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Chapter 30: Problem 7

For a doubly ionized lithium atom \(\mathrm{Li}^{2+}(Z=3)\), what is the principal quantum number of the state in which the electron has the same total energy as a ground-state electron has in the hydrogen atom?

Short Answer

Expert verified
The principal quantum number is \(n = 3\).

Step by step solution

01

Understanding the Problem

We need to find the principal quantum number (n) for an electron in a doubly ionized lithium atom (\(\text{Li}^{2+}\)) where its energy matches the ground state energy of a hydrogen atom. Lithium's atomic number is \(Z = 3\).
02

Calculate the Ground State Energy of Hydrogen

The ground-state energy for hydrogen is given by the formula \[E_n = -\frac{13.6\, \text{eV}}{n^2}\]. For hydrogen's ground state, \(n = 1\), so the energy \(E_1 = -13.6\, \text{eV}\).
03

Formula for Energy Levels in Hydrogen-like Atoms

For hydrogen-like atoms, the energy levels are given by the formula \[E_n = -Z^2 \times \frac{13.6\, \text{eV}}{n^2}\], where \(Z\) is the atomic number.
04

Equating Energies

We set the equation \(-Z^2 \times \frac{13.6\, \text{eV}}{n^2} = -13.6\, \text{eV}\). For lithium, \(Z = 3\), so the equation simplifies to \(-3^2 \times \frac{13.6\, \text{eV}}{n^2} = -13.6\, \text{eV}\).
05

Solve for Principal Quantum Number

In the equation \(-9 \times \frac{13.6\, \text{eV}}{n^2} = -13.6\, \text{eV}\), divide both sides by \(-13.6\, \text{eV}\) and solve for \(n^2\): \[9 \times \frac{1}{n^2} = 1\]. Therefore, \(n^2 = 9\), which gives us \(n = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Doubly Ionized Lithium Atom
A doubly ionized lithium atom, represented as \( \mathrm{Li}^{2+} \), is a form of the lithium atom where two electrons have been removed. Lithium usually has three electrons but when it loses two, it only has one left, moving it into a state similar to a hydrogen atom which has a single electron. This is fundamental because it makes lithium act like a "hydrogen-like" atom. In this state, the single remaining electron is therefore affected more strongly by the nuclear charge than in neutral lithium. This is because the remaining electron feels the full positive charge of the nucleus more powerfully without the two other electrons around to help shield it.
  • Nuclear Charge: For \( \mathrm{Li}^{2+} \), the electron feels a charge of \( Z = 3 \).
  • Hydrogen-like: Acts like a hydrogen atom but experiences a stronger nuclear pull.
Ground State Energy
The ground state energy of an atom corresponds to the lowest energy level that an electron can occupy. For hydrogen, the energy level can be calculated using the formula: \[ E_n = -\frac{13.6 \text{ eV}}{n^2} \] For the ground state of hydrogen, where the principal quantum number \( n = 1 \), this formula results in an energy of \(-13.6 \text{ eV}\). This is a fundamental reference point in quantum mechanics as it represents the most stable and least energetic state an electron can exist in when part of a hydrogen atom. In our problem, we relate this to a doubly ionized lithium atom to find a matching energy state.
  • Hydrogen's Ground Energy: \(-13.6 \text{ eV}\).
  • Comparison: We equate lithium's energy level to hydrogen's to find equivalent states.
Hydrogen-like Atoms
Hydrogen-like atoms refer to ions that, like hydrogen, consist of a nucleus and a single electron. However, unlike hydrogen, these ions have more protons in the nucleus, meaning a greater nuclear charge \( Z \). Examples include \( \mathrm{Li}^{2+} \) and helium ions. Energy levels in hydrogen-like atoms still follow the quantum mechanical model, however they are adjusted for the greater nuclear charge.Using the formula for energy levels: \[ E_n = -Z^2 \times \frac{13.6 \text{ eV}}{n^2} \] This formula accounts for the difference in charge by multiplying the hydrogen energy level by \(-Z^2 \). For \( \mathrm{Li}^{2+} \), this gives a stronger pulling force on the electron, as depicted by a more negative energy. Solving for the principal quantum number allows us to find a state with energy equivalent to hydrogen's ground state.
  • Energy Formula: Adjusts for different \( Z \) values.
  • Quantum Number Relation: Helps find comparable energy levels across different hydrogen-like ions.

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Most popular questions from this chapter

There are \(Z\) protons in the nucleus of an atom, where \(Z\) is the atomic number of the element. An \(\alpha\) particle carries a charge of \(+2 \mathrm{e}\). In a scattering experiment, an \(\alpha\) particle, heading directly toward a nucleus in a metal foil, will come to a halt when all the particle's kinetic energy is converted to electric potential energy. In such a situation, how close will an \(\alpha\) particle with a kinetic energy of \(5.0 \times 10^{-13} \mathrm{J}\) come to a gold nucleus \((Z=79) ?\)

A laser is used in eye surgery to weld a detached retina back into place. The wavelength of the laser beam is \(514 \mathrm{nm}\) and the power is 1.5 W. During surgery, the laser beam is turned on for 0.050 s. During this time, how many photons are emitted by the laser?

A pulsed laser emits light in a series of short pulses, each having a duration of \(25.0 \mathrm{ms} .\) The average power of each pulse is \(5.00 \mathrm{mW}\), and the wavelength of the light is \(633 \mathrm{nm} .\) Find the number of photons in each pulse.

Which of the following subshell configurations are not allowed? For those that are not allowed, give the reason why. (a) \(3 s^{1}(b) 2 d^{2}(c) 3 s^{4}\) (d) \(4 p^{8}(e) 5 f^{12}\)

When an electron makes a transition between energy levels of an atom, there are no restrictions on the initial and final values of the principal quantum number \(n .\) According to quantum mechanics, however, there is a rule that restricts the initial and final values of the orbital quantum number \(\ell\). This rule is called a selection rule and states that \(\Delta \ell=\pm 1\). In other words, when an electron makes a transition between energy levels, the value of \(\ell\) can only increase or decrease by one. The value of \(\ell\) may not remain the same nor may it increase or decrease by more than one. According to this rule, which of the following energy level transitions are allowed? (a) \(2 \mathrm{s} \rightarrow 1 \mathrm{s}\) (b) \(2 p \rightarrow 1 s\) (c) \(4 p \rightarrow 2 p(d) 4 s \rightarrow 2 p\) (e) \(3 \mathrm{d} \rightarrow 3 \mathrm{s}\)
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