Consider an electron for a hydrogen atom in an excited...

Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of electromagnetic radiation that can completely remove (ionize) the electron from the $\mathrm{H}$ atom is $1460 \mathrm{~nm} .$ What is the initial excited state for the electron $(n=?) ?$

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Key Concepts

Ionization Energy

Ionization energy is the minimum energy required to remove an electron completely from an atom, moving it from a bound state to free space. For a hydrogen atom in a particular excited state, the ionization energy is the difference between the energy level of that state and zero energy (the energy of a free electron).

Photon Energy and Wavelength Relationship

The energy of a photon is inversely proportional to its wavelength, as given by the equation E = hc/?, where h is Planck's constant and c is the speed of light. This relationship allows us to convert a given wavelength of light (such as 1460 nm) into the photon energy that can be used to determine the energy required for processes like ionization.

Hydrogen Atom Energy Levels

Hydrogen atoms have quantized energy levels described by the formula E? = -13.6 eV/n², where n is the principal quantum number. This quantization means that an electron in a hydrogen atom can only exist in specific energy states, and transitioning between these states requires absorption or emission of energy equal to the difference between the levels.

Transcript

00:01 All right, good afternoon, problem 72 of chapter 7.

00:03 So, consider an electron for hydrogen dioxide in the excitatory.

00:05 The maximum wavelength of electromagnetic variation that can completely remove or ionize the electron from hydrogen is 1 ,460 nanomeres.

00:13 What is the initial excite say for n.

00:16 So we're going to use a rip equation.

00:17 1 over lambda equals r times 1 over n2 squared minus 1 over n1 squared.

00:22 As we divide both sides by r, and then that will give us 1 over lambda, r is equal to 1 over n2 squared minus 1 over n1 squared.

00:31 N1 is going to be the highest energy.

00:36 So theoretically, it's going to be n equals, that's going to be b at, n is equal to infinity.

00:49 And the n2 is going to be initial excited c.

00:51 So we can rewrite this as 1 over lambda r plus 1 over n1 square equals 1 over n2 squared.

00:56 We take the reciprocal.

01:03 We're going to then, we're going to take the reciprocal and then the square root of both sides or we can raise both sides by negative one half.

01:17 But remember, the limit of n1 is going to be infinity.

01:21 That's going to be as far away from our nucleus as possible.

01:25 So as the limit of n1 as a proportion of infinity is of 1 over n1 squared...

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