Question

The statement that is NOT correct is:

• A. Angular quantum number signifies the shape of the orbital
• B. Energies of stationary states in hydrogen like atoms is inversely proportional to the square of the principal quantum number
• C. Total number of nodes for 3s orbital is three.
• D. The radius of the first orbit of \( \text{He}^+ \) is half that of the first orbit of hydrogen atom.

Hint:

Remember the formulas for nodes:
- Radial nodes = \( n - l - 1 \)
- Angular nodes = \( l \)
- Total nodes = radial nodes + angular nodes = \( n - 1 \)

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to examine the given statements about quantum mechanics and atomic structures to identify the incorrect statement.

Step 2: Detailed Explanation:
Let us analyze each option:
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Option (A): The angular quantum number \( l \) (orbital angular momentum quantum number) determines the shape of the orbital (e.g., \( l=0 \) for spherical s, \( l=1 \) for dumbbell p). This statement is correct.
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Option (B): The energy of stationary states in a hydrogen-like atom is given by \( E_n = -13.6 \frac{Z^2}{n^2} \, \text{eV} \). The magnitude of energy is inversely proportional to \( n^2 \). This statement is correct.
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Option (C): The total number of nodes in any orbital is given by the formula \( n - 1 \), where \( n \) is the principal quantum number. For a \( 3s \) orbital, \( n = 3 \). Therefore, the total number of nodes is \( 3 - 1 = 2 \). Thus, stating that the total number of nodes is three is incorrect.
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Option (D): The Bohr radius of the \( n \)-th orbit is given by \( r_n = 0.529 \frac{n^2}{Z} \, \text{\AA} \). For the first orbit (\( n = 1 \)) of hydrogen (\( Z = 1 \)), \( r_H \propto 1 \). For the first orbit (\( n = 1 \)) of \( \text{He}^+ \) (\( Z = 2 \)), \( r_{\text{He}^+} \propto \frac{1}{2} \). Hence, the radius of \( \text{He}^+ \) is half that of hydrogen. This statement is correct.

Step 3: Final Answer:
(C) Total number of nodes for 3s orbital is three.