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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 Andromeda
smart Reply
what is velocity
Fati Reply
displacement per unit time
Murlidhar
the ratec of displacement over time
Jamie
the rate of displacement over time
Jamie
the rate of displacement over time
Jamie
did you need it right now
Pathani
up to tomorrow
Santosh
i need a description and derivation of kinetic theory of gas
Santosh
pls the sum of change in kinetic and potential energy is always what ?
Faith
i need a description and derivation of kinetic theory of gas
Santosh
did you need it right now
Pathani
A few grains of table salt were put in a cup of cold water kept at constant temperature and left undisturbed. eventually all the water tasted salty. this is due to?
Faith Reply
Aunt Faith,please i am thinking the dissolution here from the word "solution" exposed the grains of salt to be dissolved in the water.Thankyou
Junior
dissolution please
Junior
Aunt Faith,please i am thinking the dissolution here from the word "solution" exposed the grains of salt to be dissolved in the water.Thankyou
Junior
it is either diffusion or osmosis. just confused
Faith
due to solvation....
Pathani
what is solvation pls
Faith
water molecule surround the salt molecules . solute solute attraction break in the same manner solvent solvent interaction also break. as a result solute and solvent attraction took place.
Pathani
okay thanks
Faith
my pleasure
Pathani
what is solvation pls
Faith
water act as a solvent and salt act as solute
Pathani
okay thanks
Faith
its ok
Pathani
due to solvation....
Pathani
water molecule surround the salt molecules . solute solute attraction break in the same manner solvent solvent interaction also break. as a result solute and solvent attraction took place.
Pathani
what is magnetism
Eze Reply
physical phenomena arising from force caused by magnets
Mohammad
is the phenomenon of attracting magnetic substance like iron, cobalt etc.
Faith
what is heat
John Reply
Heat is a form of energy where molecules move
saran
Can you please help me with some questions
Janet Reply
topic-- question
Salman
I know this is unrelated to physics, but how do I get the MCQs and essay to work. they arent clickable.
Jake Reply
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
Lilian Reply
why is a body moving at a constant speed able to accelerate
Lilian Reply
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
Lilian
because it changes only direction and the speed is kept constant
Justice
Why is the sky blue...?
Star Reply
It's filtered light from the 2 forms of radiation emitted from the sun. It's mainly filtered UV rays. There's a theory titled Scatter Theory that covers this topic
Mike
A heating coil of resistance 30π is connected to a 240v supply for 5min to boil a quantity of water in a vessel of heat capacity 200jk. If the initial temperature of water is 20°c and it specific heat capacity is 4200jkgk calculate the mass of water in a vessel
fasawe Reply
A thin equi convex lens is placed on a horizontal plane mirror and a pin held 20 cm vertically above the lens concise in position with its own image the space between the undersurface of d lens and the mirror is filled with water (refractive index =1•33)and then to concise with d image d pin has to
Azummiri Reply
Be raised until its distance from d lens is 27cm find d radius of curvature
Azummiri
what happens when a nuclear bomb and atom bomb bomb explode add the same time near each other
FlAsH Reply
A monkey throws a coconut straight upwards from a coconut tree with a velocity of 10 ms-1. The coconut tree is 30 m high. Calculate the maximum height of the coconut from the top of the coconut tree? Can someone answer my question
Fatinizzah Reply
v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
Ramonyai
why is the north side is always referring to n side of magnetic
sam Reply
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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