The work functions of potassium and sodium are 4.5 eV and 2.3 eV respectively. The approximate ratio of their threshold wavelength will be -

Options

A. ` 1 : 2 `
B. ` 2 : 1 `
C. `1 : 3 `
D. `3 : 1 `

Answer

To find the approximate ratio of the threshold wavelengths of potassium and sodium, we can use the relationship between the work function and the threshold wavelength. The work function (Φ) is related to the threshold wavelength (λ₀) by the equation:

\[ \Phi = \frac{hc}{\lambda_0} \]

where:
- \( \Phi \) is the work function in electron volts (eV),
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( c \) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \( \lambda_0 \) is the threshold wavelength in meters.

### Step 1: Write the equations for the threshold wavelengths of potassium and sodium.

For potassium (K):
\[ \lambda_K = \frac{hc}{\Phi_K} \]
For sodium (Na):
\[ \lambda_{Na} = \frac{hc}{\Phi_{Na}} \]

### Step 2: Substitute the work functions into the equations.

Given:
- Work function of potassium, \( \Phi_K = 4.5 \, \text{eV} \)
- Work function of sodium, \( \Phi_{Na} = 2.3 \, \text{eV} \)

The equations become:
\[ \lambda_K = \frac{hc}{4.5 \, \text{eV}} \]
\[ \lambda_{Na} = \frac{hc}{2.3 \, \text{eV}} \]

### Step 3: Find the ratio of the threshold wavelengths.

To find the ratio \( \frac{\lambda_K}{\lambda_{Na}} \):
\[ \frac{\lambda_K}{\lambda_{Na}} = \frac{\frac{hc}{4.5}}{\frac{hc}{2.3}} = \frac{2.3}{4.5} \]

### Step 4: Simplify the ratio.

Calculating the ratio:
\[ \frac{2.3}{4.5} \approx 0.511 \]

### Conclusion:

The approximate ratio of the threshold wavelengths of potassium to sodium is:
\[ \frac{\lambda_K}{\lambda_{Na}} \approx 0.511 \]