The minimum energy that must be given to a H atom in ground state so that it can emit an H, line in balmer series is

Options

A. `12.4eV`
B. `10.2eV`
C. `13.06eV`
D. `13.6eV`

Explanation

To find the minimum energy that must be given to a hydrogen atom in the ground state so that it can emit a line in the Balmer series, we can follow these steps:

### Step 1: Understand the Balmer Series
The Balmer series corresponds to electronic transitions in a hydrogen atom where the electron moves from a higher energy level (n ≥ 3) to the second energy level (n = 2). The transitions that produce visible light in the Balmer series include n = 3 to n = 2 (H-alpha), n = 4 to n = 2 (H-beta), n = 5 to n = 2 (H-gamma), and so on.

### Step 2: Identify the Transition
For the H-gamma line, the transition occurs from n = 5 to n = 2. Therefore, we need to calculate the energy difference between these two states.

### Step 3: Calculate the Energy Levels
The energy of an electron in a hydrogen atom at a given energy level n is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

For n = 5:
\[ E_5 = -\frac{13.6 \, \text{eV}}{5^2} = -\frac{13.6 \, \text{eV}}{25} = -0.544 \, \text{eV} \]

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

### Step 4: Calculate the Energy Difference
The energy difference (ΔE) required for the transition from n = 5 to n = 2 is:
\[ \Delta E = E_2 - E_5 \]
\[ \Delta E = (-3.4 \, \text{eV}) - (-0.544 \, \text{eV}) \]
\[ \Delta E = -3.4 + 0.544 \]
\[ \Delta E = -2.856 \, \text{eV} \]

### Step 5: Determine the Minimum Energy Required
The minimum energy that must be supplied to the hydrogen atom in the ground state to enable this transition is the absolute value of the energy difference calculated:
\[ \text{Minimum Energy} = 2.856 \, \text{eV} \]

### Conclusion
Thus, the minimum energy that must be given to a hydrogen atom in the ground state so that it can emit a line in the Balmer series (specifically the H-gamma line) is approximately **2.856 eV**.

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