A sodium atom (Z 11) contains 11 protons in its nucleus....

A sodium atom (Z 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value $Z$ effective rather than 11 for the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is 5.1 eV. Use the Bohr model with $Z=Z_{\text { effective }}$ to calculate a value for $Z_{\text { effective }}$ (b) Using $Z=11$ and $Z=Z_{\text { effective, determine the corresponding two values for the radius of the }}$ outermost Bohr orbit.

A sodium atom (Z 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can
apply the model to the outermost electron as an approximation, provided that we use an effective value $Z$ effective rather than 11 for the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is 5.1 eV. Use the Bohr model with $Z=Z_{\text { effective }}$ to calculate a value for $Z_{\text { effective }}$ (b) Using $Z=11$ and $Z=Z_{\text { effective, determine the corresponding two values for the radius of the }}$ outermost Bohr orbit.

Key Concepts

Electron Orbital Radius

The electron orbital radius in the Bohr model is determined by the quantized orbits defined by the principal quantum number and the effective nuclear charge. This radius indicates the average distance between the nucleus and an electron in a given energy level, helping in approximations of atomic size and electron dynamics.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion. In the context of atomic models, it relates directly to the energy difference between bound electron states and the continuum. This concept is important as it reflects the strength of the attraction between an electron and the nucleus.

Bohr Model

The Bohr Model is a quantum model for atoms that quantizes the energy levels and orbital radii of electrons. It simplifies the atom, mainly considering electrons in circular orbits around the nucleus with specific allowed energies. Although mainly applicable to hydrogen-like atoms, it provides a basis for understanding electron transitions and spectral lines.

Effective Nuclear Charge

Effective nuclear charge refers to the net positive charge experienced by an electron in a multi-electron atom. Due to electron-electron repulsion and shielding, electrons do not experience the full nuclear charge. Instead, they feel a reduced charge, which is crucial when approximating atomic properties such as ionization energy and orbital radius.

Transcript

00:01 Okay, so first of all, let's remember that according to the board model, the total energy of the electron is equal e minus 13 .6.

00:14 The multiplies the atomic number square divided by the quantum number square.

00:22 So first of all, in order to calculate the effective atomic number, which is the first item of the, this problem.

00:33 We must find a way to discover what is the quantum number for this sodium atom.

00:40 So let's remember that the sodium atom has electronic configuration of 1s1, 2 s2, 2p6 and 3s1.

01:03 Therefore, the last atom, the outermost electron in the sodium atom, is in the third shell, which means we have n equals three.

01:22 So now that we have the quantum number, the shell, what is this electron, we can calculate what is the effective atomic number.

01:36 So the effective atomic number is going to be equal, let's see, three square that multiplies the energy divided by 13 .6, all this in the square root.

01:58 So calculating this, we have nine.

02:02 This should be a nine.

02:08 That multiplies the energy, which the problem says is 5 .1.

02:14 Divided by 13 .6 in the square root.

02:21 Calculating this, we have 1 .8 of effective atomic number.

02:29 And as expected, the effective, let's put an f here, the effective atomic number is less than 11 and greater then one...

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