What are the four quantum numbers for each of the two electrons in a 4s orbital? For the first electron:
n=4,l=0,ml=0,ms=+21
For the second electron:
n=4,l=0,ml=0,ms=−21
The Correct Answer and Explanation is:
In quantum mechanics, electrons in an atom are described by a set of four quantum numbers, each of which provides specific information about the electron’s state. These quantum numbers are:
- Principal Quantum Number (n): Indicates the energy level of the electron and its average distance from the nucleus. It can take positive integer values (1, 2, 3, …). For both electrons in the 4s orbital, ( n = 4 ), indicating they are in the fourth energy level.
- Azimuthal Quantum Number (l): Relates to the shape of the orbital and can take values from 0 to ( n-1 ). The value of ( l ) for the 4s orbital is ( l = 0 ), corresponding to an s orbital (spherical shape).
- Magnetic Quantum Number (ml): Describes the orientation of the orbital in space. For s orbitals, the only value is ( m_l = 0 ) since there is no angular momentum component along any axis.
- Spin Quantum Number (ms): Refers to the intrinsic spin of the electron and can be either ( +\frac{1}{2} ) or ( -\frac{1}{2} ). This represents the two possible orientations of the electron’s spin.
Quantum Numbers for the Electrons in a 4s Orbital
For the first electron in the 4s orbital:
- ( n = 4 )
- ( l = 0 )
- ( m_l = 0 )
- ( m_s = +\frac{1}{2} )
For the second electron in the 4s orbital:
- ( n = 4 )
- ( l = 0 )
- ( m_l = 0 )
- ( m_s = -\frac{1}{2} )
Explanation
The four quantum numbers collectively describe the state of each electron in an atom. In the case of the two electrons in the 4s orbital, both share the same principal, azimuthal, and magnetic quantum numbers, indicating they occupy the same spatial orbital. However, they differ in their spin quantum numbers. According to the Pauli Exclusion Principle, no two electrons in the same atom can have identical sets of quantum numbers. This principle is why one electron has a spin of ( +\frac{1}{2} ) while the other has a spin of ( -\frac{1}{2} ).
In summary, both electrons are in the same energy level and orbital type, yet their distinct spin states allow them to coexist within the same 4s orbital. This arrangement illustrates the foundational principles of quantum mechanics in atomic structure and electron configuration.