The work function of two metals metal A and metal B are 6.5 eV and 4.5 eV respectively. If the threshold wavelength of metal A is 2500 A, the threshold wavelength on metal B will be approximately equal to

Options

A. 3611A
B. 3500 A
C. 6332 A
D. 4321 A

Answer

To solve the problem, we need to find the threshold wavelength of metal B given the work functions of metals A and B, and the threshold wavelength of metal A.

### Step 1: Understand the relationship between work function and threshold wavelength
The work function (φ) is related to the threshold wavelength (λ) by the equation:
\[ \phi = \frac{hc}{\lambda} \]
where:
- \(h\) is Planck's constant,
- \(c\) is the speed of light,
- \(\lambda\) is the threshold wavelength.

### Step 2: Write the relationship for both metals
For metal A:
\[ \phi_A = \frac{hc}{\lambda_A} \]
For metal B:
\[ \phi_B = \frac{hc}{\lambda_B} \]

### Step 3: Set up the ratio of work functions and wavelengths
From the equations above, we can set up the following ratio:
\[ \frac{\phi_A}{\phi_B} = \frac{\lambda_B}{\lambda_A} \]

### Step 4: Rearrange to find the threshold wavelength of metal B
Rearranging the equation gives us:
\[ \lambda_B = \frac{\phi_B}{\phi_A} \cdot \lambda_A \]

### Step 5: Substitute the known values
We know:
- \(\phi_A = 6.5 \, \text{eV}\)
- \(\phi_B = 4.5 \, \text{eV}\)
- \(\lambda_A = 2500 \, \text{Å}\)

Substituting these values into the equation:
\[ \lambda_B = \frac{4.5 \, \text{eV}}{6.5 \, \text{eV}} \cdot 2500 \, \text{Å} \]

### Step 6: Calculate \(\lambda_B\)
Calculating the ratio:
\[ \lambda_B = \frac{4.5}{6.5} \cdot 2500 \]
\[ \lambda_B \approx 0.6923 \cdot 2500 \]
\[ \lambda_B \approx 1730.77 \, \text{Å} \]

### Step 7: Final result
Thus, the threshold wavelength of metal B is approximately:
\[ \lambda_B \approx 1731 \, \text{Å} \text{ (rounded to three significant figures)} \]