x
ln sec x
a x (a > 0)
1
2
a −x
1
2
x + a2
2
(a > 0)
(a > 0)
sin −1
Solid of revolution about x-axis
Volume =
ex
1 ax
e
a
ax
ln a
sin x
e ax
∫ udv = uv − ∫ vdu
∫
x =b
π y 2 dx
x =a
x
a
1
x
tan −1
a
a
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Financial mathematics
In all of the following, t is the time in years and i is annual rate of interest, depreciation or growth, expressed as a
decimal or fraction (so that, for example, i = 0·08 represents a rate of 8%)1.
Compound interest: F = P(1 + i )
t
Present value: P =
F
(1 + i )t
(F = final value. P = principal)
(F = final value. P = present value)
Depreciation – reducing balance method: F = P(1 − i )
t
(P = initial value. F = later value.)
Depreciation – straight line method: Annual depreciation =
initial cost − residual value
useful economic life
Amortisation (mortgages and loans – equal repayments at equal intervals):
R=A
1
i (1 + i ) t
(1 + i ) t − 1
(R = repayment amount, A = amount advanced)
The formulae also apply when compounding at equal intervals other than years. In such cases, t is measured in the relevant
periods of time, and i is the period rate.
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Annual percentage rate (APR) – statutory formula
The APR is the value of i (expressed as a percentage) for which the sum of the present values of all advances is
equal to the sum of the present values of all repayments. That is, the value of i for which the following equation
holds:
k =M
∑
k =1
Ak
=
(1 + i )Tk
j =m
Rj
∑ (1 + i)
j =1
tj
where:
M is the number of advances
Ak is the amount of advance k
Tk is the time in years from the relevant date to advance k
m is the number of repayments
Rj is the amount of repayment j
tj is the time in years from the relevant date to instalment j.
Converting period rate to APR/AER
i = (1 + r ) m − 1
where:
i is the APR or AER (expressed as a decimal)
r is the period rate (expressed as a decimal)
m is the number of periods in one year.
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Statistics and probability
Mean
From list: μ =
Σx
n
From frequency table: μ =
The standard error of the mean is
Σfx
Σf
From list: σ =
Σ( x − μ ) 2
n
From frequency table: σ =
Σf ( x − μ ) 2
Σf
Sampling
The sample mean x is an unbiased estimator of the
population mean μ. The adjusted sample standard
Σ( x − μ ) 2
is an unbiased estimator
deviation s =
n −1
of the population standard deviation σ.
n
standard error of the proportion is
Hypothesis testing
Tukey quick test:
Significance level
Critical value of tail-count
Standard deviation
σ
t-test:
t=
χ -test:
2
pq
.
n
1%
10
0.1%
13
X −μ
⎛ s ⎞
⎜⎜
⎟⎟
⎝ n⎠
k
2
5%
7
and the
χ =
∑
i =1
(Oi − E i ) 2
Ei
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Probability distributions
Binomial distribution:
⎛n⎞
P (r ) = ⎜⎜ ⎟⎟ p r q n − r
⎝r⎠
μ = np
Binomial coefficients
⎛n⎞ n
n!
⎜⎜ ⎟⎟= C r = C (n, r ) =
r!(n − r )!
⎝r⎠
n
Some values of C r :
σ = npq
n
Poisson distribution:
P(r ) = e
−λ
λr
r!
