The ratio of the differrence between the first and second Bohr orbit energies to that between second and third Bohr orbit energies is

Options

A. `1/3`
B. `27/5`
C. `9/4`
D. `4/9`

Answer

To solve the problem, we need to find the ratio of the difference between the energies of the first and second Bohr orbits to the difference between the energies of the second and third Bohr orbits. Let's break this down step by step.

### Step 1: Calculate the energy of the first Bohr orbit (n=1)
The formula for the energy of the nth Bohr orbit is given by:
\[ E_n = -\frac{13.6 \, Z^2}{n^2} \]
For hydrogen (Z=1) in the first orbit (n=1):
\[ E_1 = -\frac{13.6 \times 1^2}{1^2} = -13.6 \, \text{eV} \]

### Step 2: Calculate the energy of the second Bohr orbit (n=2)
Using the same formula for n=2:
\[ E_2 = -\frac{13.6 \times 1^2}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]

### Step 3: Calculate the energy of the third Bohr orbit (n=3)
For n=3:
\[ E_3 = -\frac{13.6 \times 1^2}{3^2} = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \]

### Step 4: Calculate the difference between the first and second Bohr orbit energies
Now, we find the difference between the energies of the first and second orbits:
\[ E_2 - E_1 = -3.4 - (-13.6) = -3.4 + 13.6 = 10.2 \, \text{eV} \]

### Step 5: Calculate the difference between the second and third Bohr orbit energies
Next, we find the difference between the energies of the second and third orbits:
\[ E_3 - E_2 = -1.51 - (-3.4) = -1.51 + 3.4 = 1.89 \, \text{eV} \]

### Step 6: Calculate the ratio of the differences
Finally, we calculate the ratio of the differences:
\[ \text{Ratio} = \frac{E_2 - E_1}{E_3 - E_2} = \frac{10.2}{1.89} \]

### Step 7: Simplify the ratio
Calculating the ratio:
\[ \frac{10.2}{1.89} \approx 5.4 \]

To express this in a simpler form, we can multiply both the numerator and denominator by 5 to get:
\[ \frac{10.2 \times 5}{1.89 \times 5} \approx \frac{51}{9.45} \approx \frac{27}{5} \]

### Final Answer
The ratio of the difference between the first and second Bohr orbit energies to that between the second and third Bohr orbit energies is:
\[ \text{Ratio} = \frac{27}{5} \]