de-Broglie wavelength associated with an electron revolving in the ` n^(th) ` state of hydrogen atom is directily proportional to

Options

A. n
B. `1/n`
C. `n^(2)`
D. `(1)/(n^(2))`

Answer

To solve the question regarding the de-Broglie wavelength associated with an electron revolving in the nth state of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy of the Electron:**
The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n \propto -\frac{1}{n^2} \]
This indicates that the energy is inversely proportional to the square of the principal quantum number \( n \).

2. **Relating Energy to Kinetic Energy:**
The kinetic energy (K.E) of the electron can be expressed as:
\[ K.E = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Since the total energy is primarily kinetic in the case of an electron in a stable orbit, we can relate the kinetic energy to the momentum \( p \).

3. **Momentum and Kinetic Energy:**
The momentum \( p \) of the electron is given by:
\[ p = mv \]
We can also express kinetic energy in terms of momentum:
\[ K.E = \frac{p^2}{2m} \]
From the energy relation, we have:
\[ \frac{p^2}{2m} \propto -\frac{1}{n^2} \]
Thus, we can say:
\[ p^2 \propto \frac{1}{n^2} \]
Therefore, the momentum \( p \) is directly proportional to:
\[ p \propto \frac{1}{n} \]

4. **Applying de-Broglie Wavelength Formula:**
The de-Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant. Since we found that \( p \propto \frac{1}{n} \), we can substitute this into the de-Broglie wavelength formula:
\[ \lambda \propto \frac{h}{p} \propto \frac{h}{\frac{1}{n}} = h \cdot n \]

5. **Conclusion:**
Thus, we conclude that the de-Broglie wavelength \( \lambda \) associated with an electron revolving in the nth state of a hydrogen atom is directly proportional to \( n \):
\[ \lambda \propto n \]

### Final Answer:
The de-Broglie wavelength associated with an electron revolving in the nth state of a hydrogen atom is directly proportional to \( n \).

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