Step 1: Understanding the Question:
This problem uses Bohr's quantization condition for angular momentum to identify the principal shell of an atom, and then asks for the capacity of that shell in terms of orbitals.
Step 2: Key Formula or Approach:
Bohr’s Postulate states that angular momentum (\(L\)) of an electron in a stationary orbit is quantized:
\[ L = \frac{nh}{2\pi} \]
where \(n\) is the principal quantum number.
The total number of orbitals in any principal shell is given by \(n^2\).
Step 3: Detailed Explanation:
Step A: Equate the given angular momentum value to the Bohr formula.
\[ \frac{nh}{2\pi} = \frac{2h}{\pi} \]
Step B: Solve for \(n\). Cancel out \(h\) and \(\pi\) from both sides:
\[ \frac{n}{2} = 2 \implies n = 4 \]
Step C: Identify the shell. Since \(n = 4\), we are looking at the 4th principal shell (N shell).
Step D: Calculate the total number of orbitals possible in the 4th shell using the formula \(n^2\).
\[ \text{Number of orbitals} = 4^2 = 16 \]
Step E: Verify the result by subshells. The \(n=4\) shell contains the 4s, 4p, 4d, and 4f subshells.
The 4s subshell has 1 orbital.
The 4p subshell has 3 orbitals.
The 4d subshell has 5 orbitals.
The 4f subshell has 7 orbitals.
Total = \(1 + 3 + 5 + 7 = 16\).
It is important not to confuse "orbitals" with "electrons." The maximum number of electrons would be \(2n^2 = 32\), but the question asks for orbitals.
Step 4: Final Answer:
The maximum number of orbitals possible in orbit X is 16.