A photon with a wavelength in the visible region (between 400 and 700 nm ) causes a transition

A photon with a wavelength in the visible region (between...

Key Concepts

Photon Energy and Wavelength Relationship

This concept explains the correspondence between the energy of a photon and its wavelength, governed by the relation E = hc/?. The visible region of the electromagnetic spectrum corresponds to a particular range of energies, meaning that only transitions with energy differences that result in photon wavelengths between 400 nm and 700 nm will be observable as visible light. This relation underpins the calculation to determine which electron transitions fall within the visible spectrum.

Quantized Energy Transitions

In quantum mechanics, electrons in atoms occupy discrete energy states, and a change from one state to another involves absorbing or emitting energy equal to the difference between these levels. When an electron transitions between adjacent levels (from n to n+1), the energy difference can be calculated using the quantized energy level formula, highlighting the selection of specific photon energies that induce such transitions.

Hydrogen-like Atomic Systems

This concept pertains to the description of ions with a single electron orbiting a nucleus, where the energy levels are similar to those in hydrogen but scaled by the square of the nuclear charge. The energy levels are quantized according to an n^2 relation in the denominator, typically expressed by formulas such as E? = -Z²(13.6 eV)/n². This model is essential for understanding the spectral lines and transitions in ions such as doubly ionized lithium.

Transcript

00:01 For this exercise, we have a doubly ionized lithium atom, which is initially at the nth excited state, and then a photon whose wavelength lies between 400 nanometers and 700 nanometers.

00:19 This photon comes and excites the atom to the n plus 1 energy level.

00:25 And the exercise asks us to calculate, which is the smallest value of n for which this transition is possible.

00:33 So the first thing we're going to need to calculate is what are the energies of the photon that comes? what are the possible energies? so the smallest energy that the photon can have is going to happen when its wavelength is maximum.

00:49 So the smallest energy, the e -min, the minimum energy, is going to be h .c over the maximum wavelength.

01:00 And hc is 1 ,212 electrons nanometers, the maximum wavelength is 700 nanometers, and the result is 1 .77 electron volts.

01:20 This is the minimum energy, and the maximum energy is going to happen when the wavelength is minimum, which is 1240 over 400, which is 3 .1 .1 .3 .000.

01:37 Electron volts.

01:39 Okay, so all we have is that the energy of the photon lies between.

01:45 So 1 .77 is smaller than or equal to the energy of the photon, which is smaller or equal to 3 .1.

01:57 This is all in electron volts.

01:59 So let me just highlight this.

02:02 We're going to need this later.

02:05 Okay, so the energy of the photon lies in this range.

02:08 And we have to calculate what are.

02:11 Are the energies between the different levels and between the levels and n plus one.

02:26 We have to calculate the different energies for different ends and see which energy, which is the smallest value of n whose energy lies in this range between 1 .77 and 3 .1 electron volts.

02:41 So let's do it.

02:43 First, remember that the energy of a hydrogen -like atom, of the nth level of a hydrogen like atom, is given by minus the atomic number squared over n squared times 13 .6 electron volts...

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