The ratio of the difference in energy between the first and second Bohr orbits to that between the second and third Bohr orbit is

Options

A. `1//2`
B. `1//3`
C. `4//9`
D. `27//5`

Answer

To solve the problem of finding the ratio of the difference in energy between the first and second Bohr orbits to that between the second and third Bohr orbits, we will follow these steps:

### Step 1: Calculate the energy of the first Bohr orbit (E1)
The energy of an electron in the nth orbit of a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
For the first orbit (n=1):
\[ E_1 = -\frac{13.6 \, \text{eV} \cdot Z^2}{1^2} = -13.6 \, \text{eV} \cdot Z^2 \]

### Step 2: Calculate the energy of the second Bohr orbit (E2)
For the second orbit (n=2):
\[ E_2 = -\frac{13.6 \, \text{eV} \cdot Z^2}{2^2} = -\frac{13.6 \, \text{eV} \cdot Z^2}{4} = -3.4 \, \text{eV} \cdot Z^2 \]

### Step 3: Calculate the difference in energy between the first and second orbits (ΔE1-2)
\[ \Delta E_{1-2} = E_2 - E_1 = \left(-3.4 \, \text{eV} \cdot Z^2\right) - \left(-13.6 \, \text{eV} \cdot Z^2\right) \]
\[ \Delta E_{1-2} = -3.4 \, \text{eV} \cdot Z^2 + 13.6 \, \text{eV} \cdot Z^2 = 10.2 \, \text{eV} \cdot Z^2 \]

### Step 4: Calculate the energy of the third Bohr orbit (E3)
For the third orbit (n=3):
\[ E_3 = -\frac{13.6 \, \text{eV} \cdot Z^2}{3^2} = -\frac{13.6 \, \text{eV} \cdot Z^2}{9} \approx -1.51 \, \text{eV} \cdot Z^2 \]

### Step 5: Calculate the difference in energy between the second and third orbits (ΔE2-3)
\[ \Delta E_{2-3} = E_3 - E_2 = \left(-1.51 \, \text{eV} \cdot Z^2\right) - \left(-3.4 \, \text{eV} \cdot Z^2\right) \]
\[ \Delta E_{2-3} = -1.51 \, \text{eV} \cdot Z^2 + 3.4 \, \text{eV} \cdot Z^2 = 1.89 \, \text{eV} \cdot Z^2 \]

### Step 6: Calculate the ratio of the two energy differences
Now, we can find the ratio of the differences:
\[ \text{Ratio} = \frac{\Delta E_{1-2}}{\Delta E_{2-3}} = \frac{10.2 \, \text{eV} \cdot Z^2}{1.89 \, \text{eV} \cdot Z^2} \]
\[ \text{Ratio} = \frac{10.2}{1.89} \approx 5.39 \]

### Final Answer
The ratio of the difference in energy between the first and second Bohr orbits to that between the second and third Bohr orbits is approximately **5.39**.

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