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Chapter 2: Problem 85

Which of the following statements is false? (1) For azimuthal quantum number \(l=3\), the maximum number of electrons will be 14 . (2) The concept of electron spin was introduced by Uhlenbeck and Goudsmit. (3) The principal quantum number of an atom represents size of the orbit and distance of electron from nucleus. (4) The shape of an orbital is governed by magnetic quantum number.

Short Answer

Expert verified
Statement (4) is false.

Step by step solution

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01

- Understanding Electron Capacity for Azimuthal Quantum Number

For an azimuthal quantum number ( l = 3 ), the electrons occupy the f orbitals. Each f orbital can contain a maximum of 14 electrons because there are 7 orbitals and each orbital can hold 2 electrons. So, statement (1) is true.
02

- Electron Spin Concept

The concept of electron spin was indeed introduced by Uhlenbeck and Goudsmit. Hence, statement (2) is true.
03

- Principal Quantum Number

The principal quantum number ( n ) represents the size of the orbit and the average distance of the electron from the nucleus. As such, statement (3) is true.
04

- Shape of Orbitals and Magnetic Quantum Number

The shape of an orbital is actually governed by the azimuthal quantum number ( l ), not the magnetic quantum number ( m_l ), which dictates the orientation of the orbital in space. Therefore, statement (4) is false.

Key Concepts

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

Azimuthal Quantum Number
The azimuthal quantum number, also known as the angular momentum quantum number or secondary quantum number, is symbolized by the letter \( l \). This quantum number defines the shape of an electron's orbital and can take on any integer value from 0 to \( n-1 \), where \( n \) is the principal quantum number.

For each value of \( l \), there is a corresponding type of orbital:
  • \( l = 0 \) corresponds to an s orbital
  • \( l = 1 \) corresponds to a p orbital
  • \( l = 2 \) corresponds to a d orbital
  • \( l = 3 \) corresponds to an f orbital
In the given exercise, \( l = 3 \) for f orbitals. Each f orbital can hold a total of 14 electrons because there are 7 different orientations of an f orbital, each capable of holding 2 electrons (one with spin-up and one with spin-down). Thus, the statement regarding the maximum number of electrons being 14 is true.
Electron Spin
Electron spin, discovered by Samuel Goudsmit and George Uhlenbeck in 1925, is a fundamental property of electrons. This quantum number is often denoted as \( s \) or \( m_s \) and can have a value of either +1/2 or -1/2.

Electron spin describes the intrinsic angular momentum of electrons, which contributes to their magnetic moment. In simpler terms, it's as if the electron is spinning around its own axis, although this is a somewhat simplified picture.

The two possible spins (spin-up or spin-down) allow each orbital to hold two electrons, one with each spin type. This is key to understanding the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same four quantum numbers. Therefore, the statement that the concept of electron spin was introduced by Uhlenbeck and Goudsmit is true.
Principal Quantum Number
The principal quantum number, noted as \( n \), determines the overall size and energy level of an electron's orbital. It is always a positive integer (1, 2, 3, ...). The larger the value of \( n \), the larger and higher in energy the orbitals are.

Generally, the principal quantum number reflects how far the electron is from the nucleus. For example, electrons with a principal quantum number of 1 will be closer to the nucleus and lower in energy compared to electrons with \( n = 2 \) or \( n = 3 \). This quantum number is crucial for defining an electron's position within an atom.

Therefore, the statement in the exercise correctly identifies that the principal quantum number represents the size of the orbit and the average distance of the electron from the nucleus.
Magnetic Quantum Number
The magnetic quantum number, denoted as \( m_l \), specifies the orientation of an orbital in space relative to an external magnetic field. It can take on any integer value between \( -l \) and \( +l \), including zero.

For example, if \( l = 1 \) (a p orbital), \( m_l \) can be -1, 0, or +1. These values indicate that there are three different p orbitals, each oriented differently in space. However, the shape of the orbitals (spherical for s, dumbbell-shaped for p, etc.) is not defined by \( m_l \), but by the azimuthal quantum number \( l \).

In the given exercise, the statement that the shape of an orbital is governed by the magnetic quantum number is false. The correct determinant for the shape of an orbital is the azimuthal quantum number, as previously discussed.

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Most popular questions from this chapter

Which of the following statements is wrong? (1) The energy of the electron at infinite distance from the nucleus in Bohr's model is taken as zero. (2) If an electron is brought from an infinite distance close to the nucleus of the atom, the energy of the electron nucleus system decreases to a greater negative value. (3) \(\Lambda\) s the electron moves away from the nucleus its velocity increases. (4) \(\Lambda\) s the electron moves away from the nucleus its kinetic energy decreases while potential energy increases.

Which of the following statements is wrong? (1) The rule that limits the maximum number of electrons in an orbital to two is Paulis exclusion principle. (2) IIund's rule states that in a set of degenerate orbitals the electrons distribute with parallel spins. (3) The order of filling of electrons in the orbitals of an atom will be \(5 \mathrm{~s}, 4 \mathrm{p}, 3 \mathrm{~d}, 4 \mathrm{~d}, 4 \mathrm{~s}\). (4) The number of unpaired electrons in the electronic configuration \(1 \mathrm{~s}^{2} 2 \mathrm{~s}^{2} 2 \mathrm{p}^{4}\) is 2 .

Which sct of quantum numbers represent the clectron of the lowest encrgy? (1) \(n=2, l=0, m=0, s=-\frac{1}{2}\) (2) \(n=2, l=1, m=0, s=+\frac{1}{2}\) (3) \(n=4, l=1, m=0, s=+\frac{1}{2}\) (4) \(n=4, l=0, m=0, s=-\frac{1}{2}\)

Among the following the correct statement(s) is/are (1) Increase in the frequency of the incident radiation increases the kinetic energy of photoelectrons. (2) Threshold wavelength depends upon work function. (3) The study of photoclectric effect is useful in understanding quantisation of energy. (4) To cross the threshold energy intensity of the light must be increased. (1) I, II and III (2) II, III and IV (3) I, III and IV (4) I, II, III and IV

Which of the following is false about electron? (1) The negatively charged electrons in an atom are attracted by the positively charged nucleus by electrostatic attractive force. (2) An electron near the nucleus is strongly attracted by the nucleus and has low potential energy. (3) An electron distant from the nucleus is less firmly held and has high potential energy. (4) On heating an atom, all of its electrons are ejected.
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