Physics Q40 – JEE-MAIN 2025

Assuming the validity of Bohr's atomic model for hydrogen like ions the radius of

L i^{+ + +}

ion in its ground state is given by

\frac{1}{X} a_{0}

, where

X =

_______. (Where

a_{0}

is the first Bohr's radius.)

Clue 1 of 3: Bohr's Model for Hydrogen-like Ions

Recall the fundamental principles of Bohr's atomic model, specifically how it describes the electron orbits in single-electron systems.

🥷

Ninja Strategy•Direct Z-dependence

For the ground state (n=1), the radius is inversely proportional to the atomic number Z. Thus, X must be equal to Z for Lithium.

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Step 1: Identify parameters for the given ion✦ Active

For an ion to be 'hydrogen-like', it must contain only one electron. For Lithium (Li), which has an atomic number $Z = 3$ , the ion with one electron is $L i^{2 +}$ (often denoted as $L i^{+ + +}$ in some contexts to imply a highly ionized state, but specifically $L i^{2 +}$ is hydrogen-like). Thus, for this ion, the atomic number is $Z = 3$ .

The problem states the ion is in its ground state, which means the principal quantum number is $n = 1$ .

Step 2: Apply Bohr's radius formula○ Expand

According to Bohr's atomic model, the radius of the $n^{t h}$ orbit for a hydrogen-like atom is given by the formula:

r_{n} = a_{0} \frac{n^{2}}{Z}

Substitute the values $n = 1$ and $Z = 3$ into the formula:

r = a_{0} \frac{1^{2}}{3} = \frac{1}{3} a_{0}

Step 3: Determine the value of X○ Expand

The problem states that the radius of the $L i^{+ + +}$ ion in its ground state is given by $\frac{1}{X} a_{0}$ . Comparing our calculated radius with the given form:

\frac{1}{X} a_{0} = \frac{1}{3} a_{0}

From this comparison, we can conclude that $X = 3$ .

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Physics Question 40 – JEE-MAIN 2025

Clue 1 of 3: Bohr's Model for Hydrogen-like Ions

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