μ =λ
, λ = np
σ= λ
Normal (Gaussian) distribution:
f ( x) =
1
σ 2π
e
−
( x−μ )2
2σ 2
Standard normal distribution:
1
1 − 2 z2
f ( z) =
e
2π
x−μ
Standardising formula: z =
σ
r
0
1
2
3
4
5
6
7
0
1
1
1
1
2
1
2
1
3
1
3
3
1
4
1
4
6
4
5
1
5
10
10
5
1
6
1
6
15
20
15
6
7
1
7
21
35
35
21
7
1
8
1
8
28
56
70
56
28
8
8
1
1
1
9
1
9
36
84
126
126
84
36
9
10
1
10
45
120
210
252
210
120
45
11
1
11
55
165
330
462
462
330
165
12
1
12
66
220
495
792
924
792
495
13
1
13
78
286
715
1287
1716
1716
1287
14
1
14
91
364
1001
2002
3003
3432
3003
15
1
15
105
455
1365
3003
5005
6435
6435
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Area under the standard normal curve
P ( z ≤ z1 ) =
1
2π
∫
z1 − 1 z 2
e 2 dz
−∞
-3
-2
-1
0
1 z 2
1
z1
0·00
0·01
0·02
0·03
0·04
0·05
0·06
0·07
0·08
0·09
0·0
0·5000
5040
5080
5120
5160
5199
5239
5279
5319
5359
0·1
0·2
0·3
0·4
0·5398
0·5793
0·6179
0·6554
5438
5832
6217
6591
5478
5871
6255
6628
5517
5910
6293
6664
5557
5948
6331
6700
5596
5987
6368
6736
5636
6026
6406
6772
5675
6064
6443
6808
5714
6103
6480
6844
5753
6141
6517
6879
0·5
0·6915
6950
6985
7019
7054
7088
7123
7157
7190
7224
0·6
0·7
0·8
0·9
0·7257
0·7580
0·7881
0·8159
7291
7611
7910
8186
7324
7642
7939
8212
7357
7673
7967
8238
7389
7704
7995
8264
7422
7734
8023
8289
7454
7764
8051
8315
7486
7794
8078
8340
7517
7823
8106
8365
7549
7852
8133
8389
1·0
0·8413
8438
8461
8485
8508
8531
8554
8577
8599
8621
3
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Area under the standard normal curve (continued)
z1
0·00
0·01
0·02
0·03
0·04
0·05
0·06
0·07
0·08
0·09
1·1
1·2
1·3
1·4
0·8643
0·8849
0·9032
0·9192
8665
8869
9049
9207
8686
8888
9066
9222
8708
8907
9082
9236
8729
8925
9099
9251
8749
8944
9115
9265
8770
8962
9131
9279
8790
8980
9147
9292
8810
8997
9162
9306
8830
9015
9177
9319
1·5
0·9332
9345
9357
9370
9382
9394
9406
9418
9429
9441
1·6
1·7
1·8
1·9
0·9452
0·9554
0·9641
0·9713
9463
9564
9649
9719
9474
9573
9656
9726
9484
9582
9664
9732
9495
9591
9671
9738
9505
9599
9678
9744
9515
9608
9686
9750
9525
9616
9693
9756
9535
9625
9699
9761
9545
9633
9706
9767
2·0
0·9772
9778
9783
9788
9793
9798
9803
9808
9812
9817
2·1
2·2
2·3
2·4
0·9821
0·9861
0·9893
0·9918
9826
9864
9896
9920
9830
9868
9898
9922
9834
9871
9901
9925
9838
9875
9904
9927
9842
9878
9906
9929
9846
9881
9909
9931
9850
9884
9911
9932
9854
9887
9913
9934
9857
9890
9916
9936
2·5
0·9938
9940
9941
9943
9945
9946
9948
9949
9951
9952
2·6
2·7
2·8
2·9
0·9953
0·9965
0·9974
0·9981
9955
9966
9975
9982
9956
9967
9976
9982
9957
9968
9977
9983
9959
9969
9977
9984
9960
9970
9978
9984
9961
9971
9979
9985
9962
9972
9979
9985
9963
9973
9980
9986
9964
9974
9981
9986
3·0
0·9987
9987
9987
9988
9988
9989
9989
9989
9990
9990
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Chi-squared distribution – inverse values
The table gives the value of k corresponding to the
indicated area A.
That is, given a required probability A, the table gives
the value of k for which P χ 2 > k = A .
