Draw to careful scale an energy-level diagram for hydrogen...

Draw to careful scale an energy-level diagram for hydrogen for levels with $n=1$, $2,3,4, \infty .$ Show the following on the diagram: $(a)$ the limit of the Lyman series, $(b)$ the $\mathrm{H}_{\beta}$ line, $(c)$ the transition between the state whose binding energy $(=$ energy needed to remove the electron from the atom) is $1.51 \mathrm{eV}$ and the state whose excitation energy is $10.2 \mathrm{eV},$ and $(d)$ the longest wavelength line of the Paschen series.

Key Concepts

Line Transitions

Line transitions are the processes in which an electron moves between energy levels, emitting or absorbing a photon whose energy corresponds to the difference between these levels. These transitions define the spectral lines seen in emission or absorption spectra, such as the H? line in the Balmer series which represents a specific transition where the wavelength of the emitted light falls within the visible region.

Energy Levels

The energy levels in a hydrogen atom are quantized states in which the electron can exist, each characterized by a principal quantum number (n). The energy associated with each level is described by the formula E = -13.6 eV/n² for hydrogen, meaning that as n increases, the energy becomes less negative and the electron is less tightly bound to the nucleus.

Quantum Numbers

The principal quantum number (n) is the key descriptor of energy levels in an atom. It defines the main energy state of the hydrogen electron and determines the size of the orbital. In diagrams, levels are typically labeled with increasing n (1, 2, 3, ...), where n = 1 is the ground state and higher n values correspond to excited states.

Spectral Series

Spectral series in hydrogen refer to groups of electronic transitions that share a common lower energy level. For example, the Lyman series ends at n = 1 and involves transitions from higher n to n = 1, the Balmer series has a common lower level of n = 2, and the Paschen series converges at n = 3. Each series is associated with a characteristic region of the electromagnetic spectrum.

Binding and Excitation Energy

Binding energy is the energy required to remove an electron from an atom, while excitation energy is the energy absorbed to raise an electron to a higher energy level. Understanding these energies is crucial for interpreting diagrams where specific energy differences are noted, such as between states having a particular binding energy and excitation energy.

Transcript

0:00 Hi there.

00:01 So for this problem, we're told that a hydrogen atom in a state having abiding energy, this is the energy required to remove an electron of 0 .85 electron balls made a transition to a state with an excitation energy.

00:26 This is the difference in energy between the state and the ground state of 10.

00:33 Two electron balls.

00:39 Now for part a of this problem, we are asked to find the energy of the emitted photon.

00:48 Now, the energy of the state which corresponds to abiding energy of 0 .85 will be the initial energy, and that is minus 0 .85 electron volts.

01:04 And as the states of the hydrogen atom in the bourne's model have energy given by the relation that we know it is equal to the energy of the n -state is equal to minus 13 .6 and this divided by n squared, this in electron bolts, of course.

01:27 So the state with energy, with the initial energy, will correspond to the quantum number.

01:38 So we solved for n in the previous expression, so we will have that.

01:41 That is 13 .6, and this divided by 0 .85, which is the value given.

01:50 And we need to take the square root of this, so that will be to elevate this to 1 divided by 2.

01:55 So from this, we obtain that n is equal to 4.

02:00 And it is given that the hydrogen atom makes a transition to the state with excitation energy 10 .2 electron balls.

02:10 So the energy of this state will do.

02:14 So the final energy in that case will be minus 13 .6 plus 10 .2 electron balls.

02:23 So that will give us that the final energy is minus 3 .4 electron balls.

02:28 So the quantum number corresponding to this state, we do something similar as before.

02:33 The quantum number in this case is going to be 13 .6 divided by 3 .4.

02:40 This elevated to 1 over 2 because we need to take the square root of this and then this will give us a value of 2.

02:47 Now, the energy of the hydrogen which is emitted when hydrogen atomates a transition from the state with the energy of 0 .80s to the state with this energy, this final energy of 3 .4 electron balls will be the following.

03:08 So, we know that the change in the energy should be equal to the part between plam's constant and the frequency.

03:18 So we know that this is the difference between the initial energy and the final energy...

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