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Chapter 2: Problem 5

The radius of the second Bohr orbit, in terms of the Bohr radius, \(a_{0}\), in \(\mathrm{Li}^{2+}\) is: (a) \(\frac{2 a_{0}}{3}\) (b) \(\frac{4 a_{0}}{9}\) (c) \(\frac{4 a_{0}}{3}\) (d) \(\frac{2 a_{0}}{9}\)

Short Answer

Expert verified
(c) \( \frac{4a_0}{3} \)

Step by step solution

01

Identify the Formula

The radius of the nth Bohr orbit is given by the formula \( r_n = n^2 \frac{a_0}{Z} \), where \( n \) is the orbit number, \( a_0 \) is the Bohr radius, and \( Z \) is the atomic number of the nucleus.
02

Substitute Values

For \( \text{Li}^{2+} \), the atomic number \( Z \) is 3 because lithium has 3 protons. For the second Bohr orbit, \( n = 2 \). Substitute these values into the formula: \( r_2 = 2^2 \frac{a_0}{3} \).
03

Simplify the Expression

Calculate \( 2^2 \) which gives 4, and then substitute it into the expression: \( r_2 = \frac{4a_0}{3} \).
04

Compare with Options

Compare the simplified expression \( \frac{4a_0}{3} \) with the given options. The correct option is (c) \( \frac{4a_0}{3} \).

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Key Concepts

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

Bohr Radius
The Bohr radius is a fundamental physical constant that represents the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is denoted as \( a_0 \) and forms the basis for calculating the sizes of quantum orbits in the Bohr Model. This concept is crucial because:
  • It provides a specific reference point for determining the electron's orbital paths around the nucleus.
  • Its value is approximately \( 5.29 \times 10^{-11} \) meters, and it sets the scale for atomic dimensions.
In the exercise, the Bohr radius \( a_0 \) is used to calculate the radius of electron orbits in multi-electron atoms, such as the lithium ion \( \mathrm{Li}^{2+} \). By multiplying \( a_0 \) by factors determined by the orbit's quantum numbers and the atomic number, it allows us to find the size of specific orbits in different elements."
Atomic Number
The atomic number denoted as \( Z \) is a crucial integer variable that defines the number of protons in the nucleus of an atom. Each element has a unique atomic number which determines its position in the periodic table and its chemical properties:
  • It equals the number of electrons in a neutral atom, balancing the positive charge of the protons.
  • In a charged ion, such as \( \mathrm{Li}^{2+} \), the atomic number remains the same, but electron count varies.
For lithium, \( Z = 3 \) since it has three protons. This atomic number is essential in determining the radii of Bohr orbits when using the formula \( r_n = n^2 \frac{a_0}{Z} \). Understanding the atomic number is vital for visualizing electron distribution and size of atomic orbits in multi-electron systems."
Quantum Orbits
Quantum orbits in the Bohr Model describe the specific paths that electrons take as they travel around an atomic nucleus. These orbits are characterized by discrete, quantized energy levels, which means that electrons can only reside at certain distances from the nucleus.
  • The principal quantum number \( n \) describes the orbit's size and energy level; larger \( n \) values indicate higher energy and larger orbits.
  • The second Bohr orbit for an electron in a system like \( \mathrm{Li}^{2+} \) corresponds to \( n = 2 \).
In the solved exercise, the second orbit radius was calculated using \( n = 2 \) and the corresponding adjustments for the atomic number, resulting in the formula \( r_2 = \frac{4a_0}{3} \). Quantum orbits help visualize how electrons are arranged in atoms and play a significant role in determining the chemical properties of elements."

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Most popular questions from this chapter

The uncertainty principle and the concept of wave nature of matter were proposed by \(\ldots \ldots \ldots \ldots . .\) and \(\ldots \ldots \ldots \ldots \ldots\) respectively. (Heisenberg, Schrodinger, Maxwell, de Broglie)

The energy of the electron in the second and the third Bohr's orbits of the hydrogen atom is \(-5.42 \times 10^{-12}\) erg and \(-2.41 \times 10^{-12}\) erg respectively. Calculate the wavelength of the emitted radiation when the electron drops from the third to the second orbit.

Which of the following relates to photons both as wave motion and as a stream of particles? [1992-1 Mark] (a) Inference (b) \(E=m c^{2}\) (c) Diffraction (d) \(E=h v\)

Correct set of four quantum numbers for the valence (outermost) electron of rubidium \((Z=37)\) is : [1984-1 Mark] (a) \(5,0,0,+1 / 2\) (b) \(5,1,0,+1 / 2\) (c) \(5,1,1,+1 / 2\) (d) \(6,0,0,+1 / 2\)

If the de Broglie wavelength of the electron in \(\mathrm{n}^{\text {th }}\) Bohr orbit in a hydrogenic atom is equal to \(1.5 \pi \mathrm{a}_{0}\left(\mathrm{a}_{0}\right.\) is Bohr radius), then the value of \(\mathrm{n} / \mathrm{z}\) is : \(\quad\) [Main Jan. \(12,2019(\mathrm{II})]\) (a) \(0.40\) (b) \(1.50\) (c) \(1.0\) (d) \(0.75\)
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