For a hydrogen atom, the energies that an electron can have are given by the expression, `E=-13.58//n^(2)eV`, where n is an integer. The smallest amount of energy that a hydrogen atom in the ground state can absorb is:

Options

A. 1.00 eV
B. 3.39 eV
C. 6.79 eV
D. 10.19 eV

Answer

To find the smallest amount of energy that a hydrogen atom in the ground state can absorb, we need to follow these steps:

### Step 1: Identify the ground state of the hydrogen atom
The ground state of a hydrogen atom corresponds to the lowest energy level, which is when the principal quantum number \( n = 1 \).

### Step 2: Substitute \( n \) into the energy formula
The energy of the electron in a hydrogen atom is given by the formula:
\[ E = -\frac{13.58}{n^2} \text{ eV} \]
Substituting \( n = 1 \):
\[ E = -\frac{13.58}{1^2} \text{ eV} \]
\[ E = -13.58 \text{ eV} \]

### Step 3: Determine the energy required for absorption
The smallest amount of energy that the hydrogen atom can absorb to transition to a higher energy level is equal to the difference between the energy of the ground state and the energy of the first excited state. The first excited state corresponds to \( n = 2 \).

### Step 4: Calculate the energy of the first excited state
Substituting \( n = 2 \) into the energy formula:
\[ E = -\frac{13.58}{2^2} \text{ eV} \]
\[ E = -\frac{13.58}{4} \text{ eV} \]
\[ E = -3.395 \text{ eV} \]

### Step 5: Calculate the energy difference
The energy absorbed to transition from the ground state to the first excited state is:
\[ \Delta E = E_{n=2} - E_{n=1} \]
\[ \Delta E = (-3.395) - (-13.58) \]
\[ \Delta E = -3.395 + 13.58 \]
\[ \Delta E = 10.185 \text{ eV} \]

### Conclusion
Thus, the smallest amount of energy that a hydrogen atom in the ground state can absorb is **10.185 eV**.

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