9. Quantum Mechanics

Bohr Equation

Practice Problem: The Bohr Model and Photon Wavelength

Use the Balmer equation to calculate the wavelength in nano-meters of the spectral line for hydrogen when n = 6 and m = 2. What is the energy in kilojoules per mole of the radiation corresponding to this line?

Calculate the wavelength in nm of the light emitted when an electron makes a transition from an orbital in n = 5 to an orbital in n = 2 in the hydrogen atom. (LO 5.8)
(a) 2.31 * 10-3 nm 
(b) 4.34 * 10-2 nm 
(c) 231 nm
(d) 434 nm

Consider the three electronic transitions in a hydrogen atom
shown here, labeled A, B, and C.
(c) Calculate the wavelength of the photon emitted for each
transition. Do any of these transitions lead to the emission
of visible light? If so which one(s)?
<img src="https://lightcat-files.s3.amazonaws.com/problem_images/b07753e2b74d59c6-1676563976768.jpg" alt="" />
Calculate the wavelength of the photon emitted for transition B.

Louis de Broglie's explanation of Bohr's atomic model

Orbital energies in single-electron atoms or ions, such as He+, can be described with an equation similar to the Balmer–Rydberg equation: <img src="https://lightcat-files.s3.amazonaws.com/problem_images/d05f8b7e6d130f4e-1669825194486.jpg" alt="" />
where Z is the atomic number. What wavelength of light in nanometers is emitted when the electron in He+ falls from n = 3 to n = 2?

Bohr Model of the Hydrogen Atom

An atomic emission spectrum of hydrogen shows three 
wavelengths: 
1875 nm, 1282 nm, and 1093 nm. Assign these 
wavelengths to transitions in the hydrogen atom.

Consider the three electronic transitions in a hydrogen atom
shown here, labeled A, B, and C.
(b) Calculate the energy of the photon emitted for each
transition.
<img src="https://lightcat-files.s3.amazonaws.com/problem_images/eb8ca460df19dcb3-1676556500771.jpg" alt="" />
Calculate the energy of the photon emitted for transition C.

What is the wavelength of a photon (in nm) absorbed during a transition from the n = 2 to n = 5 state in the hydrogen atom?

Determine the end (final) value of n in a hydrogen atom transition, if the electron starts in n = 5 and the atom releases a photon of light with an energy of 4.5738 × 10-19 J.

Calculate the wavelength and energy in kilojoules necessary to completely remove an electron from the second shell (m = 2) of a hydrogen atom (R∞ = 1.097 * 10-2 nm-1).

An atomic emission spectrum of hydrogen shows three
wavelengths:
121.5 nm, 102.6 nm, and 97.23 nm. Assign these
wavelengths to transitions in the hydrogen atom.

The series of emission lines of the hydrogen atom for which 
nf = 4 is called the Brackett series. (b) Calculate the wavelengths 
of the first three lines in the Brackett series—those for which 
ni = 5, 6, and 7.

An electron in the n = 7 level of the hydrogen atom relaxes to a 
lower-energy level, emitting light of 397 nm. What is the value 
of n for the level to which the electron relaxed?

One of the emission lines of the hydrogen atom has a wavelength 
of 94.974 nm. (b) Determine the initial and 
final values of n associated with this emission.

Consider a transition in which the electron of a hydrogen 
atom is excited from n = 1 to n = . (a) What is the end 
result of this transition?

Calculate the frequency of the light emitted when an electron in a
hydrogen atom makes each transition:
a. n = 4¡n = 3
b. n = 5¡n = 1
c. n = 5¡n = 4
d. n = 6¡n = 5

(c) The hydrogen atom now has a single electron in the 3d subshell. What is the energy in kJ>mol required to remove this electron?

The visible emission lines observed by Balmer all involved 
nf = 2. (b) Calculate the wavelengths 
of the first three lines in the Balmer series—those for 
which ni = 3, 4, and 5—and identify these lines in the emission 
spectrum shown in Figure 6.11.

Consider a transition in which the electron of a hydrogen
atom is excited from n = 1 to n = . (b) What is the wavelength of light that must be absorbed to accomplish this process?

Deriving the Bohr Radius of the Atom

(c) The energy required to completely remove the outermost electron from the excited xenon atom is 369 kJ/mol, almost identical to that of cesium (376 kJ/mol). Explain.

