Consider atoms `H, He^(+), Li^(++)` in their ground states. Suppose `E_(1), E_(2)` and `E_(3)` are minimum energies required so that the atoms `H He^(+), Li^(++)` can achieve their first excited states respectively, then

Options

A. `E_(1)=E_(2)=E_(3)`
B. `E_(1) gt E_(2) gt E_(3)`
C. `E_(1) lt E_(2) lt E_(3)`
D. `E_(1) = E_(2) E_(3)`

Explanation

To find the minimum energies \( E_1, E_2, \) and \( E_3 \) required for the hydrogen atom (H), helium ion (He\(^+\)), and lithium ion (Li\(^{++}\)) to achieve their first excited states, we can use the formula for the energy levels of hydrogen-like atoms.

### Step 1: Understand the Energy Level Formula
The energy levels for hydrogen-like atoms are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where:
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number.

### Step 2: Calculate the Ground State Energy
For the ground state (\( n = 1 \)):
- For H (\( Z = 1 \)):
\[ E_1 = -\frac{1^2 \cdot 13.6}{1^2} = -13.6 \, \text{eV} \]
- For He\(^+\) (\( Z = 2 \)):
\[ E_2 = -\frac{2^2 \cdot 13.6}{1^2} = -54.4 \, \text{eV} \]
- For Li\(^{++}\) (\( Z = 3 \)):
\[ E_3 = -\frac{3^2 \cdot 13.6}{1^2} = -122.4 \, \text{eV} \]

### Step 3: Calculate the First Excited State Energy
For the first excited state (\( n = 2 \)):
- For H (\( Z = 1 \)):
\[ E_{1,2} = -\frac{1^2 \cdot 13.6}{2^2} = -3.4 \, \text{eV} \]
- For He\(^+\) (\( Z = 2 \)):
\[ E_{2,2} = -\frac{2^2 \cdot 13.6}{2^2} = -13.6 \, \text{eV} \]
- For Li\(^{++}\) (\( Z = 3 \)):
\[ E_{3,2} = -\frac{3^2 \cdot 13.6}{2^2} = -30.6 \, \text{eV} \]

### Step 4: Calculate the Energy Differences
Now, we find the minimum energies required to excite each atom from the ground state to the first excited state:
- For H:
\[ E_1 = E_{1,2} - E_1 = -3.4 - (-13.6) = 10.2 \, \text{eV} \]
- For He\(^+\):
\[ E_2 = E_{2,2} - E_2 = -13.6 - (-54.4) = 40.8 \, \text{eV} \]
- For Li\(^{++}\):
\[ E_3 = E_{3,2} - E_3 = -30.6 - (-122.4) = 91.8 \, \text{eV} \]

### Step 5: Compare the Energies
Now we can compare the energies:
- \( E_1 = 10.2 \, \text{eV} \) (H)
- \( E_2 = 40.8 \, \text{eV} \) (He\(^+\))
- \( E_3 = 91.8 \, \text{eV} \) (Li\(^{++}\))

### Conclusion
The order of energies required to excite the atoms is:
\[ E_1 < E_2 < E_3 \]
Thus, the correct option is that the minimum energy required to excite hydrogen is the least, followed by helium, and then lithium.