The minimum enegry required to excite a hydrogen atom from its ground state is

Options

A. `13.6 eV`
B. `-13.6 eV`
C. `3.4 eV`
D. `10.2 eV`

Explanation

To find the minimum energy required to excite a hydrogen atom from its ground state, we can use the formula for the energy levels of a hydrogen atom. The energy of an electron in a hydrogen atom at a principal quantum number \( n \) is given by:

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

### Step 1: Identify the ground state and the excited state
The ground state of a hydrogen atom corresponds to \( n_1 = 1 \). The first excited state corresponds to \( n_2 = 2 \).

### Step 2: Write the energy difference formula
The minimum energy required to excite the atom from the ground state to the first excited state can be calculated using the formula:

\[ \Delta E = E_{n_2} - E_{n_1} = -\frac{13.6 \, \text{eV}}{n_2^2} - \left(-\frac{13.6 \, \text{eV}}{n_1^2}\right) \]

### Step 3: Substitute the values of \( n_1 \) and \( n_2 \)
Substituting \( n_1 = 1 \) and \( n_2 = 2 \):

\[ \Delta E = -\frac{13.6 \, \text{eV}}{2^2} + \frac{13.6 \, \text{eV}}{1^2} \]

### Step 4: Calculate the energy values
Calculating the individual energy levels:

\[ E_{n_1} = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
\[ E_{n_2} = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

### Step 5: Find the energy difference
Now, substituting these values back into the energy difference equation:

\[ \Delta E = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = -3.4 \, \text{eV} + 13.6 \, \text{eV} = 10.2 \, \text{eV} \]

### Conclusion
Thus, the minimum energy required to excite a hydrogen atom from its ground state is:

\[ \Delta E = 10.2 \, \text{eV} \]