# 30.3 Bohr’s theory of the hydrogen atom  (Page 7/14)

 Page 7 / 14

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.

$\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)⇒\lambda =\frac{1}{R}\left[\frac{\left({n}_{\text{i}}\cdot {n}_{\text{f}}{\right)}^{2}}{{n}_{\text{i}}^{2}-{n}_{\text{f}}^{2}}\right];\phantom{\rule{0.25em}{0ex}}{n}_{\text{i}}=2,\phantom{\rule{0.25em}{0ex}}{n}_{\text{f}}=1,\phantom{\rule{0.25em}{0ex}}$ so that

$\lambda =\left(\frac{m}{1.097×{\text{10}}^{7}}\right)\left[\frac{\left(2×1{\right)}^{2}}{{2}^{2}-{1}^{2}}\right]=1\text{.}\text{22}×{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{m}=\text{122 nm}$ , 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 ${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}{}^{}$ , and verify that the Bohr radius ${a}_{\text{B}}$ is $\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}$ .

${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kZq}}_{e}^{2}}=\frac{\left(\text{6.626}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J·s}{\right)}^{2}}{{4\pi }^{2}\left(9.109×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(8.988×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}\text{·}{\text{m}}^{2}/{C}^{2}\right)\left(1\right)\left(1.602×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{C}{\right)}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}$

Verify that the ground state energy ${E}_{0}$ is 13.6 eV by using ${E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}\text{.}$

If a hydrogen atom has its electron in the $n=4$ 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 $n$ for a hydrogen atom if 0.850 eV of energy can ionize it?

Find the radius of a hydrogen atom in the $n=2$ state according to Bohr’s theory.

$\text{2.12}×{\text{10}}^{\text{–10}}\phantom{\rule{0.25em}{0ex}}\text{m}$

Show that $\left(13.6 eV\right)/\text{hc}=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m}=R$ (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 $4\text{.}\text{653 μm}$ is observed in a hydrogen spectrum for a transition that ends in the ${n}_{\text{f}}=5$ level. What was ${n}_{\text{i}}$ for the initial level of the electron?

7

A singly ionized helium ion has only one electron and is denoted ${\text{He}}^{+}$ . 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 ${\text{Be}}^{3+}$ ) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is $n$ for the ${\text{Be}}^{3+}$ 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 ${C}^{+5}$ , 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 ${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}$ and ${a}_{B}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}$ 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.

$\frac{{\text{kZq}}_{e}^{2}}{{r}_{n}^{2}}=\frac{{m}_{e}{V}^{2}}{{r}_{n}}\text{,}$ so that ${r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}{V}^{2}}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\frac{1}{{V}^{2}}\text{.}$ From the equation ${m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\text{,}$ we can substitute for the velocity, giving: ${r}_{n}=\frac{{\text{kZq}}_{e}^{2}}{{m}_{e}}\cdot \frac{{4\pi }^{2}{m}_{e}^{2}{r}_{n}^{2}}{{n}^{2}{h}^{2}}$ so that ${r}_{n}=\frac{{n}^{2}}{Z}\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\frac{{n}^{2}}{Z}{a}_{\text{B}},$ where ${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}$ .

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 $\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)$ ? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