(
Degrees of
A = 0·995
freedom
0·0000
1
0·0100
2
0·0717
3
0·2070
4
0·4117
5
0·6757
6
0·9893
7
1·3444
8
1·7349
9
2·1559
10
2·6032
11
3·0738
12
3·5650
13
4·0747
14
A
)
k
0·99
0·975
0·95
0·05
0·025
0·01
0·005
0·0002
0·0201
0·1148
0·2971
0·5543
0·8721
1·2390
1·6465
2·0879
2·5582
3·0535
3·5706
4·1069
4·6604
0·0010
0·0506
0·2158
0·4844
0·8312
1·2373
1·6899
2·1797
2·7004
3·2470
3·8157
4·4038
5·0088
5·6287
0·0039
0·1026
0·3518
0·7107
1·1455
1·6354
2·1673
2·7326
3·3251
3·9403
4·5748
5·2260
5·8919
6·5706
3·8415
5·9915
7·8147
9·4877
11·070
12·592
14·067
15·507
16·919
18·307
19·675
21·026
22·362
23·685
5·0239
7·3778
9·3484
11·143
12·833
14·449
16·013
17·535
19·023
20·483
21·920
23·337
24·736
26·119
6·6349
9·2103
11·345
13·277
15·086
16·812
18·475
20·090
21·666
23·209
24·725
26·217
27·688
29·141
7·8794
10·597
12·838
14·860
16·750
18·548
20·278
21·955
23·589
25·188
26·757
28·300
29·819
31·319
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Chi-squared distribution – inverse values (continued)
Degrees of
A = 0·995
freedom
4·6009
15
5·1422
16
5·6972
17
6·2648
18
6·8440
19
7·4338
20
8·0337
21
8·6427
22
9·2604
23
9·8862
24
10·520
25
11·160
26
11·808
27
12·461
28
13·121
29
13·787
30
20·707
40
27·991
50
35·534
60
43·275
70
51·172
80
59·196
90
67·328
100
0·99
0·975
0·95
0·05
0·025
0·01
0·005
5·2293
5·8122
6·4078
7·0149
7·6327
8·2604
8·8972
9·5425
10·196
10·856
11·524
12·198
12·879
13·565
14·256
14·953
22·164
29·707
37·485
45·442
53·540
61·754
70·065
6·2621
6·9077
7·5642
8·2307
8·9065
9·5908
10·283
10·982
11·689
12·401
13·120
13·844
14·573
15·308
16·047
16·791
24·433
32·357
40·482
48·758
57·153
65·647
74·222
7·2609
7·9616
8·6718
9·3905
10·117
10·851
11·591
12·338
13·091
13·848
14·611
15·379
16·151
16·928
17·708
18·493
26·509
34·764
43·188
51·739
60·391
69·126
77·929
24·996
26·296
27·587
28·869
30·144
31·410
32·671
33·924
35·172
36·415
37·652
38·885
40·113
41·337
42·557
43·773
55·758
67·505
79·082
90·531
101·88
113·15
124·34
27·488
28·845
30·191
31·526
32·852
34·170
35·479
36·781
38·076
39·364
40·646
41·923
43·195
44·461
45·722
46·979
59·342
71·420
83·298
95·023
106·63
118·14
129·56
30·578
32·000
33·409
34·805
36·191
37·566
38·932
40·289
41·638
42·980
44·314
45·642
46·963
48·278
49·588
50·892
63·691
76·154
88·379
100·43
112·33
124·12
135·81
32·801
34·267
35·718
37·156
38·582
39·997
41·401
42·796
44·181
45·559
46·928
48·290
49·645
50·993
52·336
53·672
66·766
79·490
91·952
104·21
116·32
128·30
140·17
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Student’s t-distribution – two-tailed inverse values
The table gives the value of k corresponding to the indicated area A
in the two tails of the distribution.
A
2
That is, the table gives the value of k for which P ( t > k ) = A .
