Energy of an electron in a hydrogen atom is calculated as `E_n=(-13.6)/n^2` eV. Is it possible for an electron in hydrogen atom to have energy of 2.8 eV

Answer

To determine if an electron in a hydrogen atom can have an energy of 2.8 eV, we will use the formula for the energy levels of the hydrogen atom, which is given by:

\[ E_n = -\frac{13.6}{n^2} \text{ eV} \]

where \( n \) is the principal quantum number (an integer: 1, 2, 3, ...).

### Step 1: Set up the equation
We want to find if there exists an integer \( n \) such that:

\[ E_n = 2.8 \text{ eV} \]

Substituting into the energy formula, we have:

\[ 2.8 = -\frac{13.6}{n^2} \]

### Step 2: Rearranging the equation
To find \( n^2 \), we rearrange the equation:

\[ n^2 = -\frac{13.6}{2.8} \]

### Step 3: Calculate \( n^2 \)
Now we will calculate \( -\frac{13.6}{2.8} \):

\[ n^2 = -\frac{13.6}{2.8} = -4.857 \]

### Step 4: Analyze the result
Since \( n^2 \) must be a positive integer (as \( n \) is a whole number), the value of \( -4.857 \) is not valid. This indicates that there is no integer \( n \) that satisfies the equation.

### Conclusion
Thus, it is not possible for an electron in a hydrogen atom to have an energy of 2.8 eV.

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