A sample of hydrogen atom gas contains `100 "atom"`. All the atoms are excited to the same `n^(th)` excited state. The total energy released by all the atoms is `(4800)/(49) Rch` ( where `Rch=13.6eV`), as they come to the ground state through various types of transitions, Find

Options

A. maximum energy of the emitted photon will be less than `(48)/(49)Rch`.
B. maximum energy of the emitted photon may be grater than `(48)/(49) Rch`
C. the value of `n=6`
D. maximum total number of photons that can be emitted by this sample may be less than 600.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have a sample of hydrogen gas containing 100 atoms, all of which are excited to the same \( n^{th} \) excited state. The total energy released as these atoms transition to the ground state is given as:

\[ E_{\text{total}} = \frac{4800}{49} R_{ch} \]

where \( R_{ch} = 13.6 \, \text{eV} \).

### Step 2: Calculate the Energy Released per Atom
The energy released by all 100 atoms can be divided by the number of atoms to find the energy released per atom:

\[ E_{\text{per atom}} = \frac{E_{\text{total}}}{100} = \frac{4800}{49 \times 100} R_{ch} = \frac{48}{49} R_{ch} \]

### Step 3: Relate Energy to Quantum States
The energy released when an atom transitions from the \( n^{th} \) excited state to the ground state (n=1) can be expressed using the formula:

\[ E = R_{ch} \left( \frac{1}{1^2} - \frac{1}{n^2} \right) = R_{ch} \left( 1 - \frac{1}{n^2} \right) \]

### Step 4: Set the Energies Equal
We can now set the energy released per atom equal to the expression derived from the quantum states:

\[ \frac{48}{49} R_{ch} = R_{ch} \left( 1 - \frac{1}{n^2} \right) \]

### Step 5: Cancel \( R_{ch} \) and Solve for \( n \)
Cancelling \( R_{ch} \) from both sides gives:

\[ \frac{48}{49} = 1 - \frac{1}{n^2} \]

Rearranging this equation:

\[ \frac{1}{n^2} = 1 - \frac{48}{49} = \frac{1}{49} \]

Taking the reciprocal:

\[ n^2 = 49 \]

Thus,

\[ n = 7 \]

### Step 6: Determine the Maximum Number of Photons
When transitioning from the \( n^{th} \) excited state to the ground state, the maximum number of photons emitted corresponds to the number of transitions that can occur. For an atom in the \( n^{th} \) state, the maximum number of transitions is \( n-1 \). Therefore, for \( n = 7 \):

\[ \text{Maximum number of photons} = 6 \]

### Conclusion
The value of \( n \) is 7, and the maximum number of photons that can be emitted is 6.

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