degrees of
freedom
A
2
-k
k
Significance level
20%
10%
5%
2%
1%
0·2%
0·1%
0·02%
0·01%
1
2
3
4
3·078
6·314
12·71
31·82
63·66
318·3
636·6
3183
6366
1·886
1·638
1·533
2·920
2·353
2·132
4·303
3·182
2·776
6·965
4·541
3·747
9·925
5·841
4·604
22·33
10·21
7·173
31·60
12·92
8·610
70·70
22·20
13·03
99·99
28·00
15·54
5
1·476
2·015
2·571
3·365
4·032
5·893
6·869
9·678
11·18
6
7
8
9
1·440
1·415
1·397
1·383
1·943
1·895
1·860
1·833
2·447
2·365
2·306
2·262
3·143
2·998
2·896
2·821
3·707
3·499
3·355
3·250
5·208
4·785
4·501
4·297
5·959
5·408
5·041
4·781
8·025
7·063
6·442
6·010
9·082
7·885
7·120
6·594
10
1·372
1·812
2·228
2·764
3·169
4·144
4·587
5·694
6·211
11
12
13
14
1·363
1·356
1·350
1·345
1·796
1·782
1·771
1·761
2·201
2·179
2·160
2·145
2·718
2·681
2·650
2·624
3·106
3·055
3·012
2·977
4·025
3·930
3·852
3·787
4·437
4·318
4·221
4·140
5·453
5·263
5·111
4·985
5·921
5·694
5·513
5·363
15
1·341
1·753
2·131
2·602
2·947
3·733
4·073
4·880
5·239
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Student’s t-distribution – two-tailed inverse values (continued)
Significance level
degrees of
freedom
20%
10%
5%
2%
1%
0·2%
0·1%
0·02%
0·01%
16
17
18
19
1·337
1·333
1·330
1·328
1·746
1·740
1·734
1·729
2·120
2·110
2·101
2·093
2·583
2·567
2·552
2·539
2·921
2·898
2·878
2·861
3·686
3·646
3·610
3·579
4·015
3·965
3·922
3·883
4·790
4·715
4·648
4·590
5·134
5·043
4·966
4·899
20
1·325
1·725
2·086
2·528
2·845
3·552
3·850
4·539
4·838
21
22
23
24
1·323
1·321
1·319
1·318
1·721
1·717
1·714
1·711
2·080
2·074
2·069
2·064
2·518
2·508
2·500
2·492
2·831
2·819
2·807
2·797
3·527
3·505
3·485
3·467
3·819
3·792
3·768
3·745
4·492
4·452
4·416
4·382
4·785
4·736
4·694
4·654
25
1·316
1·708
2·060
2·485
2·787
3·450
3·725
4·352
4·619
26
27
28
29
1·315
1·314
1·313
1·311
1·706
1·703
1·701
1·699
2·056
2·052
2·048
2·045
2·479
2·473
2·467
2·462
2·779
2·771
2·763
2·756
3·435
3·421
3·408
3·396
3·707
3·689
3·674
3·660
4·324
4·299
4·276
4·254
4·587
4·556
4·531
4·505
30
1·310
1·697
2·042
2·457
2·750
3·385
3·646
4·234
4·482
40
50
60
1·303
1·299
1·296
1·684
1·676
1·671
2·021
2·009
2·000
2·423
2·403
2·390
2·704
2·678
2·660
3·307
3·261
3·232
3·551
3·496
3·460
4·094
4·014
3·962
4·321
4·228
4·169
80
100
1·292
1·290
1·664
1·660
1·990
1·984
2·374
2·364
2·639
2·626
3·195
3·174
3·416
3·390
3·899
3·861
4·095
4·054
∞
1·282
1·645
1·960
2·326
2·576
3·090
3·290
3·719
3·891
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Units of measurement
Base units
The International System of Units (Système International d’Unités) is founded on seven base quantities,
which are assumed to be mutually independent. These base units are:
Base quantity
SI base unit
Symbol for unit
length (l)
metre
m
mass (m)
kilogram
kg
time (t)
second
s
electric current (I)
ampere
A
temperature (T)
kelvin
K
amount of substance (n)
mole
mol
luminous intensity
candela
cd
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Prefixes
Prefixes are used to form decimal multiples and submultiples of SI units. The common prefixes are:
Factor
Prefix
Symbol
Factor
Prefix
Symbol
1012
tera
T
10–12
pico
p
109
giga
G
10–9
nano
n
106
mega
M
10–6
micro
μ
103
kilo
k
10–3
milli
m
h
–2
centi
c
–1
deci
d
10
2
10
1
hecto
deka
da
10
10
The symbol for a prefix is combined with the unit symbol to which it is attached to form a new unit symbol,
e.g. kilometre (km), milligram (mg), microsecond (μs).
