Sodium and copper have work functions of 2.3 eV and 4.5 eV. The ratio of their corresponding threshold wavelengths will be approximately equal to

Options

A. `2:1`
B. `1:2`
C. `sqrt(2):1`
D. `1:4`

Explanation

To solve the problem, we need to find the ratio of the threshold wavelengths for sodium and copper based on their work functions. The work function (W) is related to the threshold wavelength (λ) by the equation:

\[ W = \frac{hc}{\lambda} \]

Where:
- \( W \) is the work function (in eV),
- \( h \) is Planck's constant (\( 4.1357 \times 10^{-15} \) eV·s),
- \( c \) is the speed of light (\( 3 \times 10^8 \) m/s),
- \( \lambda \) is the threshold wavelength (in meters).

### Step-by-Step Solution:

1. **Write the formula for the threshold wavelength**:
The threshold wavelength can be expressed as:
\[ \lambda = \frac{hc}{W} \]

2. **Calculate the threshold wavelength for sodium (λ_S)**:
Using the work function for sodium (\( W_S = 2.3 \) eV):
\[ \lambda_S = \frac{hc}{W_S} = \frac{hc}{2.3} \]

3. **Calculate the threshold wavelength for copper (λ_C)**:
Using the work function for copper (\( W_C = 4.5 \) eV):
\[ \lambda_C = \frac{hc}{W_C} = \frac{hc}{4.5} \]

4. **Find the ratio of the threshold wavelengths (λ_S / λ_C)**:
To find the ratio of the threshold wavelengths, we can write:
\[ \frac{\lambda_S}{\lambda_C} = \frac{\frac{hc}{W_S}}{\frac{hc}{W_C}} = \frac{W_C}{W_S} \]

5. **Substitute the values of the work functions**:
Now substitute the values:
\[ \frac{\lambda_S}{\lambda_C} = \frac{4.5 \, \text{eV}}{2.3 \, \text{eV}} \]

6. **Calculate the ratio**:
\[ \frac{\lambda_S}{\lambda_C} \approx \frac{4.5}{2.3} \approx 1.9565 \]

### Final Answer:
The ratio of the corresponding threshold wavelengths for sodium and copper is approximately \( 1.96 \).