Wavelength of electron waves in two Bohr orbits is in ratio3:5 the ratio of kinetic energy of electron is 25 : x, hence x is :

Options

A. `25:9`
B. `5:3`
C. `9:25`
D. `3:5`

Explanation

To solve the problem, we need to determine the value of \( x \) based on the given ratios of the wavelengths and the kinetic energies of electrons in two Bohr orbits.

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and kinetic energy**:
The de Broglie wavelength (\( \lambda \)) of an electron is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

2. **Express kinetic energy in terms of wavelength**:
The kinetic energy (\( KE \)) of an electron is given by:
\[ KE = \frac{1}{2} mv^2 \]
From the de Broglie equation, we can express \( v \) in terms of \( \lambda \):
\[ v = \frac{h}{m\lambda} \]
Substituting this expression for \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{h}{m\lambda}\right)^2 = \frac{h^2}{2m\lambda^2} \]

3. **Establish the relationship between kinetic energy and wavelength**:
From the above equation, we can see that kinetic energy is inversely proportional to the square of the wavelength:
\[ KE \propto \frac{1}{\lambda^2} \]

4. **Set up the ratio of kinetic energies**:
Given that the wavelengths in two Bohr orbits are in the ratio \( 3:5 \), we can express this as:
\[ \frac{\lambda_1}{\lambda_2} = \frac{3}{5} \]
Therefore, the inverse ratio of the squares of the wavelengths will be:
\[ \frac{KE_1}{KE_2} = \frac{\lambda_2^2}{\lambda_1^2} = \frac{(5)^2}{(3)^2} = \frac{25}{9} \]

5. **Relate the kinetic energy ratio to the given values**:
We know from the problem statement that the kinetic energy ratio is given as \( 25:x \). Thus, we can set up the equation:
\[ \frac{25}{x} = \frac{25}{9} \]

6. **Solve for \( x \)**:
Cross-multiplying gives:
\[ 25 \cdot 9 = 25 \cdot x \]
Simplifying this, we find:
\[ x = 9 \]

### Final Answer:
Thus, the value of \( x \) is \( 9 \).