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 :

Answer

To solve the problem, we need to use the relationship between the de Broglie wavelength and the kinetic energy of an electron in Bohr orbits. Here’s a step-by-step solution:

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

### Step 2: Relate kinetic energy to wavelength
The kinetic energy (\( KE \)) of the electron is given by:
\[ KE = \frac{1}{2} mv^2 \]
Using the relationship between momentum and kinetic energy, we can express kinetic energy in terms of wavelength:
\[ KE = \frac{p^2}{2m} = \frac{(mv)^2}{2m} = \frac{m v^2}{2} \]
Substituting \( v \) from the de Broglie wavelength formula:
\[ KE = \frac{h^2}{2m\lambda^2} \]
This shows that kinetic energy is inversely proportional to the square of the wavelength:
\[ KE \propto \frac{1}{\lambda^2} \]

### Step 3: Set up the ratios
Given that the ratio of the wavelengths in two Bohr orbits is \( \frac{\lambda_1}{\lambda_2} = \frac{3}{5} \), we can express this as:
\[ \lambda_1 = 3k \quad \text{and} \quad \lambda_2 = 5k \]
for some constant \( k \).

### Step 4: Calculate the ratio of kinetic energies
From the inverse proportionality, we have:
\[ \frac{KE_1}{KE_2} = \frac{\lambda_2^2}{\lambda_1^2} = \frac{(5k)^2}{(3k)^2} = \frac{25k^2}{9k^2} = \frac{25}{9} \]
This means:
\[ \frac{KE_1}{KE_2} = \frac{25}{x} \]
where \( KE_1 = 25 \) and \( KE_2 = x \).

### Step 5: Equate the ratios
From the above, we can set up the equation:
\[ \frac{25}{x} = \frac{25}{9} \]
Cross-multiplying gives:
\[ 25 \cdot 9 = 25 \cdot x \]
This simplifies to:
\[ 225 = 25x \]

### Step 6: Solve for \( x \)
Dividing both sides by 25:
\[ x = \frac{225}{25} = 9 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{9} \]