b) Using Moseley’s law, obtain the frequency of an X-ray line when L to K transition
takes place in silver atom. It is given that б = 3.
Answers
Answer:
The frequency
ν
of a characteristic X-ray of an element is related to its atomic number
Z
by
√
ν
=
a
(
Z
−
b
)
,
where
a
and
b
are constants called proportionality and screening (or shielding) constants. For
K
series, the value of
a
is
√
3
R
c
/
4
and that of
b
is 1. Here
R
is Rydberg's constant and
c
is speed of light (as in Bohr's model). For
L
series, the value of
a
is
√
5
R
c
/
36
and
b
is 7.4. The relation and values of
a
and
b
are experimentally determined by Henry Moseley.
Explanation:
Solved Problems on Moseley's Law
Problem from IIT JEE 2003
Characteristic X-rays of frequency
4.2
×
10
18
Hz are produced when transitions from
L
-shell to
K
-shell take place in a certain target material. Use Moseley's law to determine the atomic number of the target material. (Rydberg's constant
=
1.1
×
10
7
m
−
1
.)
Solution:
The characteristic X-ray is emitted when an electron in
L
shell makes a transition to the vacant state in
K
shell. In Moseley's equation,
√
ν
=
a
(
Z
−
b
)
,
the parameter
b
≈
1
for this transition because electron from
L
shell finds nuclear charge
Z
e
shielded by remaining one electron in
K
shell i.e., effective nuclear charge is
(
Z
−
1
)
e
. Thus, by substituting values,
1
λ
=
ν
c
=
4.2
×
10
18
3
×
10
8
=
R
(
Z
−
1
)
2
[
1
n
2
1
−
1
n
2
2
]
=
1.1
×
10
7
(
Z
−
1
)
2
[
1
1
2
−
1
2
2
]
,
which gives,
Z
=
42
. Moseley's law played key role in arrangement of elements in the periodic table and to find many new (missing) elements.
Problem from IIT JEE 2014
If
λ
C
u
is the wavelength of
K
α
X-ray line of copper (atomic number 29) and
λ
M
o
is the wavelength of the
K
α
X-ray line of molybdenum (atomic number 42), the the ratio
λ
C
u
/
λ
M
o
is close to
1.99
2.14
0.50
0.48
Solution:
The wavelength of
K
α
X-ray line is related to atomic number
Z
by Moseley's Formula
1
λ
=
R
(
Z
−
1
)
2
[
1
1
2
−
1
2
2
]
=
3
4
R
(
Z
−
1
)
2
.
Substitute the value of
Z
to get
λ
C
u
λ
M
o
=
(
Z
M
o
−
1
)
2
(
Z
C
u
−
1
)
2
=
(
41
)
2
(
28
)
2
=
2.14.
The elements with higher atomic number (molybdenum in this example) gives high energy X-rays (short wavelengths).
Problem from IIT JEE 2008
Which of the following statements is wrong in the context of X-rays generated from a X-ray tube?
Wavelength of characteristic X-rays decreases when the atomic number of the target increases.
Cut-off wavelength of the continuous X-rays depends on the atomic number of the target.
Intensity of the characteristic X-rays depends on the electrical power given to the X-ray tube.
Cut-off wavelength of the continuous X-rays depends on the energy of the electrons in the X-ray tube.
Solution:
The frequency
ν
of characteristic X-rays is related to atomic number
Z
by Moseley's law,
√
ν
=
a
(
Z
−
b
)
,
which gives
λ
=
c
ν
=
c
a
2
(
Z
−
b
)
2
.
Thus, the wavelength of emitted X-rays decreases with increase in
Z
. The cut-off wavelength of continuous X-rays corresponds to maximum energy of electron in X-ray tube. It is given by
h
c
/
λ
=
e
V
,
where
V
is the accelerating potential. The intensity of X-rays depends on the number of electrons striking the target per second, which, in turn, depends on the electrical power given to the X-ray tube as energy of each electron is
e
V
.
Questions on Moseley's Law
Question: If 178.5 pm is the wavelength of X-ray line of copper (atomic number 29) and 71 pm is the wavelength of the X-ray line of molybdenum (atomic number 42) then the value of a and b in Moseley's equation are
A.
a
=
7
×
10
7
H
z
1
/
2
, b=1.0
B.
a
=
5
×
10
7
H
z
1
/
2
, b=7.4
C.
a
=
5
×
10
7
H
z
1
/
2
, b=1.4
D.
a
=
7
×
10
7
H
z
1
/
2
, b=7.4
Question: Moseley's Law for characteristic X-rays is
√
ν
=
a
(
Z
−
b
)
. In this formula
A. both a and b are independent on the material.
B. a is independent but b depends on the material.
C. b is independent but a depends on the material.
D. both a and b depend on the material.
Related Topic
Characteristic and Continuous X-rays | Problems | IIT JEE
References and External Links
Concepts of Physics Part 2 by HC Verma (Link to Amazon)
IIT JEE Physics by Jitender Singh and Shraddhesh Chaturvedi
X-ray Fluorescence and Moseley’s Law (pdf article on X-rays, Moseley's