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Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.

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What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?

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Problems&Exercises

By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

1 λ = R 1 n f 2 1 n i 2 λ = 1 R ( n i n f ) 2 n i 2 n f 2 ; n i = 2, n f = 1, size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right ) drarrow λ= { {1} over {R} } left [ { { \( n rSub { size 8{i} } cdot n rSub { size 8{f} } \) rSup { size 8{2} } } over {n rSub { size 8{i} } rSup { size 8{2} } - n rSub { size 8{f} } rSup { size 8{2} } } } right ]; n rSub { size 8{i} } =2" , "n rSub { size 8{f} } =1" ,"} {} so that

λ = m 1.097 × 10 7 ( 2 × 1 ) 2 2 2 1 2 = 1 . 22 × 10 7 m = 122 nm size 12{λ= left ( { {m} over {1 "." "097" times "10" rSup { size 8{7} } } } right ) left [ { { \( 2 "." 1 \) rSup { size 8{2} } } over {4 - 1} } right ]=1 "." "22" times "10" rSup { size 8{ - 7} } " m"="122"" nm"} {} , which is UV radiation.

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Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.

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Look up the values of the quantities in a B = h 2 4 π 2 m e kq e 2 , and verify that the Bohr radius a B is 0.529 × 10 10 m size 12{0 "." "529" times "10" rSup { size 8{ - "10"} } " m"} {} .

a B = h 2 4 π 2 m e kZq e 2 = ( 6.626 × 10 34 J·s ) 2 4 π 2 ( 9.109 × 10 31 kg ) ( 8.988 × 10 9 N · m 2 / C 2 ) ( 1 ) ( 1.602 × 10 19 C ) 2 = 0.529 × 10 10 m

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Verify that the ground state energy E 0 size 12{E rSub { size 8{0} } } {} is 13.6 eV by using E 0 = 2 π 2 q e 4 m e k 2 h 2 . size 12{E rSub { size 8{o} } = { {2π rSup { size 8{2} } q rSub { size 8{e} } rSup { size 8{4} } m rSub { size 8{e} } k rSup { size 8{2} } } over {h rSup { size 8{2} } } } "." } {}

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If a hydrogen atom has its electron in the n = 4 size 12{n=4} {} state, how much energy in eV is needed to ionize it?

0.850 eV

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A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is n size 12{n} {} for a hydrogen atom if 0.850 eV of energy can ionize it?

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Find the radius of a hydrogen atom in the n = 2 size 12{n=2} {} state according to Bohr’s theory.

2.12 × 10 –10 m size 12{2 "." "12" times "10" rSup { size 8{"-10"} } " m"} {}

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Show that 13.6 eV / hc = 1.097 × 10 7 m = R (Rydberg’s constant), as discussed in the text.

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What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

365 nm

It is in the ultraviolet.

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Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.

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Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

No overlap

365 nm

122 nm

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(a) Which line in the Balmer series is the first one in the UV part of the spectrum?

(b) How many Balmer series lines are in the visible part of the spectrum?

(c) How many are in the UV?

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A wavelength of 4 . 653 μm size 12{4 "." "653 μm"} {} is observed in a hydrogen spectrum for a transition that ends in the n f = 5 size 12{n rSub { size 8{f} } =5} {} level. What was n i size 12{n rSub { size 8{i} } } {} for the initial level of the electron?

7

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A singly ionized helium ion has only one electron and is denoted He + size 12{"He" rSup { size 8{+{}} } } {} . What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?

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A beryllium ion with a single electron (denoted Be 3 + size 12{"Be" rSup { size 8{3+{}} } } {} ) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is n size 12{n} {} for the Be 3 + size 12{"Be" rSup { size 8{3+{}} } } {} ion?

(b) How much energy in eV is needed to ionize the ion from this excited state?

(a) 2

(b) 54.4 eV

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Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is C + 5 size 12{C rSup { size 8{+5} } } {} , a carbon atom with only a single electron.

(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?

(b) What is the wavelength of the first line in this ion’s Paschen series?

(c) What type of EM radiation is this?

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Verify Equations r n = n 2 Z a B and a B = h 2 4 π 2 m e kq e 2 = 0.529 × 10 10 m using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.

kZq e 2 r n 2 = m e V 2 r n , so that r n = kZq e 2 m e V 2 = kZq e 2 m e 1 V 2 . From the equation m e vr n = n h 2 π , we can substitute for the velocity, giving: r n = kZq e 2 m e 4 π 2 m e 2 r n 2 n 2 h 2 so that r n = n 2 Z h 2 4 π 2 m e kq e 2 = n 2 Z a B , where a B = h 2 2 m e kq e 2 .

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The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by 1 λ = R 1 n f 2 1 n i 2 size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {} ? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

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Questions & Answers

what is ohm's law
Pamilerin Reply
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
Pamilerin Reply
what is distribution of trade
Grace Reply
what's acceleration
Joshua Reply
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
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Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
Laura
acceleration it is the rate of change in velocity with time
Stephen
acceleration is change in velocity per rate of time
Noara
what is ohm's law
Stephen
Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
Laura
how i don understand
Willam Reply
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
Savannah Reply
How do you determine the magnitude of force
Peace Reply
mass × acceleration OR Work done ÷ distance
Seema
Which eye defect is corrected by a lens having different curvatures in two perpendicular directions?
Valentina Reply
acute astigmatism?
the difference between virtual work and virtual displacement
Noman Reply
How do you calculate uncertainties
Ancilla Reply
What is Elasticity
Salim Reply
using a micro-screw gauge,the thickness of a piece of a A4 white paper is measured to be 0.5+or-0.05 mm. If the length of the A4 paper is 26+or-0.2 cm, determine the volume of the A4 paper in: a). Cubic centimeters b). Cubic meters
Ancilla Reply
what is module
Alex Reply
why it is possible for an object(man) to stay on air without falling down?
akande Reply
its impossible, what do you mean exactly?
Ryan
Exactly
Emmanuella
it's impossible
Your
Why is it not possible to stand in air?
bikko
the air molecules are very light enough to oppose the gravitational pull of the earth on the man..... hence, freefall occurs
Arzail
because of gravitational forces
Pamilerin
this mostly occur in space
Stephen
what is physics
Joshua Reply
no life without physics ....that should tell you something
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Emmanuella
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E=MC^2
study of matter and energy and an inter-relation between them.
Minahil
that's how the mass and energy are related in stationery frame
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Kinetic energy is the energy due to montion of waves,electrons,atoms, molecule,substances an object s.
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Hilal
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Hilal
meteor are the glowy (i.e. heated when the enter into our atmosphere) parts of meteoroids. now, meteoroids are the debris resulting from the collision of asteroids or comets. yes, it occurs in daytime too, but due to the daylight, we cant observe it as clearly as in night
Arzail
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Waka
wich method we use to find the potential on a grounded sphere
Noman
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Pamilerin
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Laura
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Divine
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Laura
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Alpha
Physics generally is the study of everything around us.
Steven
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Practice Key Terms 7

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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