Orbital Quantum Numbers And Total Electron Energy

In summary, the smallest possible value for the total energy of an electron with an orbital quantum number of 4 in a hydrogen atom is 0.544 eV, according to the quantum mechanical model. This value is independent of the angular momentum quantum number, but there is a constraint that l must be less than n.
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Hello :smile:

Homework Statement



The orbital quantum number for the electron in the hydrogen atom is l = 4. What
is the smallest possible value (in eV) for the total energy of this electron? (Use the
quantum mechanical model of the hydrogen atom.)

Homework Equations


The Attempt at a Solution



I know that the angular momentum of the electron is given by;

[itex]L = \sqrt{l(l + 1)}\frac{h}{2 \pi}[/itex]

[itex]L = \sqrt{20} \frac{h}{2 \pi}[/itex]

L = 4.64x10-33 Kgm2s-1

My textbook doesn't really discuss the QM picture of the atom, so I don't know how to relate this to the energy of the electron.

I know how to do it for the Bohr model, but clearly that's no good.

I appreciate any help you can give,

thanks!

<EDIT>

Oops.

" In fact, calculating the energy from the quantum mechanical wave function gives the expression Bohr derived for the energy:"

This thread can be ignored/deleted. sorry.
 
Last edited:
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  • #2
The energy of the electron is given by the Bohr equation
$$
E_n = - \frac{\mathrm{Ry}}{n^2}
$$
where ##\mathrm{Ry} \approx 13.6\ \mathrm{eV}## is the Rydberg constant (expressed in units of energy).

This energy is independent of the angular momentum quantum number ##l##. However, there is the constraint that ##l < n##. Therefore, if ##l = 4##, the lowest value of ##n## is 5, and hence the lowest energy is
$$
E_5 = - \frac{13.6\ \mathrm{eV}}{5^2} = 0.544\ \mathrm{eV}
$$
 
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Related to Orbital Quantum Numbers And Total Electron Energy

What is an orbital quantum number?

An orbital quantum number is a value that represents the shape of an electron's orbital in an atom. It is denoted by the letter l and can have values from 0 to n-1, where n is the principal quantum number.

How does the orbital quantum number affect the energy of an electron?

The orbital quantum number, along with the principal quantum number, determines the energy of an electron in an atom. The higher the value of l, the higher the energy of the electron and the further it is from the nucleus. This is because higher l values correspond to orbitals with more complex shapes, which require more energy to maintain.

What is the relationship between the orbital quantum number and the shape of an electron orbital?

The orbital quantum number indicates the shape of an electron orbital. It has values from 0 to n-1, and each value corresponds to a different orbital shape. For example, l=0 corresponds to an s orbital, l=1 corresponds to a p orbital, l=2 corresponds to a d orbital, and so on.

What is the maximum number of electrons that can occupy an orbital with a given orbital quantum number?

The maximum number of electrons that can occupy an orbital with a given orbital quantum number is given by the formula 2(2l+1). For example, an s orbital (l=0) can hold a maximum of 2 electrons, a p orbital (l=1) can hold a maximum of 6 electrons, and a d orbital (l=2) can hold a maximum of 10 electrons.

How does the total electron energy relate to the principal quantum number and the orbital quantum number?

The total electron energy is determined by the sum of the principal quantum number (n) and the orbital quantum number (l). This means that two electrons with the same principal quantum number but different orbital quantum numbers will have different total energies.

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