Which enrgy state of doubly ionized lithium `(Li^(++)` has the same energy as that of the gorund state of hydrogen?

Options

A. n=1
B. n=2
C. n =3
D. n=4

Explanation

To find which energy state of doubly ionized lithium (Li\(^{++}\)) has the same energy as that of the ground state of hydrogen, we can follow these steps:

### Step 1: Understand the energy of the ground state of hydrogen
The energy of the ground state of hydrogen (n=1) is given by:
\[ E_H = -13.6 \, \text{eV} \]

### Step 2: Write the formula for the energy levels of hydrogen-like atoms
The energy levels of a hydrogen-like atom can be expressed as:
\[ E_n = -\frac{Z^2}{n^2} \times 13.6 \, \text{eV} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 3: Identify the atomic number for Li\(^{++}\)
For doubly ionized lithium (Li\(^{++}\)), the atomic number \( Z \) is 3.

### Step 4: Set the energy of Li\(^{++}\) equal to the energy of hydrogen
We want to find the value of \( n \) such that:
\[ E_n = -\frac{3^2}{n^2} \times 13.6 \, \text{eV} = -13.6 \, \text{eV} \]

### Step 5: Simplify the equation
Substituting the values, we have:
\[ -\frac{9}{n^2} \times 13.6 = -13.6 \]

### Step 6: Cancel out the negative signs and simplify
Cancelling out \( -13.6 \) from both sides gives:
\[ \frac{9}{n^2} = 1 \]

### Step 7: Solve for \( n^2 \)
Multiplying both sides by \( n^2 \) gives:
\[ 9 = n^2 \]

### Step 8: Find \( n \)
Taking the square root of both sides results in:
\[ n = 3 \]

### Conclusion
The energy state of doubly ionized lithium (Li\(^{++}\)) that has the same energy as that of the ground state of hydrogen is:
\[ n = 3 \]

### Final Answer
The correct option is **n = 3**.

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