Draw an energy-level diagram, similar to Figure 39.24, for...

Draw an energy-level diagram, similar to Figure $39.24,$ for the He $^{+}$ ion. On your diagram: a. Show the first five energy levels. Label each with the values of $n$ and $E_{n}$ b. Show the ionization limit. c. Show all possible emission transitions from the $n=4$ energy level. d. Calculate the wavelengths (in $\mathrm{nm}$ ) for each of the transitions in part $c$ and show them alongside the appropriate arrow.

Draw an energy-level diagram, similar to Figure $39.24,$ for the He $^{+}$ ion. On your diagram:
a. Show the first five energy levels. Label each with the values of $n$ and $E_{n}$
b. Show the ionization limit.
c. Show all possible emission transitions from the $n=4$ energy level.
d. Calculate the wavelengths (in $\mathrm{nm}$ ) for each of the transitions in part $c$ and show them alongside the appropriate arrow.

Key Concepts

Energy-Level Diagram

An energy-level diagram is a visual representation of the discrete energy states available to an atom or ion. It typically shows the quantized energy levels as horizontal lines, with transitions between these levels depicted by arrows. This diagram helps in understanding how electrons move between different energy states and how energy is absorbed or emitted during these transitions.

Quantization of Energy in Hydrogen-Like Systems

In hydrogen-like systems, which include ions with a single electron, the energy levels are quantized and can be described by a formula that depends on the principal quantum number (n) and the nuclear charge (Z). The energies are inversely proportional to the square of n, indicating that as electrons reside in higher levels (larger n), the energy differences become smaller.

Ionization Limit

The ionization limit refers to the threshold energy required to remove an electron completely from an atom or ion, causing it to become ionized. In an energy-level diagram, this limit is represented as the highest level or the continuum boundary above which the electron is no longer bound to the nucleus.

Electron Emission Transitions

Emission transitions occur when an electron moves from a higher energy level to a lower one, releasing energy in the form of a photon. Each transition corresponds to a specific photon energy (and thus wavelength) determined by the difference in energy between the initial and final states.

Rydberg Formula and Wavelength Calculation

The Rydberg formula is used to calculate the wavelengths of the photons emitted or absorbed during electron transitions in hydrogen-like atoms. It relates the difference between the reciprocals of the squares of the principal quantum numbers of the initial and final levels to the wavelength of the photon, making it fundamental for determining spectral lines.

Transcript

00:01 For this question we are looking at the ion helium plus.

00:05 Now helium plus, its nuclei contains two protons and its original atom is two electrons, but since it is a plus, it means it has lost an electron.

00:22 So it only has one electron in the orbit.

00:25 And we can see that it is actually a hydrogen -like ion.

00:29 So we can use the equation 39 .37, which tells us the energy levels for hydrogen -like ions.

00:41 You can estimate them to be negative z squared times 13 .6 ev, 5xn square.

00:51 In our case, the z is the number of protons that the nucleus has, and for helium plus, that is 2.

01:03 So what we get over here, we get 2 square, so this everything must multiply by 4.

01:11 We get...

01:17 Right, so this will describe the energy levels for our helium plus ion.

01:24 Now we just have to substitute the correct value of n into the...

01:30 To get our different energy levels.

01:34 So what we want is the first 5 energy levels.

01:39 So we have to substitute n equals to 1, 2, 3, 4, 5.

01:55 And the values that we get, i'm going to just put it over here.

02:05 We have negative 13 .6.

02:19 And finally, substituting n equals to 5, we should get minus 2 .2.

02:26 All of these energies are in terms of electron volts.

02:32 Right, then the quantum number, i label them on the left.

02:35 Left.

02:39 Alright, so the ionization energy, because all of these values are with respect to the ionization level, that is the ionization level is 0, bv.

02:53 So the ionization energy, if you were to start from the ground state, it would just be 0 minus negative 54 .4, which would give us 504...

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