Ultraviolet light of wavelengths `lambda_(1)` and `lambda_(2)` when allowed to fall on hydrogen atoms in their ground state is found to liberate electrons with kinetic energy `1.8 eV` and `4.0 eV` respectively. Find the value of `(lambda_(1))/(lambda_(2))`.

Options

A. `(7)/(8)`
B. `(8)/(7)`
C. `(9)/(20)`
D. `(20)/(9)`

Answer

To solve the problem, we need to find the ratio of the wavelengths \( \frac{\lambda_1}{\lambda_2} \) given the kinetic energies of the electrons liberated by ultraviolet light of wavelengths \( \lambda_1 \) and \( \lambda_2 \) when they fall on hydrogen atoms in their ground state.

### Step-by-Step Solution:

1. **Understanding the Energy of Electrons**:
The energy of the incident photons can be expressed in terms of the kinetic energy of the emitted electrons and the ionization energy of hydrogen. The energy of a photon is given by:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

2. **Setting Up the Equations**:
For the first wavelength \( \lambda_1 \) with kinetic energy \( K_1 = 1.8 \, \text{eV} \):
\[ \frac{hc}{\lambda_1} = E_0 + K_1 \]
where \( E_0 \) is the ionization energy of hydrogen (approximately 13.6 eV).
Thus, we can write:
\[ \frac{hc}{\lambda_1} = 13.6 + 1.8 = 15.4 \, \text{eV} \]
(Equation 1)

For the second wavelength \( \lambda_2 \) with kinetic energy \( K_2 = 4.0 \, \text{eV} \):
\[ \frac{hc}{\lambda_2} = E_0 + K_2 \]
Thus, we can write:
\[ \frac{hc}{\lambda_2} = 13.6 + 4.0 = 17.6 \, \text{eV} \]
(Equation 2)

3. **Finding the Ratio of Wavelengths**:
Now, we can divide Equation 2 by Equation 1:
\[ \frac{\frac{hc}{\lambda_2}}{\frac{hc}{\lambda_1}} = \frac{17.6}{15.4} \]
The \( hc \) terms cancel out:
\[ \frac{\lambda_1}{\lambda_2} = \frac{17.6}{15.4} \]

4. **Calculating the Ratio**:
To simplify \( \frac{17.6}{15.4} \):
\[ \frac{17.6}{15.4} = \frac{176}{154} = \frac{88}{77} = \frac{20}{9} \]

5. **Final Result**:
Therefore, the ratio of the wavelengths is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{20}{9} \]

### Conclusion:
The value of \( \frac{\lambda_1}{\lambda_2} \) is \( \frac{20}{9} \).