If a suface has work function of 3.00eV, the longest wavelength of light which will cause the emission of electrons is

Options

A. `4.8xx10^-7`m
B. `5.99xx10^-7`m
C. `4.13xx10^-7`m
D. `6.84xx10^-7`m

Explanation

To solve the problem of finding the longest wavelength of light that can cause the emission of electrons from a surface with a work function of 3.00 eV, we will follow these steps:

### Step 1: Understand the relationship between work function and wavelength
The work function (W) of a material is the minimum energy required to remove an electron from the surface. The energy of a photon can be related to its wavelength (λ) using the equation:
\[ W = \frac{hc}{\lambda_0} \]
where:
- \( W \) is the work function in joules,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^{8} \, \text{m/s} \)),
- \( \lambda_0 \) is the wavelength in meters.

### Step 2: Convert the work function from eV to joules
The work function is given as 3.00 eV. To convert electron volts to joules, we use the conversion factor:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Thus, the work function in joules is:
\[ W = 3.00 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 4.8 \times 10^{-19} \, \text{J} \]

### Step 3: Rearrange the equation to solve for the wavelength
From the equation \( W = \frac{hc}{\lambda_0} \), we can rearrange it to solve for \( \lambda_0 \):
\[ \lambda_0 = \frac{hc}{W} \]

### Step 4: Substitute the known values into the equation
Now we substitute the values of \( h \), \( c \), and \( W \):
\[ \lambda_0 = \frac{(6.626 \times 10^{-34} \, \text{Js}) \times (3.00 \times 10^{8} \, \text{m/s})}{4.8 \times 10^{-19} \, \text{J}} \]

### Step 5: Perform the calculation
Calculating the numerator:
\[ 6.626 \times 10^{-34} \times 3.00 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{Js m} \]
Now, divide by the work function:
\[ \lambda_0 = \frac{1.9878 \times 10^{-25}}{4.8 \times 10^{-19}} \]
\[ \lambda_0 \approx 4.13 \times 10^{-7} \, \text{m} \]

### Step 6: Final Result
The longest wavelength of light that will cause the emission of electrons from the surface is approximately:
\[ \lambda_0 \approx 4.13 \times 10^{-7} \, \text{m} \]

### Conclusion
Thus, the correct answer is \( \lambda_0 \approx 4.13 \times 10^{-7} \, \text{m} \).

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