Energy and radius of first Bohr orbit of and are [Given ]
- A
- B
- C
- D
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Correct answer: C
The energy and radius of the first Bohr orbit for a hydrogen-like atom is given by:
Energy:
$ E_n = \frac{-2.18 \times 10^{-18} \times Z^2}{n^2} \, \text{J} $
Radius:
$ r_n = \frac{52.9 \times n^2}{Z} \, \text{pm} $
Where:
$ Z $ is the atomic number.
$ n $ is the orbit number (or principal quantum number).
Let's calculate these values for the ions $\text{He}^+$ and $\text{Li}^{2+}$:
For $\text{He}^+$:
The atomic number, $ Z = 2 $
Principal quantum number, $ n = 1 $
Energy, $ E_{\text{He}^+} $:
$ E_{\text{He}^+} = -2.18 \times 10^{-18} \times 2^2 = -8.72 \times 10^{-18} \, \text{J} $
Radius, $ r_{\text{He}^+} $:
$ r_{\text{He}^+} = \frac{52.9 \times 1^2}{2} = 26.45 \, \text{pm} $
For $\text{Li}^{2+}$:
The atomic number, $ Z = 3 $
Principal quantum number, $ n = 1 $
Energy, $ E_{\text{Li}^{2+}} $:
$ E_{\text{Li}^{2+}} = -2.18 \times 10^{-18} \times 3^2 = -19.62 \times 10^{-18} \, \text{J} $
Radius, $ r_{\text{Li}^{2+}} $:
$ r_{\text{Li}^{2+}} = \frac{52.9 \times 1^2}{3} = 17.63 \, \text{pm} $
These calculations show the energy and radius for the first Bohr orbit of $\text{He}^+$ and $\text{Li}^{2+}$.
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