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Chapter 3: Problem 81

If \(l=1\), what values can \(m_{l}\) have? What values can \(l\) have if \(n=2 ?\)

Short Answer

Expert verified
When \( l=1 \), \( m_l \) can be \(-1, 0, 1\). When \( n=2 \), \( l \) can be \(0\) or \(1\).

Step by step solution

01

Understanding Quantum Numbers

Quantum numbers are values that describe the properties of electrons in an atom. The quantum number \( l \) is the azimuthal quantum number (angular momentum quantum number), which can take values from 0 to \( n-1 \), where \( n \) is the principal quantum number.
02

Determine Possible Values of \( m_l \) when \( l = 1 \)

\( m_l \) is the magnetic quantum number, which describes the orientation of the orbital in space. It can take integral values from \( -l \) to \( +l \). With \( l = 1 \), \( m_l \) can take the values \(-1, 0, \) and \( 1 \).
03

Determine Possible Values of \( l \) when \( n = 2 \)

\( l \) can take values from 0 to \( n-1 \). If \( n = 2 \), \( l \) can be either 0 or 1. Therefore, \( l \) can have the values \( 0 \) and \( 1 \).

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Key Concepts

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

Azimuthal Quantum Number
The azimuthal quantum number, often denoted as \( l \), is fundamental to understanding the three-dimensional space in which electrons move around the nucleus of an atom. This quantum number essentially defines the shape of the electron's orbital, which can dramatically impact chemical properties and interactions.
The values of \( l \) range from \( 0 \) up to \( n-1 \), where \( n \) is the principal quantum number of the electron's shell. For instance:
  • If \( n = 2 \), the possible values for \( l \) are \( 0 \) and \( 1 \).
  • When \( l = 0 \), the orbital is spherical, often referred to as an 's' orbital.
  • If \( l = 1 \), the orbital takes on a dumbbell shape, known as a 'p' orbital.

As the azimuthal quantum number increases, so does the complexity of the shapes, influencing the distribution of electrons around an atom.
Magnetic Quantum Number
The magnetic quantum number, represented by \( m_l \), delves into the specific orientation of the electron's orbital around the nucleus. It's essential for determining how orbitals are oriented in space under the influence of a magnetic field.
For any given azimuthal quantum number \( l \), the magnetic quantum number \( m_l \) can have integral values ranging from \(-l\) to \(+l\).
  • This means if \( l = 1 \), \( m_l \) could be \(-1, 0, \) or \( 1 \).
  • These values determine the various orientations of the 'p' orbitals (like \( p_x, p_y, \) and \( p_z \)).

Thus, the magnetic quantum number is crucial in identifying the precise orientation of electron orbitals, particularly when subjected to external magnetic conditions.
Principal Quantum Number
The principal quantum number, denoted as \( n \), is one of the most significant quantum numbers as it primarily determines the energy level and size of the electron's orbital. It essentially indicates how far an electron is from the nucleus.
The values of \( n \) are positive integers starting from 1:
  • As \( n \) increases, the electron is further away from the nucleus, which implicates higher energy levels.
  • This means if \( n = 2 \), the potential values for \( l \) are \( 0 \) and \( 1 \), allowing more complex orbital shapes.

The concept of the principal quantum number helps establish the groundwork for the structure of electronic shells and subshells, fundamentally shaping the arrangement and behavior of electrons within an atom.

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Most popular questions from this chapter

Which of the following ions has the largest radius? Explain. (a) \(\mathrm{Na}^{+}\) (b) \(\mathrm{Mg}^{2+}\) (c) \(\mathrm{S}^{2-}\) (d) \(\mathrm{Cl}^{-}\) (e) \(\mathrm{Se}^{2-}\)

Calculate the maximum number of unpaired electrons that can be placed in a \(5 d\) subshell.

Draw orbital diagrams for the following atoms. (a) \(\mathrm{Si}\) (b) \(\mathrm{V}\) (c) \(\mathrm{Ga}\) (d) \(\mathrm{Cl}\) (e) \(\mathrm{Na}\)

Some elements, such as tin and lead, have more than one common ion. Use the electron configurations to predict the most likely ions of \(\mathrm{Sn}\) and \(\mathrm{Pb}\).

Look up the covalent radii for magnesium and sulfur atoms and the ionic radii of \(\mathrm{Mg}^{2+}\) and \(\mathrm{S}^{2-}\) ions in Appendix B.4. Explain why \(\mathrm{Mg}^{2+}\) ions are smaller than \(\mathrm{S}^{2-}\) ions even though magnesium atoms are larger than sulfur atoms.
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