Instead of principle quantum number (n), azimuthal quantum number (L) &magnetic quqntum number m,
a set of new quantum number s, t & u was introduce with similar logic but different value as defind
below
`s=1,2,3,...........oo` all positive integral values.
`t=(s^(2)-1^(2)),(s^(2)-2^(2)),(s^(2)-3^(2))`......... No negative value
`u=((t+1))/2"to"+((t+1))/2`(including zero , if any ) in integaral step .
Each orbital can have maximum four electrons .
(s+t) rule is defined ,similar to (n+l) rule.
Number of electrons that can be accommodated in s=2 and s=3 shell.

Options

A. 14, 38
B. 28, 76
C. 8, 28
D. None of these

Explanation

To solve the problem, we need to calculate the number of electrons that can be accommodated in the shells defined by the new quantum numbers \( s \), \( t \), and \( u \) for \( s = 2 \) and \( s = 3 \).

### Step-by-Step Solution:

**Step 1: Determine the values of \( t \) for \( s = 2 \)**

Using the formula for \( t \):
\[ t = s^2 - n^2 \quad \text{for } n = 1, 2, 3, \ldots \]
For \( s = 2 \):
- \( t_1 = 2^2 - 1^2 = 4 - 1 = 3 \)
- \( t_2 = 2^2 - 2^2 = 4 - 4 = 0 \)
- \( t_3 = 2^2 - 3^2 = 4 - 9 = -5 \) (not valid)

Thus, the permissible values of \( t \) for \( s = 2 \) are \( t = 3 \) and \( t = 0 \).

**Step 2: Determine the values of \( u \) for \( s = 2 \)**

Using the formula for \( u \):
\[ u = \left(-t + 1\right)/2 \text{ to } \left(t + 1\right)/2 \]

For \( t = 0 \):
\[ u = \left(-0 + 1\right)/2 \text{ to } \left(0 + 1\right)/2 \implies u = 0 \text{ to } 0 \quad \text{(1 value)} \]

For \( t = 3 \):
\[ u = \left(-3 + 1\right)/2 \text{ to } \left(3 + 1\right)/2 \implies u = -1 \text{ to } 2 \quad \text{(4 values: -1, 0, 1, 2)} \]

**Total orbitals for \( s = 2 \):**
- From \( t = 0 \): 1 orbital
- From \( t = 3 \): 4 orbitals

Total orbitals = \( 1 + 4 = 5 \)

**Step 3: Calculate the number of electrons for \( s = 2 \)**

Each orbital can hold a maximum of 4 electrons:
\[ \text{Total electrons} = 5 \text{ orbitals} \times 4 \text{ electrons/orbital} = 20 \text{ electrons} \]

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**Step 4: Determine the values of \( t \) for \( s = 3 \)**

For \( s = 3 \):
- \( t_1 = 3^2 - 1^2 = 9 - 1 = 8 \)
- \( t_2 = 3^2 - 2^2 = 9 - 4 = 5 \)
- \( t_3 = 3^2 - 3^2 = 9 - 9 = 0 \)
- \( t_4 = 3^2 - 4^2 = 9 - 16 = -7 \) (not valid)

Thus, the permissible values of \( t \) for \( s = 3 \) are \( t = 8, 5, 0 \).

**Step 5: Determine the values of \( u \) for \( s = 3 \)**

For \( t = 0 \):
\[ u = \left(-0 + 1\right)/2 \text{ to } \left(0 + 1\right)/2 \implies u = 0 \text{ to } 0 \quad \text{(1 value)} \]

For \( t = 5 \):
\[ u = \left(-5 + 1\right)/2 \text{ to } \left(5 + 1\right)/2 \implies u = -2 \text{ to } 3 \quad \text{(6 values: -2, -1, 0, 1, 2, 3)} \]

For \( t = 8 \):
\[ u = \left(-8 + 1\right)/2 \text{ to } \left(8 + 1\right)/2 \implies u = -3.5 \text{ to } 4.5 \quad \text{(9 values: -3, -2, -1, 0, 1, 2, 3, 4)} \]

**Total orbitals for \( s = 3 \):**
- From \( t = 0 \): 1 orbital
- From \( t = 5 \): 6 orbitals
- From \( t = 8 \): 9 orbitals

Total orbitals = \( 1 + 6 + 9 = 16 \)

**Step 6: Calculate the number of electrons for \( s = 3 \)**

\[ \text{Total electrons} = 16 \text{ orbitals} \times 4 \text{ electrons/orbital} = 64 \text{ electrons} \]

### Final Answers:
- For \( s = 2 \): 20 electrons
- For \( s = 3 \): 64 electrons

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