The de-Brogile wavelength associated with electron in the `n = 4` level is :-

Options

A. two times the de-Brogile wavelength of the electron in the ground state
B. `1//4^(th)` of the de-Brogile wavelength of the electron in the ground state
C. four times the de-Brogile wavelength of the electron in the grounds state
D. half of the de-Brogile wavelength of the electron in the ground state

Explanation

To find the de Broglie wavelength associated with an electron in the \( n = 4 \) level, we can follow these steps:

### Step 1: Understand the de Broglie Wavelength Formula
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the electron.

### Step 2: Relate Momentum to Velocity
The momentum \( p \) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Therefore, we can rewrite the de Broglie wavelength as:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Understand the Relationship Between Velocity and Principal Quantum Number
In quantum mechanics, the velocity of an electron in an atom is inversely related to the principal quantum number \( n \). This means:
\[ v \propto \frac{1}{n} \]
As \( n \) increases, the velocity \( v \) decreases.

### Step 4: Establish the Proportionality of Wavelength to Quantum Number
From the relationship established in Step 3, we can say:
\[ \lambda \propto n \]
This indicates that the wavelength is directly proportional to the principal quantum number.

### Step 5: Set Up the Ratio of Wavelengths
Using the proportionality, we can write the ratio of wavelengths for different quantum levels:
\[ \frac{\lambda_1}{\lambda_4} = \frac{n_1}{n_4} \]
where \( n_1 = 1 \) (ground state) and \( n_4 = 4 \).

### Step 6: Solve for \( \lambda_4 \)
From the ratio, we can express \( \lambda_4 \):
\[ \lambda_4 = \frac{n_4}{n_1} \cdot \lambda_1 = \frac{4}{1} \cdot \lambda_1 = 4 \lambda_1 \]
This means the wavelength associated with the electron in the \( n = 4 \) level is four times that of the wavelength in the ground state.

### Conclusion
Thus, the de Broglie wavelength associated with the electron in the \( n = 4 \) level is:
\[ \lambda_4 = 4 \lambda_1 \]