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Common quantities, their symbols and units of measurement
Quantity
Symbol
SI unit
Symbol for
SI unit
Non-SI unit used
absorbed dose
D
gray
Gy = J kg–1
acceleration
a
metre per second squared
m s–2
acc. due to gravity
g
metre per second squared
m s–2
activity
A
becquerel
Bq
amount of substance
n
mole
mol
amplitude
A
metre
m
angle
θ
radian
rad
degree (º)
minute (')
second ('')
angular velocity
ω
radian per second
rad s–1
rpm
area
A
metre squared
m2
are (a) = 100 m2
hectare (ha)
=100 a = 10 000 m2
atomic number
Z
capacitance
C
farad
F = C V–1
charge
q
coulomb
C=As
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Quantity
Symbol for
SI unit
Non-SI unit used
mole per litre
mol l–1
ppm; %(w/v), %(v/v)
g cm–3
Symbol
SI unit
concentration
c
critical angle
C
density
ρ
kilogram per metre cubed
kg m–3
displacement
s
metre
m
dose equivalent
H
sievert
Sv = J kg–1
electric current
I
ampere
A
electric field strength
E
volt per metre
V m–1 = N C–1
electronic charge
e
coulomb
C=As
energy (electrical)
W
joule
J=Nm
energy (heat)
Q
joule
J
energy (kinetic)
Ek
joule
J
energy (potential)
Ep
joule
J
joule
J
energy (food)
enthalpy
h
joule
J
focal length
f
metre
m
force
F
newton
N = kg m s–2
kW h
kcal = 4182 J = 1 Cal
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Quantity
Symbol
frequency
f
SI unit
Symbol for
SI unit
hertz
Hz = s–1
half-life
T1/2
second
s
length (distance)
l, s
metre
m
magnetic flux
Φ
weber
Wb
magnetic flux density
B
tesla
T = Wb m–2
magnification
m
mass
m
kilogram
kg
mass number
A
mole per litre
mol l–1
molarity
moment of a force
M
newton metre
Nm
moment of inertia
I
kilogram metre squared
kg m2
momentum
p
kilogram metre per second
kg m s–1
permittivity
ε
farad per metre
F m–1
periodic time
T
second
s
power
P
watt
W = J s–1
pascal
Pa = N m–2
pressure
P, p
Non-SI unit used
tonne (t) = 1000 kg
bar = 105 Pa
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Quantity
Symbol
SI unit
Symbol for
SI unit
refractive index
n
resistance
R
ohm
Ω = V A–1
resistivity
ρ
ohm metre
Ωm
metre per second
m s–1
watt per metre squared
W m–2
speed
sound intensity
sound intensity level
u, v
I
knot = 0.514 m s–1
bel (1 B = 10 dB)
I.L.
specific heat capacity
c
strain
ε
stress
Non-SI unit used
–1
–1
joule per kilogram per kelvin
J kg K
σ
Newton per metre squared
Pa = N m –2
temperature
T
kelvin
K
temperature
t, θ
degree Celsius
°C
change in temperature
Δθ
kelvin
K
tension
T
newton
N
thermal conductivity
k
watt per metre per kelvin
W m–1 K–1
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Quantity
Symbol
SI unit
Symbol for
SI unit
time
t
second
s
torque
T
newton metre
Nm
metre per second
m s–1
volt
V = J C–1
watt per metre squared per
kelvin
W m–2 K–1
velocity
voltage
potential difference
u, v
V
U-value
volume
V
metre cubed
m3
wavelength
λ
metre
m
weight
W
newton
N = kg m s–2
work
W
joule
J=Nm
Young’s modulus
E
Newton per metre squared
Pa = N m –2
Non-SI unit used
minute (min)
day (d)
hour (h)
year (y)
litre (l) = 1000 cm3
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Frequently used constants
Constant
alpha particle mass
Avogadro constant
Boltzmann constant
electron mass
electron volt
electronic charge
Faraday constant
gravitational constant
neutron mass
permeability of free space
Symbol
mα
NA
k
me
eV
e
F
G
mn
μ0
permittivity of free space
ε0
Planck constant
proton mass
proton-electron mass ratio
speed of light in vacuo
h
mp
mp/me
c, c0
universal gas constant
R
Value
6.644 6565 × 10–27 kg
6.022 1415 × 1023 mol–1
1.380 6505 × 10–23 J K–1
9.109 3826 × 10–31 kg
1.602 176 53 × 10–19 J
1.602 176 53 × 10–19 C
96 485.3383 C mol–1
6.6742 × 10–11 m3 kg–1 s–2
1.674 927 28 × 10–27 kg
4π × 10–7 N A–2
8.854 187 817 × 10–12 F m–1
6.626 0693 × 10–34 J s
1.672 621 71 × 10–27 kg
1836.182 672 16
2.997 924 58 × 108 m s–1
8.314 472 J K–1 mol–1
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Particle physics
Class
Leptons
Symbol
electron
e
1
stable
ν
–3
stable
neutrino
muon
tau
pi meson
(mass of electron = 1)
< 10
+
μ μ
–
+
–
+
–
τ τ
π π
207
3500
πo
Mesons
Baryons
Mass
Name
+
–
o
Mean life
Year of
discovery
1897
1956
–6
2 × 10 s
–12
1 × 10
s
–8
1937
1975
273
2.6 × 10 s
1947
264
8.4 × 10–17 s
1947
?
