Question

Select the correct answer and explain your reasoning. An electron having \(n=3\) and \(m_{\ell}=0\) (a) must have \(m_{s}=+\frac{1}{2} ;(\mathbf{b})\) must have \(\ell=1 ;(\mathbf{c})\) may have \(\ell=0,1\) or \(2 ;\) (d) must have \(\ell=2\).

Short answer

The correct option is (c), because when \(n=3\), the electron can have \(\ell=0,1\) or \(2\).

Step-by-step solution

Analyze Option A

Option (a) refers to the spin quantum number \(m_{s}\). It does not depend on either \(n\) or \(m_{\ell}\), so we can't say it must have \(m_{s}=+\frac{1}{2}\). It can have either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Therefore, option (a) is incorrect.

Analyze Option B

Option (b) states the electron must have \(\ell=1\). However, since \(n=3\), the value of \(\ell\) can range from 0 to 2. Therefore, it doesn't have to be that \(\ell=1\). So, option (b) is also incorrect.

Analyze Option C

Option (c) suggests the electron may have \(\ell=0,1\) or \(2\). This is logically correct because for \(n=3\), the possible values of \(\ell\) can indeed range from 0 to 2. Therefore, option (c) is correct. Nevertheless, we should analyze option (d) to complete our examination.

Analyze Option D

Option (d) states that the electron must have \(\ell=2\). However, as in option (b), we have already established that the value of \(\ell\) can range from 0 to 2 for \(n=3\). So, option (d) is incorrect.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a key quantum number representing the energy level of an electron in an atom. It is always a positive integer (1, 2, 3,...). The higher the value of \(n\), the further an electron is, on average, from the nucleus of an atom.

For instance, an electron with \(n=1\) is in the first energy level, closest to the nucleus, whereas \(n=3\), like in our original exercise, refers to an electron within the third energy level.

This number not only indicates how far an electron is from the nucleus but also influences the size and energy of the orbital. Larger \(n\) values mean higher energy and larger orbitals. In the context of quantum numbers, \(n\) is crucial because it determines the number of allowed subshells; each energy level contains \(n\) possible subshells.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \(\ell\), determines the shape of the electron's orbital. The value of \(\ell\) ranges from 0 to \(n-1\), where \(n\) is the principal quantum number. For example, if \(n=3\), \(\ell\) can have values of 0, 1, or 2.

Each \(\ell\) value corresponds to a specific type of orbital:

• \(\ell = 0\): "s" orbital (spherical shape)
• \(\ell = 1\): "p" orbital (dumbbell shape)
• \(\ell = 2\): "d" orbital (complex, cloverleaf shape)

These shapes describe the region in space where the probability of finding an electron is highest. Angular momentum quantum number gives us an insight into the electron's path around the nucleus, helping us understand chemical bonding and reactivity.

Magnetic Quantum Number

The magnetic quantum number, represented by \(m_{\ell}\), specifies the orientation of an electron's orbital in space for a given value of \(\ell\). In simpler terms, it tells us how these orbitals are oriented relative to a magnetic field.

The possible values of \(m_{\ell}\) range from \(-\ell\) to \(+\ell\), including zero. For example, if \(\ell=1\), \(m_{\ell}\) can have values of -1, 0, or +1. In our exercise, \(m_{\ell}=0\) suggests the electron's orbital is in a neutral orientation.

Understanding the magnetic quantum number is crucial for visualizing the electron clouds' orientation and how they interact in a magnetic environment. This quantum number plays a key role in determining the arrangement of electrons in multi-electron atoms, directly influencing the atom's magnetic properties.

Spin Quantum Number

The spin quantum number, denoted by \(m_{s}\), indicates the direction of an electron's spin. Electron spin is a fundamental property, much like charge or mass, and it can be either +1/2 or -1/2.

Two electrons in the same orbital must have opposite spins. This is due to the Pauli Exclusion Principle, which states no two electrons can have the same set of four quantum numbers within an atom.

In the context of our exercise, the electron spin does not depend on the other quantum numbers such as \(n\) or \(m_{\ell}\). Hence option (a) was incorrect, as an electron could have either spin orientation. Recognizing the electron's spin is vital for understanding atomic structures and magnetic properties, especially when determining how elements interact with each other.