Question

What are the values for the four quantum numbers of an electron in the 3d orbital?

Answer 1

Hint :There are four quantum numbers, each representing a characteristic of the electron. The 3d orbital is in the 3rd energy level. And the azimuthal quantum number corresponding to d orbital is 2.

Complete Step By Step Answer:
We know that quantum numbers are a series of numbers that define the position and energy of an electron in an atom. Principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number are the four types of quantum numbers.
We will first look into the four quantum numbers briefly. The principal quantum number represents the energy level or shell in which the electron is present. And the shell corresponding to the quantum number $ n $ can occupy a maximum of $ 2{n^2} $ electrons. We know that azimuthal quantum number $ l $ specifies the subshell in which the electron resides. We are aware that the electrons in a shell with energy level $ n $ can take the values $ 1,2,...,n - 1 $ for their $ l $ and it represents $ s,p,d,f,g,h,... $ sub-shells. Each subshell has a different shape. Also, the magnetic quantum number $ {m_l} $ can take the values $ \left\{ { - l, - (l - 1),..., - 1,0,1,...,l - 1,l} \right\} $ and corresponds to each orientation of orbitals in each subshell. The spin quantum number $ {m_s} $ determines the direction of spin of an electron.
Now, we are asked to find the quantum numbers of electrons in $ 3d $ orbital. Observe that the electrons reside in the third energy level. So, principal quantum number $ n = 3 $ . Also, since the electrons are in $ d $ orbital, their azimuthal quantum number $ l = 2 $ . Now their magnetic quantum number $ {m_l} $ will be $ \left\{ { - 2, - 1,0,1,2} \right\} $ as $ l = 2 $ . Now they can take any one of the spin quantum number $ {m_s} = \left\{ {\dfrac{1}{2}, - \dfrac{1}{2}} \right\} $ .
Hence, the set of quantum numbers for electrons in 3d orbital is $ n = 3 $ , $ l = 2 $ , $ {m_l} = \left\{ { - 2, - 1,0,1,2} \right\} $ and $ {m_s} = \left\{ {\dfrac{1}{2}, - \dfrac{1}{2}} \right\} $ .

Note :
For an electron in an atom, it is uniquely described by these four quantum numbers. That is, no two different electrons can have the same set of quantum numbers. Also, the orbitals are half-filled before they are completely filled.