The ground state ionization energy for the one electron ion X^m+ is 4.72 ×10^4 kJ / mol. Identi

The ground state ionization energy for the one electron ion...

Key Concepts

Hydrogen?like Ions

Hydrogen?like ions are species that have only one electron, regardless of the number of protons in the nucleus. Because they have a single electron, their energy levels can be calculated exactly using formulae derived from the Bohr model, making them an ideal system for understanding atomic structure and quantum mechanics.

Quantum Energy Levels and the Bohr Model

According to the Bohr model, the energy levels of electrons in a hydrogen?like ion are quantized and depend on both the principal quantum number and the square of the nuclear charge. The ground state energy (n=1) is proportional to the square of the atomic number (Z²), which explains why an ion with a larger Z will have a higher ionization energy.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in its ground state. For hydrogen?like ions, this energy is directly related to the square of the nuclear charge and can be calculated using the relation derived from the Bohr model. This concept helps in identifying the element and its ionic charge when the ionization energy is known.

Relationship between Nuclear Charge and Ionization Energy

In hydrogen?like ions, the ionization energy increases with the square of the nuclear charge because the electron is more strongly attracted to a nucleus with more protons. This quadratic relationship allows one to determine the atomic number of the ion if the ionization energy is given, which in turn helps identify the element and the degree of ionization.

Transcript

00:01 To solve this problem, we need to have an understanding of two things.

00:05 The first is what ionization energy is, and the second is how we can find energy at any given energy level.

00:12 So ionization energy is the amount of energy it takes to remove an electron from an atom completely.

00:18 So in this problem, they tell us that the ionization energy is 4 .72 times 10 to the 4 kilojoules per mole.

00:32 Call this the ionization energy, which is the energy when the electron is completely removed, so we call that e infinity, minus the energy at the very first energy level.

00:48 The energy at e infinity is zero, so the energy at my first energy level is negative 4 .72 times 10 to the 4.

01:14 Since this is in kilojoules per mole, and the problem is talking about one ion, we have to change it, change our units from kilojoules per mole to joules per ion.

01:27 So that's a simple conversion we do before we move on.

01:37 We know that there are 1 ,000 joules in every kilojoule, and that in every mole there are 6 .02 times 10 to the 23rd.

01:55 Ions are atoms.

01:56 So my energy at my given energy level is equal to negative 7 .84 times 10 to the minus 17 joules.

02:13 And we use this information to help us find what the atom is by looking at the bore model.

02:21 So the bore model tells us that at any given energy level, you can find its particular energy.

02:29 And that e or the energy is equal to a constant negative 2 .178 times 10 to the minus 18 times z squared over n squared...

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