The ratio of energies of first excited state of `He^+` ion and ground state of `H^−`atom is

Options

A. 1:1
B. 1:4
C. 4:1
D. 16:1

Explanation

To find the ratio of the energies of the first excited state of the `He^+` ion and the ground state of the `H^-` atom, we can use the formula for the energy levels of hydrogen-like atoms:

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

Where:
- \(E_n\) is the energy of the level,
- \(Z\) is the atomic number,
- \(n\) is the principal quantum number.

### Step 1: Calculate the energy of the first excited state of `He^+`

For the `He^+` ion:
- Atomic number \(Z = 2\)
- Principal quantum number for the first excited state \(n = 2\)

Using the formula:
\[ E_{He^+} = -\frac{13.6 \times 2^2}{2^2} = -\frac{13.6 \times 4}{4} = -13.6 \, \text{eV} \]

### Step 2: Calculate the energy of the ground state of `H^-`

For the `H^-` atom:
- Atomic number \(Z = 1\)
- Principal quantum number for the ground state \(n = 1\)

Using the formula:
\[ E_{H^-} = -\frac{13.6 \times 1^2}{1^2} = -\frac{13.6 \times 1}{1} = -13.6 \, \text{eV} \]

### Step 3: Calculate the ratio of the energies

Now, we can find the ratio of the energies of `He^+` and `H^-`:
\[ \text{Ratio} = \frac{E_{He^+}}{E_{H^-}} = \frac{-13.6}{-13.6} = 1 \]

### Final Answer

Thus, the ratio of the energies of the first excited state of `He^+` ion to the ground state of `H^-` atom is:
\[ \text{Ratio} = 1:1 \]