The ratio of maximum to minimum possible radiation energy Bohr's hypothetical hydrogen atom is equal to

Options

A. 2
B. 4
C. `4//3`
D. `3//2`

Explanation

To solve the problem of finding the ratio of maximum to minimum possible radiation energy in Bohr's hypothetical hydrogen atom, we can follow these steps:

### Step 1: Understand the Concept of Energy Levels
In a hydrogen atom, the energy of an electron in a particular orbit (energy level) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number (n = 1, 2, 3, ...).

### Step 2: Determine Maximum Energy
The maximum energy corresponds to the ionization energy, which is the energy required to remove the electron completely from the atom. This occurs when the electron is at infinity (n = ∞). Thus, the maximum energy is:
\[ E_{\text{max}} = E_{\infty} = 0 \, \text{eV} \text{ (since it is defined as zero energy at infinity)} \]
However, for practical purposes in this context, we consider the energy required to remove the electron from the ground state (n = 1) to infinity:
\[ E_{\text{max}} = 13.6 \, \text{eV} \]

### Step 3: Determine Minimum Energy
The minimum energy corresponds to the transition from the second energy level (n = 2) to the first energy level (n = 1). We can calculate this energy using the energy level formula:
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
The energy difference (minimum energy) for the transition from n = 2 to n = 1 is:
\[ E_{\text{min}} = E_1 - E_2 = (-13.6) - (-3.4) = -13.6 + 3.4 = -10.2 \, \text{eV} \]

### Step 4: Calculate the Ratio
Now we can find the ratio of maximum to minimum energy:
\[ \text{Ratio} = \frac{E_{\text{max}}}{E_{\text{min}}} = \frac{13.6 \, \text{eV}}{10.2 \, \text{eV}} \]

### Step 5: Simplify the Ratio
To simplify this ratio:
\[ \text{Ratio} = \frac{13.6}{10.2} \approx 1.333 \]
This can be expressed as:
\[ \text{Ratio} = \frac{4}{3} \]

### Final Answer
Thus, the ratio of maximum to minimum possible radiation energy in Bohr's hypothetical hydrogen atom is:
\[ \frac{4}{3} \]