Calculate the frequency, energy (in keV), and wavelength of the Kαx ray for the elements (a) cal

Calculate the frequency, energy (in keV), and wavelength of...

Key Concepts

Characteristic X-rays

Characteristic x-rays are produced when an electron makes a transition from a higher energy level to fill a vacancy in an inner shell of an atom. The K? x-ray specifically results from an electron dropping from the L shell (n=2) to the K shell (n=1), releasing energy characteristic of that element.

Electron Energy Level Transitions

Within the Bohr model and quantum theory, electrons occupy quantized energy levels. When an electron transitions between these levels, the energy difference is emitted as a photon. The energy, frequency, and wavelength of the photon correspond directly to the difference in energy levels, making these transitions crucial for understanding characteristic x-ray emissions.

Planck’s Relation and Photon Properties

The energy (E), frequency (f), and wavelength (?) of a photon are interrelated by Planck’s equation, E = hf, and by the relationship between frequency and wavelength, f = c/?, where h is Planck's constant and c is the speed of light. This set of equations allows us to convert between the energy of the emitted x-ray, its frequency, and its wavelength.

Atomic Number and Effective Nuclear Charge

The energy of the emitted x-ray is influenced by the atomic number (Z) of the element, as a higher nuclear charge increases the binding energy of electrons in inner shells. However, the effective nuclear charge experienced by the electrons is reduced due to electron shielding, a concept that is encapsulated by empirical laws such as Moseley’s law in x-ray spectroscopy.

Transcript

00:01 For the following, we're asked to find the frequency, the energy in terms of k -e -v, and the wavelength.

00:09 So frequency is given by the moseley's law, which says that frequency f is equal to 2 .4a times 10 to 15 hertz times the atomic number of z -minus 1 squared.

00:21 Once we know frequency, we can get energy because energy e is equal to planks constant times the frequency.

00:27 Once we know the frequency, we can also get the wavelength, because we know the frequency.

00:30 Wavelength is equal to the speed of light, c divided by the frequency f.

00:34 So for z equals 20, we simply have to plug in 20 for a value of z into the equation for frequency.

00:40 And in this case, we find that the frequency is equal to 3 .7 times 10 to the third hertz, which is the same as 3 .7, or excuse me, 3 .7 times 10 to the third electron volts, which is the same as 3 .7 keb, kile electron volts.

00:55 We're asked to put things in units of kev, so let's write it in terms of kb.

01:01 Next, calculating the energy, the energy is equal to planck's constant, which is equal to 4 .136 times 10 to the minus 15 electron volts seconds times the frequency.

01:17 Oh, i'm sorry, and i made a mistake.

01:19 So i said frequency is 3 .7 keb.

01:22 That's actually the energy.

01:23 I got a little ahead of myself, and you probably notice that when you plug that in that that didn't work.

01:29 So again, calculating frequency 2 .48 times 10 to 15 hertz times z minus 1 z is 20 so this is 20 minus 1 squared gives you 8 .95 times 10 to the 17 hertz sorry about that so 8 .95 times 10 to the 17 hertz is the frequency that we're dealing with here so now that we know frequency in units of hertz we can calculate the energy because energy is equal to planks constant times the frequency planks constant is 4 .136 times 10 to the minus 15 electron volt seconds.

02:15 So this gives us our 3 .7 times 10 to the third electron volts, or 3 .7 k .eb.

02:26 And we can also calculate the wavelength from the frequency, since wavelength lambda is equal to the speed of light divided by the frequency, the speed of light is three times 10 to the 8 meters per second.

02:36 So this gives us 3 .35 times 10 to the minus 10 meters.

02:57 We can box that in as our solution for a, the final part.

03:02 Now for b, we, we'll be, we're going to we're doing the exact same thing, but this time z is equal to 27...

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