Sodium and copper have work functions `2.3 eV and 4.5 eV` respectively . Then the ratio of the wavelength is nearest

Options

A. `1:2`
B. `4:1`
C. `2:1`
D. `1:4`

Answer

To find the ratio of the wavelengths of sodium and copper based on their work functions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Function**: The work function (Φ) is the minimum energy required to remove an electron from a material. For sodium (Na), the work function is 2.3 eV, and for copper (Cu), it is 4.5 eV.

2. **Relate Work Function to Wavelength**: The work function can be expressed in terms of the threshold wavelength (λ₀) using the equation:
\[ \Phi = \frac{hc}{\lambda_0} \]
where:
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
- \(c\) = speed of light (\(3 \times 10^8 \, \text{m/s}\))
- \(\lambda_0\) = threshold wavelength

3. **Write the Equations for Sodium and Copper**:
- For sodium:
\[ \Phi_{Na} = \frac{hc}{\lambda_{Na}} \implies \lambda_{Na} = \frac{hc}{\Phi_{Na}} \]
- For copper:
\[ \Phi_{Cu} = \frac{hc}{\lambda_{Cu}} \implies \lambda_{Cu} = \frac{hc}{\Phi_{Cu}} \]

4. **Find the Ratio of Wavelengths**:
- The ratio of the wavelengths of sodium to copper can be expressed as:
\[ \frac{\lambda_{Na}}{\lambda_{Cu}} = \frac{\Phi_{Cu}}{\Phi_{Na}} \]

5. **Substitute the Work Function Values**:
- Substitute the given work functions:
\[ \frac{\lambda_{Na}}{\lambda_{Cu}} = \frac{4.5 \, \text{eV}}{2.3 \, \text{eV}} \]

6. **Calculate the Ratio**:
- Performing the division:
\[ \frac{4.5}{2.3} \approx 1.9565 \]
- This can be approximated to \(2\) (nearest whole number).

7. **Final Ratio**:
- Therefore, the ratio of the wavelengths is:
\[ \lambda_{Na} : \lambda_{Cu} \approx 2 : 1 \]

### Conclusion:
The ratio of the wavelengths of sodium to copper is approximately \(2 : 1\).