Question

Possible set of quantum numbers for which: \(n=4, l=3\) and \(s=+\frac{1}{2}\) is (a) 14 (b) 7 (c) 5 (d) 10

Short answer

7 possible sets

Step-by-step solution

Understanding Quantum Numbers

The principal quantum number () determines the energy level of an electron in an atom. The value of atomic orbital angular momentum quantum number () must be in the range from 0 to n-1. The magnetic quantum number (), can have values from - to +, including zero. The spin quantum number () can be +1/2 or -1/2 representing the two possible spin states of an electron.

Determine the Possible Values for the Magnetic Quantum Number ()

Since the orbital angular momentum quantum number () is 3, the possible values for the magnetic quantum number () will be the integers from -3 to +3. There are 7 possible values in total: -3, -2, -1, 0, 1, 2, and 3.

Count the Number of Possible Sets

Each of these 7 values of () can be paired with either +1/2 or -1/2 spin quantum number (). However, we are only considering the case where () is +1/2. Therefore, we only count each of the 7 values of () once, instead of twice. This gives us 7 possible sets of quantum numbers.

Key concepts

Principal Quantum Number (n)

The principal quantum number (), an integral value starting from 1, depicts the electron's energy level in an atom. It plays a key role in determining the electron's energy and, in part, the distance of the electron from the nucleus. Think of it as the address' floor number, where higher values of indicate higher energy levels—like moving to a higher floor in a building.

As increases, the orbitals become larger, increasing the possibility of an electron being further from the nucleus. It's critical to remember that only positive integers (1,2,3,...) can be used for , aligning with the quantized nature of electron energy levels. The maximum number of electrons that can fit in a particular energy level is given by the formula 2.

Orbital Angular Momentum Quantum Number (l)

The orbital angular momentum quantum number (), ranging from 0 up to ( - 1), defines the shape of an electron's orbital. It's like specifying the type of apartment within the building's floor. The orbitals, with distinctive shapes such as spherical (s), dumbbell (p), cloverleaf (d), and complex multi-lobed (f) shapes, are represented by = 0, 1, 2, and 3, respectively.

Understanding the possible values of is crucial for visualizing where an electron is likely to be found within the atom. The different shapes imply different modes of electron cloud distribution in space, greatly influencing chemical bonding and electron-pair repulsion theories.

Magnetic Quantum Number (ml)

The magnetic quantum number (), whose values range from - to + (including zero), specifies the orbital's orientation in space relative to the other orbitals. Think of it as the apartment number that gives a unique identity to each orbital within a sublevel. For an of 3, as in our example, values can be -3, -2, -1, 0, 1, 2, or 3, resulting in a total of 7 possible orientations.

This quantum number becomes highly relevant in the presence of a magnetic field, which is where the term 'magnetic' originates from. It can affect the energy of the electrons slightly when the atom is in such an environment, a phenomenon known as the Zeeman effect.

Spin Quantum Number (s)

The spin quantum number () indicates the intrinsic spin of an electron within its orbital. This binary property provides each electron with one of two possible spin states: +1/2 or -1/2. You can imagine this as the electron spinning on its own axis, clockwise or counterclockwise.

Electrons behaving like tiny magnets due to their spin introduce the concept of 'spin pairs'. In order for two electrons to occupy the same orbital, they must have opposite spins, a principle called the Pauli exclusion principle. This quantum number not only determines magnetic properties of an electron but is also fundamental for the structure of the periodic table and the arrangement of electrons in an atom.