Question

Which of the following statements is (are) correct for an electron with \(n=4\) and \(m_{\ell}=2 ?\) Explain. (a) The electron is in the fourth principal shell. (b) The electron may be in a \(d\) orbital. (c) The electron may be in a \(p\) orbital. (d) The electron must have \(m_{s}=+\frac{1}{2}\)

Short answer

The correct statements are (a) The electron is in the fourth principal shell, and (b) The electron may be in a d orbital.

Step-by-step solution

Understand Quantum Numbers

Quantum numbers are numbers assigned to atomic orbitals that describe various properties of the orbitals and the electrons they contain. Here in this exercise, three quantum numbers are mentioned: n, m_l, and m_s.\nThe principal quantum number n describes the energy level of an electron, the apparent 'shell' the electron is in.\nThe magnetic quantum number m_l can take on any integer values between -n and +n, and it tells us about the orientation of an orbital.\nThe spin quantum number m_s, however, only has two possible values: +1/2 or -1/2, representing the two possible spin states of an electron: spin up and spin down.

Analyze each statement

(a) The electron is in the fourth principal shell. This statement is correct. Here, n=4, which means that the electron is located in the fourth principal shell (or energy level).\n(b) The electron may be in a d orbital. This is correct as well. m_l could take values from -n to +n. For n=4, m_l could be from -4 to +4. The value given here, m_l=2, indicates that the electron could be in a d orbital because d orbitals correspond to m_l values from -2 through +2.\n(c) The electron may be in a p orbital. This statement is incorrect. p orbitals correspond to m_l values of -1, 0, and +1. Since m_l=2, it means the electron cannot be in a p orbital.\n(d) The electron must have m_s=+1/2. This statement is incorrect. The spin quantum number m_s is independent of the values of n and m_l. Regardless of its principal quantum number or its magnetic quantum number, an electron may have a spin quantum number of either +1/2 (spin up) or -1/2 (spin down). These spin states are not determined by the value of n or m_l.

Conclusion

After analyzing each statement, it can be concluded that Statements A and B are correct, while Statements C and D are incorrect. These conclusions are based on the definition and implications of the principal quantum number (n), the magnetic quantum number (m_l), and the spin quantum number (m_s).

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a fundamental concept in quantum mechanics that describes the energy level of an electron within an atom. The principal quantum number can take on any positive integer value: 1, 2, 3, and so on. Each value of \(n\) corresponds to a different electron shell, or energy level, around the nucleus of an atom. Understanding the principal quantum number helps us know the size and energy of an electron's orbit. For instance, when \(n=4\), the electron is in the fourth shell, indicating that it is farther from the nucleus and has higher energy compared to lower \(n\) values like \(n=1\) or \(n=2\).

• The higher the value of \(n\), the greater the energy and the larger the orbital.
• Electrons with higher \(n\) values are generally further from the nucleus.

Knowing the principal quantum number is crucial for predicting the chemical behavior of atoms since it affects electron configurations and bonding.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \(m_\ell\), provides information about the orientation of an electron's orbital within a given subshell. For a particular value of the angular momentum quantum number \(l\), \(m_\ell\) can range from \(-l\) to \(+l\), including zero. This means that for each subshell, there can be multiple possible orientations.In the exercise example, \(m_\ell=2\) and \(n=4\) helps determine that the electron is neither in an \(s\) nor in a \(p\) orbital because these would not allow for an \(m_\ell\) of 2. Instead, it suggests that the electron could reside in a \(d\) orbital, as \(d\) orbitals have \(l=2\) where \(m_\ell\) values span from -2 to +2.

• \(m_\ell\) identifies the specific orbital within a subshell.
• It plays a role in explaining how electrons are distributed in magnetic fields.

Understanding \(m_\ell\) helps to explain the placement of electrons around a nucleus, particularly in multi-electron atoms where electron interactions become complex.

Spin Quantum Number

The spin quantum number, given by \(m_s\), reflects an intrinsic property of electrons known as spin. Each electron has a spin that can be either +1/2 or -1/2, often referred to as "spin up" or "spin down." This quantum number is independent of the principal () and magnetic (\(m_\ell\)) quantum numbers, thus it provides additional information about the electron spin state. The significance of the spin quantum number is crucial in explaining the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of all four quantum numbers. This principle is fundamental in understanding atomic structure and electron pairing in orbitals.

• \(m_s\) is the sole quantum number associated with the electron's spin direction.
• It plays a critical role in determining an atom's magnetic properties.

Recognizing how \(m_s\) works is important for understanding electron arrangement and chemical bonding, as it affects how electrons pair within orbitals.