what is thermodynamics
Are the antimatters of Hadrons also Hadrons?!Does the same rule apply to Leptons?
yes. Hadrons are the elementary particles that take part in stong, electromagnetic and weak interactions. Infact only Hadrons are involved in Strong interactions and when an anti-particle of any hadron is produced, it would be a hadron-conservations laws. Leptons are involved in weak int and follow
Lalita
what is physics
physic is a pure science that deal with behavior of matter,energy & how it related to other physical properties
Ridwan
Owk. But am are Art student.
Hussaini
What happens when an aeroplanes window is opened at cruise altitude?
what is the minimum speed for any object to travel in time?
as per theory of relativity, minimum speed will be the speed of light
Mr.
what is physics
it is just a branch of science which deals with the reasons behind the daily activities taking place everyday in our lives. it clearly states the reason in the form of laws.
sandhya
?
lkpostpost2000@yahoo
like Newton's laws , Kepler's laws etc....
sandhya
physics is the study of motion or moving things. Usually the moving things are normal items like vars or planets but sometimes it's electricity or heat that moves.
Jake
what happens when an aeroplane takes off?
it flies
Mr.
the lift generated by the wing overcome the weight of the plane(in Newton)and a net force of upward is created
Phebilia
it is a direct application of Magnus effect (which helps in throwing curve balls) the wings of plane are made in such a way that the net flow of air is more below them rather than on their upper side. So when the plane accelerates, the flaps produce the upward lift when enough velocity is obtained
Mr.
then due to lower pressure on upper part of wings helps producing an additional lift because air flows from areaof lower to the area of higher pressure
Mr.
The engines located under the wings generate thrust .. in relation thrust is a force ... which ovwrcomes or becomes greater than the weight of the plane.. remember weight is a force Weight = m x g-2 So therefore F(thrust) becomes greater than F(weight) Even if by 1Newton the plane starts lifting o
Theophilus
what happens when a ship moves
Williams
What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?
If it reduces the magnitude of the velocity, the acceleration sign is the opposite compared to the velocity.
Nicolas
yes
Williams
what is accerelation
an objects tendency to speed up over time
RayRay
acceleration is the change in velocity over the change in time it would be written delta-v over delta-t.
Shii
the change in velocity V over a period of time T.
Matthew
Delta means "change in"...not period of
Shii
just kidding. it all works mathematically
Shii
except doesn't time really only change if the instantaneous speeds vary...?
Shii
and I assume we are all talking average acceleration
Shii
Hey shiii 😀
the rate of change of velocity is callaed acceleration
Amna
a=delta v/delta t
Amna
the rate of change in velocity with respect to time is acceleration
Nana
nana you r right
Indrajit
good
oguji
what is meant by lost volt
Lost volt. Lol. It is the electrical energy lost due to the nature or the envirommental conditions (temperature and pressure) that affect the cable across which the potential difference is measured.
Theophilus
What is physics?
physics is brance science concerned with nature and properties of matter and energy
George
sure
Okpara
yah....
kashif
physics is study of the natural phenomenon on the basis of certain laws and principles. it's like watching a game of chess and trying to understand its rules how it's played.
Ajit
awesome
Okpara
physics is study of nature and it's law
AMRITA
physics is a branch of science that deals with the study of matter ,properties of matter and energy
Lote
Branch of science (study) of matter, motion and energy
Theophilus
what is a double-slit experiment?Explain.
when you pass a wave of any kind ie sound water light ect you get an interface pattern forming on a screen behind it, where the peaks and troughs add and cancel out due to the diffraction caused by a wave traveling through the slits
Luke
double slit experiment was done by YOUNG. And it's to give out monochromatic coherent, if an incoherent wave is passing through it. And then the waves form interference fringes. The screen placed in front of the double slit is preferably a film and then in the middle where "p=0" a brighter color
navid
is formed and then the constructive interferences occur at 0 (which is the brightest band)... then a sequence of bright band (constructive interference) and dark band (destructive interference) happens and the further from the central band the lower the intensity of bright band(constructive interfe
navid
what is photoelectric effect
the emission of electrons in some materials when light of suitable frequency falls on them
Hardeyyemih
The phenomenon that involves the emission of electrons (photoelectrons) when light of appropriate wavelength and frequency is incident on the surface of a metal.
ibrahim
what is regelation
is the process of melting under pressure and freezing when pressure is reduce
bawire
poisons ratio is which chapter
STREET_
Regelation is the phenomenon of melting under pressure and freezing again when the pressure is reduced
Theophilus
how do i convert energy in MeV/c2 to GeV/c2 and vice versa?
And also from J/s to MeV?I don't quite understand what is in the book,particle physics just in case.
Daniel
what happen to the air molecules in the space between the prongs of the tunning fork?
the air molecules collide each other to make vibration then musical sound will produce.
Abbas