Understanding Modified Spin Quantum Numbers and Period 4 Elements

In standard quantum mechanics, the spin quantum number ($m_s$) has two possible values: $-\frac{1}{2}$ and $+\frac{1}{2}$. Here, we are given a modified scenario with three possible spin values: $-\frac{1}{2}$, 0, and $+\frac{1}{2}$. All other quantum numbers and orbital filling rules remain unchanged.

Step 1: Determine Electron Capacity per Orbital

In standard quantum mechanics, each orbital can hold a maximum of 2 electrons due to two possible spin states. With three spin states, each orbital can now hold 3 electrons instead of 2.

Step 2: Identify Orbitals in the 4th Period

The 4th period corresponds to the principal quantum number $n=4$. The subshells and their orbital counts are:

• 4s: 1 orbital
• 4p: 3 orbitals
• 4d: 5 orbitals
• 4f: 7 orbitals

However, note that the 4f subshell is not filled until the 6th period (lanthanides). In the 4th period, the order of filling is:

4s → 3d → 4p

So, the relevant subshells for the 4th period are:

• 4s: 1 orbital
• 3d: 5 orbitals (though n=3, it fills during the 4th period)
• 4p: 3 orbitals

Total orbitals = 1 (4s) + 5 (3d) + 3 (4p) = 9 orbitals.

Step 3: Calculate Maximum Electrons in 4th Period

With each orbital holding 3 electrons (due to three spin states), the maximum number of electrons in the 4th period is:

$9\times 3=27$ electrons.

Step 4: Relate Electrons to Number of Elements

Each element in a period corresponds to adding one electron (and one proton). Therefore, the number of elements in the 4th period equals the maximum number of electrons it can hold, which is 27.

Final Answer

27

Related Topics

• Quantum Numbers: Principal (n), azimuthal (l), magnetic (m_l), and spin (m_s) quantum numbers define an electron's state in an atom.
• Aufbau Principle: Electrons fill orbitals in order of increasing energy.
• Hund's Rule: Every orbital in a subshell is singly occupied before any orbital is doubly occupied.
• Periodic Table Structure: Periods correspond to the highest principal quantum number of electrons in elements.

Key Formulae and Theory

Maximum electrons per orbital: Normally 2 (due to two spin states), but here 3.

Maximum electrons in a subshell: $2\times (2l+1)$ for standard case; with three spins, it becomes $3\times (2l+1)$.

Orbital counting: s subshell has 1 orbital, p has 3, d has 5, f has 7.

Elements in a period: Equal to the total number of electrons in the outermost shell when filled.