Find the binding energy of a hydrogen atom in the state `n = 2`

Explanation

To find the binding energy of a hydrogen atom in the state \( n = 2 \), we can follow these steps:

### Step 1: Understand the formula for binding energy
The binding energy \( E_n \) of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Substitute the value of \( n \)
For the state \( n = 2 \), we substitute \( n \) into the formula:
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} \]

### Step 3: Calculate \( 2^2 \)
Calculate \( 2^2 \):
\[ 2^2 = 4 \]

### Step 4: Substitute back into the formula
Now substitute \( 4 \) back into the equation:
\[ E_2 = -\frac{13.6 \, \text{eV}}{4} \]

### Step 5: Perform the division
Now, perform the division:
\[ E_2 = -3.4 \, \text{eV} \]

### Step 6: State the final answer
Thus, the binding energy of a hydrogen atom in the state \( n = 2 \) is:
\[ E_2 = -3.4 \, \text{eV} \]

### Summary of the Solution
The binding energy of a hydrogen atom in the state \( n = 2 \) is \( -3.4 \, \text{eV} \).

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