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

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

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.

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

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

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} } } } "." } {}

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

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?

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"} {}

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

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.

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.

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

(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?

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

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?

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

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?

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 .

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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, College physics -- hlca 1104. OpenStax CNX. May 18, 2013 Download for free at http://legacy.cnx.org/content/col11525/1.1
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