The hydrogen ATOMS are excited to `n=3 .` During de-excitation, whatever photons are emitted are being used to eject photoelectrons from a metal surface. What should be the maximum work function.of the metal (in eV ) so that the photon of minimum energy can also eject photoelectron from the metal?

Answer

To solve the problem, we need to determine the minimum energy of the photons emitted during the de-excitation of hydrogen atoms from the n=3 level. This energy must be compared to the work function of the metal to ensure that photoelectrons can be ejected.

### Step 1: Determine the energy levels of the hydrogen atom
The energy levels of a hydrogen atom can be calculated using the formula:

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

Where \( n \) is the principal quantum number.

### Step 2: Calculate the energy at n=3
For \( n=3 \):

\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]

### Step 3: Determine the energy of the photon emitted during de-excitation
When the hydrogen atom de-excites from \( n=3 \) to \( n=2 \) or \( n=1 \), we need to calculate the energy difference.

1. **From n=3 to n=2**:
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]
\[ \Delta E_{3 \to 2} = E_2 - E_3 = -3.4 \, \text{eV} - (-1.51 \, \text{eV}) = -3.4 + 1.51 = -1.89 \, \text{eV} \]

2. **From n=3 to n=1**:
\[ E_1 = -13.6 \, \text{eV} \]
\[ \Delta E_{3 \to 1} = E_1 - E_3 = -13.6 \, \text{eV} - (-1.51 \, \text{eV}) = -13.6 + 1.51 = -12.09 \, \text{eV} \]

### Step 4: Identify the minimum photon energy
The minimum energy photon emitted during de-excitation will be the transition from \( n=3 \) to \( n=2 \) because it has the smallest energy difference. Thus, the minimum photon energy is:

\[ E_{\text{min}} = 1.89 \, \text{eV} \]

### Step 5: Relate photon energy to work function
For photoelectrons to be ejected, the energy of the photon must be greater than or equal to the work function \( \phi \) of the metal:

\[ E_{\text{min}} \geq \phi \]

### Step 6: Calculate the maximum work function
Thus, the maximum work function \( \phi \) of the metal can be calculated as:

\[ \phi = E_{\text{min}} = 1.89 \, \text{eV} \]

### Final Answer
The maximum work function of the metal should be **1.89 eV**.

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