The ratio of energies of first two excited states hydrogen atom is

Options

A. `3//1`
B. `1//4`
C. `4//9`
D. `9//4`

Explanation

To find the ratio of the energies of the first two excited states of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Energy Formula
The energy of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Identify the Quantum Numbers
- For the **first excited state**, the principal quantum number \( n \) is **2**.
- For the **second excited state**, the principal quantum number \( n \) is **3**.

### Step 3: Calculate the Energies
Using the formula, we can calculate the energies for both states:

1. **Energy for the first excited state (n = 2)**:
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

2. **Energy for the second excited state (n = 3)**:
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]

### Step 4: Find the Ratio of Energies
To find the ratio of the energies of the first and second excited states, we can use:
\[ \text{Ratio} = \frac{E_3}{E_2} \]
Substituting the values we calculated:
\[ \text{Ratio} = \frac{-1.51 \, \text{eV}}{-3.4 \, \text{eV}} = \frac{1.51}{3.4} \]

### Step 5: Simplify the Ratio
To simplify the ratio:
\[ \text{Ratio} = \frac{1.51}{3.4} = \frac{9}{4} \]

### Conclusion
Thus, the ratio of the energies of the first two excited states of the hydrogen atom is:
\[ \frac{E_3}{E_2} = \frac{9}{4} \]