Question

\(\mathrm{K}\) and \(\mathrm{L}\) shell of an element are completely filled and there are 16 electrons in M-shell and 2 -electrons in N-shell. The atomic number of the element is (a) 18 (b) 28 (c) 22 (d) 26

Short answer

The atomic number of the element is (b) 28.

Step-by-step solution

Remember the electron capacity of shells

The maximum number of electrons that can be accommodated in the shells of an atom are given by the formula 2n^2, where n is the principal quantum number for shells K, L, M, N... corresponding to n values 1, 2, 3, 4 respectively. Therefore the electron capacity for K-shell is 2(1)^2=2, for L-shell is 2(2)^2=8, for M-shell is 2(3)^2=18, and for N-shell is 2(4)^2=32.

Calculate the total number of electrons

The K-shell (n=1) is completely filled, so there are 2 electrons in the K-shell. The L-shell (n=2) is also completely filled, so there are 8 electrons in the L-shell. According to the problem, there are 16 electrons in the M-shell, and 2 electrons in the N-shell. Therefore, the total number of electrons is 2 + 8 + 16 + 2 = 28.

Determine the atomic number

The atomic number of an element is equal to the number of protons in its nucleus. Since atoms are neutral, the number of protons is equal to the number of electrons. Therefore, the atomic number is also 28.

Key concepts

Electron Shell Capacity

Understanding the electron shell capacity is crucial to grasp the structure of atoms in various elements. The shells of an atom are akin to concentric circles around the nucleus, each capable of holding a set number of electrons. These capacities are determined by a simple formula: \( 2n^2 \) where \( n \) is the principal quantum number, representing the shell level with values starting from 1 for the innermost shell (K-shell) and increasing for outer shells.

For instance, the first shell can hold a maximum of 2(1)^2 = 2 electrons, while the second shell can carry up to 2(2)^2 = 8, reflecting the filled electron configurations for both K and L shells mentioned in the exercise. Understanding this concept allows students to calculate the number of electrons that a completely filled shell can accommodate and plays a critical role in identifying an element's atomic structure.

Principal Quantum Number

At the heart of electron shell capacity lies the concept of the principal quantum number, designated by the symbol \( n \). It's a positive integer that denotes the energy level and size of an electron orbital. As \( n \) increases, the electrons are further from the nucleus, which thereby increases the shell capacity as well. The principal quantum number helps explain the properties of various shells inside an atom. For example, in our exercise, the distinction between the K (\( n=1 \) ), L (\( n=2 \) ), M (\( n=3 \) ), and N (\( n=4 \) ) shells are critical for accurately counting the total number of electrons, which in turn aids in the determination of an element's atomic number. Moreover, quantum numbers have extensive applications in quantum mechanics for describing electron configurations in more detail.

Chemical Elements Atomic Structure

The atomic structure of chemical elements is a meticulously ordered arrangement led by the atomic number, essential for distinguishing one element from another. Each atom consists of a dense nucleus surrounded by rapidly moving electrons within shells. The atomic number, which equals the number of protons and by extension the number of electrons in a neutral atom, determines the element's identity.

For clarity, take the provided exercise as an example. Once the calculation of electrons in the K, L, M, and N shells is done, the sum reflects the atomic number of the element. With 2 electrons in the K-shell (closest to the nucleus), 8 in the L-shell, 16 in the M-shell, and 2 in the N-shell, this adds up to 28. Consequently, the atomic number is 28, which identifies the element as nickel (Ni). This clear relationship helps students to not only relate atomic structure with the periodic table but also illustrates the underlying order of the nature of matter.