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 absorbed at all
B. some of atoms will move to the first excited state
C. all atoms will be excited to the `n=2` state
D. all 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 effect of the radiation frequency on these levels. The energy of the nth level of the hydrogen atom is given by:

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

This can also be expressed in electron volts as:

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

### Step 1: Calculate the energies of the first two levels (n=1 and n=2)

1. For \( n = 1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

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

### Step 2: Calculate the energy difference between the two levels

The energy difference \( E_2 - E_1 \) is calculated as follows:

\[ E_2 - E_1 = (-3.4 \, \text{eV}) - (-13.6 \, \text{eV}) = -3.4 + 13.6 = 10.2 \, \text{eV} \]

### Step 3: Determine the frequency of the radiation

The frequency \( \nu \) of the radiation that corresponds to this energy difference can be calculated using the formula:

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

### Step 4: Analyze the statements regarding the absorption of radiation

1. **Statement A**: "It will not be absorbed at all." - This statement is false because the energy of the radiation (10.2 eV) matches the energy required for the transition from the ground state (n=1) to the first excited state (n=2).

2. **Statement B**: "Some of the atoms will move to the first excited state." - This statement is true because the energy of the radiation is exactly equal to the energy difference between the two states.

3. **Statement C**: "All atoms will be excited to n=2 state." - This statement is false because not all atoms will absorb the radiation due to various factors, including energy distribution and collisions.

4. **Statement D**: "All atoms will make a transition to n=3 state." - This statement is also false because the energy required for a transition to n=3 is greater than the energy of the radiation (12.09 eV).

### Conclusion

The correct statement is **B**: "Some of the atoms will move to the first excited state."

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