Bohr's Equation to Find Photon Energy

In the Bohr model of atomic structure, electrons are constrained to orbit a nucleus at specific distances, given by the equation <img src="https://lightcat-files.s3.amazonaws.com/problem_images/958cf389fcdace33-1670000265914.jpg" alt="" />
where r is the radius of the orbit, Z is the charge on the nucleus, a0 is the Bohr radius and has a value of 5.292 * 10-11 m, and n is a positive integer (n = 1, 2, 3...) like a principal quantum number. Furthermore, Bohr concluded that the energy level E of an electron in a given orbit is <img src="https://lightcat-files.s3.amazonaws.com/problem_images/43efb32fd154214d-1670000261192.jpg" alt="" /> where e is the charge on an electron. Derive an equation that will let you calculate the difference ∆E between any two energy levels. What relation does your equation have to the Balmer–Rydberg equation?

Bohr Equation Example 1

(c) The laser pointer 
emits light because electrons in the material are excited (by 
a battery) from their ground state to an upper excited state. 
When the electrons return to the ground state, they lose the 
excess energy in the form of 532-nm photons. What is the 
energy gap between the ground state and excited state in the 
laser material?

Calculate the wavelength of the light emitted when an electron in a
hydrogen atom makes each transition and indicate the region of the
electromagnetic spectrum (infrared, visible, ultraviolet, etc.) where
the light is found.
a. n = 2¡n = 1
b. n = 3¡n = 1
c. n = 4¡n = 2
d. n = 5¡n = 2

(b) What wavelength of light is emitted by the process?

The Lyman series of emission lines of the hydrogen atom are
those for which nf = 1. (b) Calculate the wavelengths of the first three lines in the Lyman series—those for which ni = 2, 3, and 4.

Consider a transition in which the electron of a hydrogen 
atom is excited from n = 1 to n = . (d) How are the results 
of parts (b) and (c) related to the plot shown in Exercise 6.88?<img src="https://lightcat-files.s3.amazonaws.com/problem_images/87198d946a99e935-1664371544952.jpg" alt="" />

How to... Draw Bohr Models

An electron in a hydrogen atom relaxes to the n = 4 level, emitting
light of 114 THz. What is the value of n for the level in
which the electron originated?

An electron releases energy as it moves from the 6th shell to the 3rd shell. If it releases 4.25 x 109 kJ of energy at a wavelength of 915.7 nm, how many photons were released in the process?

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[BOOK CHEM] Clutch Version

Wavelength and Frequency

Wavelength&nbsp;is the distance from one crest of a wave to another, whereas frequency is the number of waves within a second.&nbsp;

Quantum Numbers: Spin Quantum Number

The Spin Quantum Number deals with the spin of an electron within an orbital.

Speed of Light

The Speed of Light in a vacuum is related to the wavelength and frequency of a light particle.

In the Bohr Equation, the energy and wavelength a photon is related to its energy shell transitions.

Quantum Numbers: Angular Momentum Quantum Number

The&nbsp;Angular Momentum Quantum Number&nbsp;gives the shape of the atomic orbitals.

Emission Spectrum

Emission Spectra is a series of lines formed when emitted light is focused by a slit and passed through a prism. &nbsp;

Introduction to Quantum Mechanics

The 4 Quantum Numbers provide us the coordinates to find the theoretical location of an electron.

Quantum Numbers: Nodes

A node is the region in an atom with zero electron density and where an electron is least likely to exist.

The Energy of Light

The Energy of Light&nbsp;involves the use of a new variable Planck's Constant.

Electromagnetic Spectrum

The Electromagnetic Spectrum is the different forms of radioactive energy that exists in our universe.

Heisenberg Uncertainty Principle

Heisenberg's Uncertainty Principle&nbsp;tries to explain the potential duality of an electron behaving as either a particle or wave.&nbsp;

Quantum Numbers: Number of Electrons

The Quantum Numbers can give both the quantity and location of electrons within a given atom.&nbsp;

Quantum Numbers: Principal Quantum Number

The Principal Quantum Number gives both the size and energy of an electron shell.&nbsp;

Photoelectric Effect

What happens when light strikes the surface of a metal? The Photoelectric Effect&nbsp;can help to explain that.&nbsp;

Quantum Numbers: Magnetic Quantum Number

The Magnetic Quantum Number gives the location of an electron within a specific orbital.

Bohr Model

In the Bohr Model, electrons can move up and down to different orbitals or shells based on absorbing or releasing of energy.&nbsp;

De Broglie Wavelength

The&nbsp;De Broglie Wavelength equation relates wavelength to velocity or speed.&nbsp;