The work function of a substance is 3.68 eV. The longest wavelength of light that can cause photoelectrons emission from this substance is approximately

Options

A. `337 nm`
B. `400 nm `
C. `310 nm `
D. `220 nm`

Answer

To find the longest wavelength of light that can cause photoelectron emission from a substance with a work function of 3.68 eV, we can follow these steps:

### Step 1: Understand the relationship between work function and wavelength
The work function (φ) is the minimum energy required to remove an electron from the surface of a material. The energy of a photon (E) is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \))
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \))

### Step 2: Convert the work function from eV to joules
The work function is given in electronvolts (eV). To convert eV to joules, we use the conversion factor:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Thus, the work function in joules is:
\[ \phi = 3.68 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 5.888 \times 10^{-19} \, \text{J} \]

### Step 3: Rearrange the energy-wavelength equation to find wavelength
From the energy-wavelength relationship, we can rearrange it to find the wavelength:
\[ \lambda = \frac{hc}{E} \]
Substituting \( E \) with the work function \( \phi \):
\[ \lambda = \frac{hc}{\phi} \]

### Step 4: Substitute the values of \( h \), \( c \), and \( \phi \)
Using the values:
- \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)
- \( \phi = 5.888 \times 10^{-19} \, \text{J} \)

We can calculate:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{Js}) \times (3 \times 10^8 \, \text{m/s})}{5.888 \times 10^{-19} \, \text{J}} \]

### Step 5: Perform the calculation
Calculating the numerator:
\[ 6.626 \times 10^{-34} \times 3 \times 10^8 = 1.9878 \times 10^{-25} \, \text{Js m} \]
Now, dividing by the work function:
\[ \lambda = \frac{1.9878 \times 10^{-25}}{5.888 \times 10^{-19}} \approx 3.375 \times 10^{-7} \, \text{m} \]

### Step 6: Convert the wavelength to nanometers
To convert meters to nanometers, we multiply by \( 10^9 \):
\[ \lambda \approx 3.375 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \approx 337.5 \, \text{nm} \]

### Conclusion
The longest wavelength of light that can cause photoelectron emission from this substance is approximately **337 nm**.

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