The energies for an electron in the K, L, and M shells of...

The energies for an electron in the $K, L,$ and $M$ shells of the tungsten atom are $-69,500 \mathrm{eV},-12,000 \mathrm{eV},$ and $-2200 \mathrm{eV},$ respectively. Calculate the wavelengths of the $K_{a}$ and $K_{\beta}$ x rays of tungsten.

Key Concepts

Atomic Energy Levels

Atoms consist of electrons arranged in discrete energy levels or shells (such as K, L, and M). The specific energies of these levels determine the binding energy of electrons and play a crucial role in phenomena like X?ray emission when vacancies are created in the inner shells.

Characteristic X-rays

Characteristic X-rays are produced when an electron from a higher energy level fills a vacancy in a lower energy shell. The nature of the transition leads to specific X-ray lines (for example, K? and K?), each corresponding to a particular energy difference between the shells involved.

Electronic Transitions in X-ray Production

When a vacancy is created in an inner electron shell, an electron from a higher shell transitions down to fill that vacancy. The energy lost in this transition is emitted as an X-ray photon, with its energy equal to the difference between the two energy levels involved.

Photon Energy-Wavelength Relationship

The energy of an emitted photon is directly related to its wavelength by the equation E = hc/?, where h is Planck's constant and c is the speed of light. This relationship is fundamental in calculating the wavelength of X-rays when the energy difference between the electron shells is known.

Transcript

00:02 Okay, so in this problem, we have described the energies in three different shells.

00:10 So we have the energy for the k shell, which is negative 69.

00:18 This is the energy of the k shell, okay? i just putting the letter k, but this is the energy of the k shell.

00:32 0 .5 calves.

00:38 For the l shell is minus 12 calves and for the m shell is 2 .2 calves.

00:53 Okay and we must find what is the k alpha and k beta beta weta wavelength for the tungsten.

01:12 For tungsten.

01:13 Okay, so again, it's a little confused.

01:15 We have the element tungsten, and we must find the k -alpha and k -beta x -ray wavelength knowing only the energies of the shells.

01:30 Okay, so we must find these two wavelength quantities here.

01:36 So, first of all, what is k -alpha? and what is k beta kbeta so k alpha is an x -ray that is produced in a transition from the l shell to the k -shell okay so when we have this type of transition here we have a k -alpha x -ray being produced and the k -beta x -ray is produced when we have a transition from m shell to the k -shell.

02:19 Okay, so keep that in mind that this is the type of transition that we must reproduce to find the x -ray that we want, the k -alpha and k -beta.

02:31 Therefore, let's begin with k alpha.

02:35 So, k alpha represents this energy transition, which means the k alpha, let's call delta e, which is the energy of this x -ray, is basically the transition from l -e minus the energy of the k -shell.

03:02 The energy of l minus the energy of the k shell.

03:07 And we know the energy of each one of the shells.

03:12 So we just need to do the subtraction.

03:15 And we find that the k alpha has an energy of 57 .5 calves...

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