Light from balmer series of hydrogen is able to eject photoelectron from a metal what can be the maximum work function of the metal?

Explanation

To solve the problem of determining the maximum work function of a metal that can be ejected by light from the Balmer series of hydrogen, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Balmer Series**:
The Balmer series corresponds to the transitions of electrons in a hydrogen atom from higher energy levels (n ≥ 3) to the second energy level (n = 2). The wavelengths of light emitted in this series fall within the visible spectrum.

2. **Identify the Energy Formula**:
The energy of the emitted photon during the transition can be calculated using the formula:
\[ E = 13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( n_1 \) is the lower energy level (2 for Balmer series) and \( n_2 \) is the higher energy level (which can go to infinity for maximum energy).

3. **Set Values for the Formula**:
For the maximum energy in the Balmer series:
- Set \( n_1 = 2 \)
- Set \( n_2 = \infty \)

4. **Calculate the Energy**:
Substitute the values into the energy formula:
\[ E = 13.6 \, \text{eV} \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) \]
Since \( \frac{1}{\infty^2} = 0 \), this simplifies to:
\[ E = 13.6 \, \text{eV} \left( \frac{1}{4} - 0 \right) = 13.6 \, \text{eV} \cdot \frac{1}{4} = 3.4 \, \text{eV} \]

5. **Determine the Maximum Work Function**:
The maximum work function (\( \phi \)) of the metal is equal to the maximum energy of the emitted photon:
\[ \phi = 3.4 \, \text{eV} \]

### Final Answer:
The maximum work function of the metal is \( 3.4 \, \text{eV} \).

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