Question

The fourth electron of Be atom will have which of the following quantum numbers? (a) \(1,0,0, \frac{1}{2}\) (b) \(2,0,0,-\frac{1}{2}\) (c) \(2,1,0,+\frac{1}{2}\) (d) \(1,1,1,+\frac{1}{2}\)

Short answer

The correct answer is (b) \(2,0,0,-\frac{1}{2}\).

Step-by-step solution

Understand the Electron Configuration

Beryllium (Be) has an atomic number of 4, which means it has 4 electrons. To understand the position of the electrons, we need to consider the electronic configuration. Beryllium's electron configuration is:\[ 1s^2 \, 2s^2 \]This means the first two electrons fill the 1s orbital, and the next two fill the 2s orbital.

Identify Quantum Numbers

Quantum numbers describe the properties of an electron in an atom. For the fourth electron of Beryllium which is in the 2s orbital, we determine its quantum numbers:- Principal quantum number \( n = 2 \) as it's in the second shell.- Azimuthal quantum number \( l = 0 \) as it is in an 's' orbital.- Magnetic quantum number \( m_l = 0 \) since for \( l = 0 \), \( m_l = 0 \) by default.- Spin quantum number \( m_s = -\frac{1}{2} \) or \( +\frac{1}{2} \). For paired electrons in the same orbital, one electron has \( m_s = +\frac{1}{2} \), and the other has \( m_s = -\frac{1}{2} \). The first electron in 2s is usually considered as \( m_s = +\frac{1}{2} \), making the fourth electron \( m_s = -\frac{1}{2} \).

Match the Quantum Numbers

Now, compare the derived quantum numbers for the fourth electron \( (n = 2, l = 0, m_l = 0, m_s = -\frac{1}{2}) \) with the given options:- Option (a) \(1,0,0, \frac{1}{2}\) - Primarily incorrect due to the value of \( n \).- Option (b) \(2,0,0,-\frac{1}{2}\) - Matches perfectly with the quantum numbers we determined.- Option (c) \(2,1,0,+\frac{1}{2}\) - Incorrect due to the value of \( l \) and \( m_s \).- Option (d) \(1,1,1,+\frac{1}{2}\) - Incorrect because all quantum numbers are wrong.

Key concepts

Electron Configuration

Electron configuration is a way to represent the distribution of electrons in an atom's orbitals. It gives us insight into the location and behavior of electrons around the nucleus. For Beryllium, which has an atomic number of 4, it means that there are 4 electrons. Understanding this concept is fundamental for determining the quantum numbers of any atom's electrons. You write electron configurations by noting down the sequence of orbital filling:

• Electrons fill the lowest energy levels first, following the Aufbau principle.
• The order of filling is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on.

Thus, the electron configuration for Beryllium is \(1s^2 \, 2s^2\), meaning two electrons fill the 1s orbital, and two electrons fill the 2s orbital.

Principal Quantum Number

The Principal Quantum Number, denoted as \(n\), is one of the primary quantum numbers important in describing an electron in an atom. This number indicates the electron's main energy level or shell. It can take any positive integer value (1, 2, 3, ...). For instance:

• \(n = 1\) indicates the first shell, the closest to the nucleus, containing the 1s orbital.
• \(n = 2\) tells us the electron is in the second shell, which includes the 2s and 2p orbitals.

In our exercise, the fourth electron of Beryllium is in the 2s orbital, so it has a principal quantum number of 2, reflecting its energy level and relative distance from the nucleus.

Azimuthal Quantum Number

The Azimuthal Quantum Number, represented by \(l\), determines the shape of the orbital in which the electron resides. It gives insight into the angular momentum of an electron. The value of \(l\) depends on the principal quantum number \(n\), and can range from 0 to \(n - 1\). The common values and corresponding orbital types are:

• \(l = 0\): s orbital, spherical shape
• \(l = 1\): p orbital, dumbbell shape
• \(l = 2\): d orbital, more complex shapes

In the case of Beryllium's fourth electron, it's in an 's' orbital (2s), so the azimuthal quantum number \(l\) is 0. This describes the spherical shape of the 2s orbital where the electron is located.

Spin Quantum Number

The Spin Quantum Number, symbolized by \(m_s\), describes the intrinsic spin of the electron within an orbital. Electrons behave like small magnetic dipoles, having a characteristic spin, usually conceptualized as being 'up' or 'down'. Therefore, \(m_s\) only has two possible values:

• \(+\frac{1}{2}\) often indicates an "up" spin
• \(-\frac{1}{2}\) usually denotes a "down" spin

In any filled orbital, electrons must have opposite spins as per the Pauli Exclusion Principle. Therefore, for Beryllium, in the 2s orbital containing two electrons, if the first electron spin value is taken as \(+\frac{1}{2}\), then naturally the second, or in this case, the fourth electron would have \(m_s = -\frac{1}{2}\). This accounts for the balance in electron orientation within the atom.