1947
K meson
K K K
≈ 970
proton
p
1836
> 1032 y
1897
neutron
n
1839
960 s
1932
lambda
Λo
2183
2.6 × 10–10 s
1947
sigma
Σ
o
2327
~ 10
s
1953
chi
Ξ+ Ξ– Ξo
2573
~ 10–10 s
1954
omega
+
Ω
–
–
Σ Σ
3272
~ 10
–10
–10
s
1964
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quark
u
s
t
d
c
b
up
strange
top
down
charmed
bottom
antiquark
ū
s̄
t̄
d̄
c̄
b̄
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Mechanics
Linear motion with constant acceleration
v = u + at
s = ut +
2
1
2
2
at
2
v = u +2as
⎛u + v⎞
⎟t
⎝ 2 ⎠
s= ⎜
average speed =
Relative motion
distance
time
sbc = sb − sc
vbc = vb − vc
abc = ab − ac
Momentum of a particle
mv
Newton’s experimental law (NEL)
v1 – v2 = – e(u1 – u2)
Conservation of momentum
m1 u 1 + m2 u 2 = m1 v 1 + m2 v 2
Impulse (change in momentum)
mv – mu
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s
r
Angle in radians
θ=
Angular velocity
ω=
Linear velocity and angular velocity
v = rω
Centripetal acceleration
a = rω 2=
Centripetal force
F = mrω =
Newton’s law of gravitation
F=
Force and acceleration
F = ma
Weight and acceleration due to gravity
W = mg = Vρg ;
Period of a satellite
T2 =
Moment of a force
M = Fd
Torque of a couple
T = Fd
θ
t
v2
r
2
mv 2
r
Gm1m2
d2
g=
GM
R2
4π 2 R 3
GM
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Centres of gravity
Hemisphere, radius r
3
8
r from centre
Hemispherical shell, radius r
1
2
r from centre
Right circular cone
1
4
h from the base
Triangular lamina
1
3
Arc, radius r, angle 2θ
Sector of disc, radius r, angle 2θ
⎛ x + x 2 + x3 y1 + y 2 + y 3 ⎞
,
from base along median = ⎜ 1
⎟
3
3
⎠
⎝
r sin θ
θ
2r sin θ
3θ
Moments of Inertia
Uniform rod, length 2l
Centre:
1
3
ml 2 One end:
4
3
ml 2
Uniform disc, radius r
Centre:
1
2
mr 2 Diameter:
1
4
mr 2
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Uniform hoop, radius r
Centre: mr 2 Diameter:
Uniform solid sphere, radius r
Diameter:
Parallel axis theorem
2 2
mr
5
2
I b = I c + m bc
Perpendicular axis theorem
IZ = I X + I y
Hooke’s law
F = – ks
Simple harmonic motion
a=–ωs
1
2
mr 2
2
T=
1
f
=
2π
ω
s = A sin (ω t + α )
2
2
v = ω ( A 2 – s2)
Simple pendulum
T = 2π
l
g
Compound pendulum
T = 2π
I
mgh
Work
W = Fs =
∫ Fds
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Potential (gravitational) energy
Ep = mgh
Kinetic energy
Ek =
Principle of conservation of mechanical energy
Ep + Ek = constant
Mass-energy equivalence
E = mc 2
Power
P=
1
2
mv 2
W
= Fv
t
power output × 100
Percentage efficiency
power input
Young’s modulus
E=
σ
ε
Stress
σ=
F
A
Strain
ε=
Δl
l
Density
ρ=
m
V
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F
A
Pressure
p=
Pressure in a fluid
p = ρgh
Thrust on an immersed plane surface
T = A pcentroid
Boyle’s law
pV = constant
Heat and temperature
Celsius temperature
t / ˚C = T /K – 273.15
Energy needed to change temperature
∆ E = mc ∆ θ
∆E = C∆θ
Energy needed to change state
∆ E = ml
∆E = L
Waves
Velocity of a wave
c = fλ
Doppler effect
f ′=
Fundamental frequency of a stretched string
f =
fc
c±u
1
T
2l
μ
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Mirror and lens formula
1
f
=
1
u
+
1
v
v
u
Magnification
m=
Power of a lens
P=
Two lenses in contact
P = P1 + P2
Refractive index
n=
n=
Diffraction grating
1
f
sin i
real depth
=
sin r apparent depth
1
sin C
=
c1
c2
nλ = d sin θ
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Electricity
Coulomb’s law
F=
q1 q2
4 πε d 2
Electric field strength
