Find the excitation energy of n = 3 level of He atom

Options

A. 48.4 eV
B. 2.4 eV
C. 3.8 eV
D. 24.2 eV

Answer

To find the excitation energy of the n = 3 level of the helium atom, we can follow these steps:

### Step 1: Understand the concept of excitation energy
Excitation energy is the energy required to move an electron from its ground state (lowest energy level) to a higher energy level. For helium, we will consider the transition from the n = 1 level to the n = 3 level.

### Step 2: Use the formula for the energy levels of hydrogen-like atoms
The energy levels of hydrogen-like atoms can be calculated using the formula:
\[ E_n = -\frac{Z^2 \cdot E_0}{n^2} \]
where:
- \( E_n \) is the energy of the level n,
- \( Z \) is the atomic number (for helium, \( Z = 2 \)),
- \( E_0 = 13.6 \, \text{eV} \) (the ionization energy of hydrogen),
- \( n \) is the principal quantum number.

### Step 3: Calculate the energy of the n = 1 level (E1)
For helium in the ground state (n = 1):
\[ E_1 = -\frac{2^2 \cdot 13.6}{1^2} = -\frac{4 \cdot 13.6}{1} = -54.4 \, \text{eV} \]

### Step 4: Calculate the energy of the n = 3 level (E3)
For helium at the n = 3 level:
\[ E_3 = -\frac{2^2 \cdot 13.6}{3^2} = -\frac{4 \cdot 13.6}{9} = -\frac{54.4}{9} \approx -6.04 \, \text{eV} \]

### Step 5: Find the excitation energy
The excitation energy (ΔE) required to move from n = 1 to n = 3 is given by:
\[ \Delta E = E_3 - E_1 \]
Substituting the values we calculated:
\[ \Delta E = -6.04 - (-54.4) = -6.04 + 54.4 = 48.36 \, \text{eV} \]

### Step 6: Conclusion
The excitation energy of the n = 3 level of the helium atom is approximately:
\[ \Delta E \approx 48.4 \, \text{eV} \]