Energy Of An Electron In A Quantum Energy Level

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{3}A_{32} Base Sequence = 12735 String Sequence = 12735 - 3 - 32 **

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Energy Of An Electron In A Quantum Energy Level

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Figure 22.12 presents four quantum energy levels of an arbitrary quantum mechanical system:

(a) Indicate the quantum energy levels associated with the *transition* that requires the most energy.

(b) Indicate the quantum energy levels associated with the *transition* that requires the least energy.

(c) Which *transition* will absorb or emit light with the longest wavelength?

(d) Which *transition* will absorb or emit light with the shortest wavelength?

(e) Suppose the electron of this quantum system is from a nucleus with atomic number 3. Compare the energy of its n = 4 to n = 3 transition with the energy of a n = 2 to n = 1 transition of an electron from a nucleus with atomic number 2.

**The strings**:
S_{7}P_{3}A_{32} (Force - Push)
**The math**:

Pj Problem of Interest is of type *force* (push). Electron is pushed to higher energy level when it absorbs energy and pushed back to the lower energy when it emits energy.

(a) The energy required for the transition from one quantum energy level to another is the difference in energy between the energy levels. The farther the energy level is from the nucleus the greater its energy.

So, in general, the vertical distance between energy levels can be used to rank transition energy.

So, the transitions between n =1 and n = 4 requires the most energy

(b) The transitions between n = 1 and n =2 requires the least energy.

(c) A quantum of energy E = νh = ch/λ is absorbed or emitted when an electron transits from one energy level to another.

So, light with the least energy will have the longest wavelength while light with the most energy will have the shortest wavelength.

So, light absorbed or emitted in transition associated with n =1 and n = 4 will have the shortest wavelength.

(d) Light absorbed or emitted in transition associated with n =1 and n =2 will have the longest wavelength.

(e) The energy E, at quantum energy level n is:

E = -(Z^{2}e^{2})/(2a_{o}n^{2}) ----------(1)

Where Z is atomic number; e is electron charge; a_{o} is Bohr radius; n is quantum energy level.

For Z = 3, and transition from n = 4 to n = 3:

Substituting in equation (1), we have:

E_{4} - E_{3} = 9e^{2}/[2a_{o}(1/9 - 1/16)]

For Z = 2, and transition from n = 2 to n = 1:

Substituting in equation (1), we have:

E_{2} - E_{1} = 4e^{2}/[2a_{o}(1/1 - 1/4)]

Comparing the transition energies:

9e^{2}/2a_{o}[1/9 - 1/16] : 4e^{2}/2a_{o}[1/1 - 1/4]

= 3(1/9 -1/16) = 7/48 = 14.58%.

So, the energy emitted in the n = 4 to n = 3 transition by an electron from a nucleus with atomic number 3 is 14.58% more than the energy emitted in the n = 2 to n = 1 transition by an electron from a nucleus with atomic number 2.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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