E=
F
q
Potential difference
V=
W
q
Ohm’s law
V = IR
Resistivity
ρ=
Resistors in series
R = R1 + R2
Resistors in parallel
1
R
=
1
RA
l
1
R1
+
Wheatstone bridge
R 1 R3
=
R2 R4
Joule’s law
P = RI 2
1
R2
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Force on a current-carrying conductor
F = IlB
Force on a charged particle
F = qvB
Induced emf
E=−
Alternating voltage and current
Vrms =
Capacitance
C=
q
V
Parallel-plate capacitor
C=
Aε 0
d
Energy stored in capacitor
W=
Power
P = VI
Magnetic flux
Φ = BA
Transformer
Vi N p
=
Vo N s
dΦ
dt
1
2
V0
2
I rms =
I0
2
CV 2
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Modern physics
Energy of a photon
E = hf
Einstein’s photoelectric equation
2
h f = Φ + mvmax
Law of radioactive decay
A = λN
Half-life
T1 2 =
Mass-energy equivalence
1
2
ln 2
λ
2
E = mc
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Electrical circuit symbols
Conductors
junction of conductors
conductors crossing with
no connection
Switches
push-to-make switch
push-to-break switch
two-way switch
(SPDP)
dual on-off switch
(DPST)
normally open
on-off switch (SPST)
normally closed
on-off switch (SPST)
dual two-way switch
(DPDT)
relay
(SPST)
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Power
cell
photovoltaic cell
battery
d.c. supply
a.c. supply
transformer
fuse
earth
variable resistor
(rheostat)
preset variable resistor
potential divider
Resistors
fixed resistor
thermistor
light-dependent resistor
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Capacitors
capacitor
electrolytic capacitor
(polarised capacitor)
variable capacitor
preset variable capacitor
(trimmer)
diode
Zener diode
photodiode
light-emitting diode
(LED)
Diodes
Meters
Ω
voltmeter
galvanometer
ammeter
ohmmeter
oscilloscope
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Amplification
pnp-junction transistor
light-sensitive transistor
amplifier
(block symbol)
microphone
earphone
loudspeaker
bell
buzzer
piezoelectric transducer
aerial (antenna)
filament lamp
signal lamp
neon lamp
npn-junction transistor
Audio
Lamps
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Other devices
M
motor
heater
inductor
inductor with
ferromagnetic core
AND
NOT
(inverter)
NAND
Logic gates
OR
NOR
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The elements
Periodic table of the elements
1
18
1
2
H
He
1.008
2
13
14
15
16
17
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
6.941
9.012
10.81
12.01
14.01
16.00
19.00
20.18
11
12
13
14
15
16
17
18
4.003
Na
Mg
A1
Si
P
S
Cl
Ar
22.99
24.31
3
4
5
6
7
8
9
10
11
12
26.98
28.09
30.97
32.07
35.45
39.95
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
39.10
40.08
44.96
47.87
50.94
52.00
54.94
55.85
58.93
58.69
63.55
65.41
69.72
72.64
74.92
78.96
79.90
83.80
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
85.47
87.62
88.91
91.22
92.91
95.94
(97.90)
101.1
102.9
106.4
107.9
112.4
114.8
118.7
121.8
127.6
126.9
131.3
55
56
57
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
132.9
137.3
138.9
178.5
180.9
183.8
186.2
190.2
192.2
195.1
197.0
200.6
204.4
207.2
209.0
87
88
89
104
105
106
107
108
109
110
111
112
113
114
115
Fr
Ra
Ac
Rf
Db
Sg
Bh
Hs
Mt
Ds
Rg
(209.0) (210.0) (222.0)
116
117
118
Uub Uut* Uuq Uup* Uuh Uus* Uuo
(223.0) (226.0) (227.0) (261.1) (262.1) (266.6) (264.1) (277.0) (268.1) (271.0) (272.2) (285.0)
(289.0)
(289.0)
(293.0)
* These elements have not yet been detected (2008).
