The ratio of the energies of the hydrogen atom in its first to second excited state is

Options

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

Answer

To find the ratio of the energies of the hydrogen atom in its first to second excited state, we can follow these steps:

### Step 1: Understand the Energy Formula
According to Bohr's theory, the energy of an electron in a hydrogen atom at a given energy level \( n \) 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 Energy Levels
- The first excited state corresponds to \( n = 2 \).
- The second excited state corresponds to \( n = 3 \).

### Step 3: Calculate the Energy for the First Excited State
Using the formula for \( n = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

### Step 4: Calculate the Energy for the Second Excited State
Using the formula for \( n = 3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]

### Step 5: Find the Ratio of Energies
To find the ratio of the energies \( E_2 \) to \( E_3 \):
\[ \text{Ratio} = \frac{E_2}{E_3} = \frac{-3.4 \, \text{eV}}{-1.51 \, \text{eV}} = \frac{3.4}{1.51} \]

### Step 6: Simplify the Ratio
Calculating the ratio:
\[ \frac{3.4}{1.51} = \frac{3.4 \times 9}{1.51 \times 9} = \frac{30.6}{13.59} \approx 2.25 \]
However, we can also express it in terms of squares of the quantum numbers:
\[ \frac{E_2}{E_3} = \frac{3^2}{2^2} = \frac{9}{4} \]

### Final Answer
The ratio of the energies of the hydrogen atom in its first to second excited state is:
\[ \frac{E_2}{E_3} = \frac{9}{4} \]