The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately. (a) 540 nm (b ) 400nm (c ) 310 nm (d) 220 nm

Explanation

To solve the problem of finding the longest wavelength of light that can cause photoelectron emission from a substance with a work function of 4.0 eV, we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) is the wavelength in meters.

### Step 2: Convert the work function from eV to joules
The work function (\(\phi\)) is given as 4.0 eV. To convert this to joules, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):
\[ \phi = 4.0 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 6.4 \times 10^{-19} \, \text{J} \]

### Step 3: Set the energy equal to the work function
For photoelectron emission to occur, the energy of the incident photon must be equal to or greater than the work function:
\[ E \geq \phi \]
This means we need to find the wavelength \(\lambda\) for which \(E = \phi\).

### Step 4: Substitute the work function into the energy formula
Using the energy formula and substituting \(\phi\):
\[ \phi = \frac{hc}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{hc}{\phi} \]

### Step 5: Calculate the wavelength
Substituting the values of \(h\), \(c\), and \(\phi\):
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{6.4 \times 10^{-19} \, \text{J}} \]
Calculating this gives:
\[ \lambda \approx \frac{1.9878 \times 10^{-25}}{6.4 \times 10^{-19}} \approx 3.11 \times 10^{-7} \, \text{m} = 311 \, \text{nm} \]

### Step 6: Conclusion
The longest wavelength of light that can cause photoelectron emission from this substance is approximately **310 nm**.

### Final Answer
(c) 310 nm

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