The maximum kinetic energy of photoelectrons emitted from a metal surface increses from 0.4 eV to 1.2 eV when the frequency of the incident radiation is increased by 40% . What is the work function (in eV) of the metal surface?

Answer

To find the work function of the metal surface, we can use the photoelectric effect equation:

\[ K.E. = hf - \phi \]

where:
- \( K.E. \) is the maximum kinetic energy of the emitted photoelectrons,
- \( h \) is Planck's constant,
- \( f \) is the frequency of the incident radiation,
- \( \phi \) is the work function of the metal.

### Step 1: Define the initial and final conditions
Let:
- \( K.E_1 = 0.4 \, \text{eV} \) (initial kinetic energy)
- \( K.E_2 = 1.2 \, \text{eV} \) (final kinetic energy)

### Step 2: Calculate the change in kinetic energy
The change in kinetic energy when the frequency is increased can be expressed as:

\[ K.E_2 - K.E_1 = 1.2 \, \text{eV} - 0.4 \, \text{eV} = 0.8 \, \text{eV} \]

### Step 3: Determine the frequency change
Given that the frequency is increased by 40%, we can express the initial and final frequencies as follows:

Let \( f_1 \) be the initial frequency. Then, the final frequency \( f_2 \) is:

\[ f_2 = f_1 + 0.4 f_1 = 1.4 f_1 \]

### Step 4: Write the equations for kinetic energy
Using the photoelectric effect equation for both frequencies, we have:

1. For the initial frequency:
\[ K.E_1 = hf_1 - \phi \]
\[ 0.4 = hf_1 - \phi \quad \text{(1)} \]

2. For the final frequency:
\[ K.E_2 = hf_2 - \phi \]
\[ 1.2 = h(1.4 f_1) - \phi \quad \text{(2)} \]

### Step 5: Substitute and solve for work function
From equation (1):
\[ \phi = hf_1 - 0.4 \]

Substituting \( \phi \) into equation (2):
\[ 1.2 = h(1.4 f_1) - (hf_1 - 0.4) \]
\[ 1.2 = 1.4hf_1 - hf_1 + 0.4 \]
\[ 1.2 = 0.4hf_1 + 0.4 \]

Now, isolate \( hf_1 \):
\[ 1.2 - 0.4 = 0.4hf_1 \]
\[ 0.8 = 0.4hf_1 \]
\[ hf_1 = \frac{0.8}{0.4} = 2 \, \text{eV} \]

### Step 6: Calculate the work function
Now, substitute \( hf_1 \) back into the equation for \( \phi \):
\[ \phi = hf_1 - 0.4 \]
\[ \phi = 2 - 0.4 = 1.6 \, \text{eV} \]

### Final Answer:
The work function of the metal surface is \( \phi = 1.6 \, \text{eV} \).