To which orbit the electron in ground state in the hydrogen atom will jump after absorbing 12.75 eV energy.

Options

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

Answer

To determine to which orbit the electron in the ground state of a hydrogen atom will jump after absorbing 12.75 eV of energy, we can follow these steps:

### Step 1: Understand the Energy Levels of Hydrogen Atom
The energy of an electron in a hydrogen atom at a specific energy level \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number (1, 2, 3, ...).

### Step 2: Calculate the Energy of the Ground State
For the ground state (where \( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Determine the Total Energy After Absorption
When the electron absorbs 12.75 eV of energy, the total energy \( E \) becomes:
\[ E = E_1 + \text{Energy absorbed} = -13.6 \, \text{eV} + 12.75 \, \text{eV} = -0.85 \, \text{eV} \]

### Step 4: Set Up the Energy Equation for the New Orbit
The energy of the electron in the new orbit \( n_2 \) can also be expressed using the energy formula:
\[ E_{n_2} = -\frac{13.6 \, \text{eV}}{n_2^2} \]
Setting this equal to the total energy after absorption:
\[ -\frac{13.6 \, \text{eV}}{n_2^2} = -0.85 \, \text{eV} \]

### Step 5: Solve for \( n_2^2 \)
Rearranging the equation gives:
\[ \frac{13.6}{n_2^2} = 0.85 \]
Multiplying both sides by \( n_2^2 \) and then rearranging gives:
\[ n_2^2 = \frac{13.6}{0.85} \]

### Step 6: Calculate \( n_2^2 \)
Calculating the right side:
\[ n_2^2 = \frac{13.6}{0.85} \approx 16 \]

### Step 7: Find \( n_2 \)
Taking the square root of both sides:
\[ n_2 = \sqrt{16} = 4 \]

### Conclusion
The electron in the ground state of the hydrogen atom will jump to the 4th orbit after absorbing 12.75 eV of energy.

### Final Answer
The answer is \( n = 4 \).

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