See page 54 for the Lanthanoid and the Actinoid Series.
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First ionisation energies of the elements
(in kJ mol–1)
1
18
1
2
H
He
1312
2
13
14
15
16
17
2372
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
520.2
899.5
800.6
1087
1402
1314
1681
2081
11
12
13
14
15
16
17
18
Na
Mg
495.8
737.7
3
4
5
6
7
8
9
10
11
12
A1
Si
P
S
Cl
Ar
577.5
786.5
1012
999.6
1251
1521
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
418.8
589.8
633.1
658.8
650.9
652.9
717.3
762.5
760.4
737.1
745.5
906.4
578.8
762.0
947.0
941.0
1140
1351
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
403.0
549.5
600.0
640.1
652.1
684.3
702.0
710.2
719.7
804.4
731.0
867.8
558.3
708.6
834.0
869.3
1008
1170
55
56
57
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
375.7
502.9
538.1
658.5
761.0
770.0
760.0
840.0
880.0
870.0
890.1
1007
589.4
715.6
703.0
812.1
890±40
1037
87
88
89
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
Fr
Ra
Ac
Rf
Db
Sg
Bh
Hs
Mt
Ds
Rg
380.0
509.3
499.0
580.0
––
––
––
––
––
––
––
Uub Uut* Uuq Uup* Uuh Uus* Uuo
––
––
––
––
* These elements have not yet been detected (2008).
See page 54 for the Lanthanoid and the Actinoid Series.
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Electronegativity values of the elements
(Using the Pauling scale)
1
18
1
2
H
He
2.20
2
13
14
15
16
17
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
0.98
1.57
2.04
2.55
3.04
3.44
3.98
––
11
12
13
14
15
16
17
18
––
Na
Mg
A1
Si
P
S
Cl
Ar
0.93
1.31
3
4
5
6
7
8
9
10
11
12
1.61
1.90
2.19
2.58
3.16
––
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
0.82
1.00
1.36
1.54
1.63
1.66
1.55
1.83
1.88
1.91
1.90
1.65
1.81
2.01
2.18
2.55
2.96
––
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
0.82
0.95
1.22
1.33
1.60
2.16
2.10
2.20
2.28
2.20
1.93
1.69
1.78
1.96
2.05
2.10
2.66
2.60
55
56
57
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
0.79
0.89
1.10
1.30
1.50
1.70
1.90
2.20
2.20
2.20
2.40
1.90
1.80
1.80
1.90
2.00
2.20
––
87
88
89
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
Fr
Ra
Ac
Rf
Db
Sg
Bh
Hs
Mt
Ds
Rg
0.70
0.90
1.10
––
––
––
––
––
––
––
––
Uub Uut* Uuq Uup* Uuh Uus* Uuo
––
––
––
––
* These elements have not yet been detected (2008).
See page 54 for the Lanthanoid and the Actinoid Series.
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Periodic table of the elements
Lanthanoid Series
Actinoid Series
58
59
60
61
62
63
64
65
66
67
68
69
70
71
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
140.1
140.9
144.2
(144.9)
150.4
152.0
157.3
158.9
162.5
164.9
167.3
168.9
173.0
175.0
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
232.0
231.0
238.0
(237.0)
(244.1)
(243.1)
(247.1)
(247.1)
(251.1)
(252.1)
(257.1)
(256.0)
67
68
69
(259.1) (262.1)
First ionisation energies of the elements
(in kJ mol–1)
58
Lanthanoid Series
Actinoid Series
59
60
61
62
63
64
65
66
70
71
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
534.4
527.0
533.1
540.0
544.5
547.1
593.4
565.8
573.0
581.0
589.3
596.7
603.4
523.5
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
587.0
568.0
597.6
604.5
584.7
578.0
581.0
601.0
608.0
619.0
627.0
635.0
642.0
470.0
66
67
68
69
70
71
Electronegativity values of the elements
(Using the Pauling scale)
58
Lanthanoid Series
Actinoid Series
59
60
61
62
63
64
65
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
1.12
1.13
1.14
––
1.17
––
1.20
––
1.22
1.23
1.24
1.25
––
1.00
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
1.30
1.50
1.70
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
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