Calculate the minimum energy that must be given to a hydrogen atom so that it can emit the `H_(beta)` line of Balmer series.

Answer

To calculate the minimum energy that must be given to a hydrogen atom so that it can emit the H-beta line of the Balmer series, we can follow these steps:

### Step 1: Identify the energy levels involved
The H-beta line corresponds to the transition from the n=4 level to the n=2 level in the hydrogen atom. In the Balmer series, the transitions are from higher energy levels (n ≥ 3) to n=2.

### Step 2: Use the formula for energy levels in hydrogen
The energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 3: Calculate the energy of the n=4 level
For \( n=4 \):
\[ E_4 = -\frac{13.6 \, \text{eV}}{4^2} = -\frac{13.6 \, \text{eV}}{16} = -0.85 \, \text{eV} \]

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

### Step 5: Calculate the energy difference for the transition
The energy difference \( \Delta E \) between the n=4 and n=2 levels is:
\[ \Delta E = E_2 - E_4 \]
Substituting the values:
\[ \Delta E = (-3.4 \, \text{eV}) - (-0.85 \, \text{eV}) = -3.4 \, \text{eV} + 0.85 \, \text{eV} = -2.55 \, \text{eV} \]

### Step 6: Determine the minimum energy required
Since energy must be supplied to the atom to reach the n=4 level from the ground state (n=1), we need to find the energy required to excite the atom from n=1 to n=4:
1. Calculate \( E_1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
2. The energy required to excite from n=1 to n=4 is:
\[ E_{\text{required}} = E_4 - E_1 = -0.85 \, \text{eV} - (-13.6 \, \text{eV}) = 12.75 \, \text{eV} \]

### Step 7: Conclusion
The minimum energy that must be given to a hydrogen atom so that it can emit the H-beta line of the Balmer series is:
\[ \text{Minimum Energy} = 12.75 \, \text{eV} \]

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