Let `E=(-1me^(4))/(8epsilon_(0)^(2)n^(2)h^(2))` be the energy of the `n^(th)` level of H-atom state and radiation of frequency `(E_(2)-E_(1))//h` falls on it ,

Options

A. it will not be absored at all
B. some of atoms will move to the first excited state
C. all atoms wil be excited to the `n = 2` state
D. no atoms will make a transition to the `n = 3` state

Explanation

To solve the problem, we need to analyze the energy levels of the hydrogen atom and the radiation frequency falling on it. We will follow these steps:

### Step 1: Identify the energy levels of the hydrogen atom.
The energy of the nth level of the hydrogen atom is given by the formula:

\[ E_n = -\frac{1 m^4}{8 \epsilon_0^2 n^2 h^2} \]

This can also be expressed in electron volts as:

\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

### Step 2: Calculate the energy for the first three levels (n=1, n=2, n=3).
1. For \( n = 1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

2. For \( n = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]

3. For \( n = 3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \]

### Step 3: Calculate the energy differences between the levels.
1. Energy difference from \( n = 1 \) to \( n = 2 \):
\[ E_2 - E_1 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \]

2. Energy difference from \( n = 1 \) to \( n = 3 \):
\[ E_3 - E_1 = (-1.51) - (-13.6) = 12.09 \, \text{eV} \]

### Step 4: Determine the frequency of the radiation.
The frequency of the radiation that falls on the atom is given by:

\[ \nu = \frac{E_2 - E_1}{h} = \frac{10.2 \, \text{eV}}{h} \]

### Step 5: Analyze the absorption of radiation.
Since the energy of the radiation (10.2 eV) is equal to the energy required for the transition from \( n = 1 \) to \( n = 2 \), we can conclude:

- Some atoms will absorb this energy and transition to the first excited state (\( n = 2 \)).
- Not all atoms will be excited to \( n = 2 \) because the energy is exactly equal to the required energy for the transition, which means only a fraction of the atoms will absorb the energy.
- No atoms will transition to \( n = 3 \) since the energy required for that transition (12.09 eV) is greater than the energy of the radiation (10.2 eV).

### Conclusion:
- Some atoms will move to the first excited state (\( n = 2 \)).
- No atom will make a transition to \( n = 3 \).