If the de Broglie wavelength of the electron in `n^(th)` Bohr orbit in a hydrogenic atom is equal to `1.5pia_(0)(a_(0)` is bohr radius), then the value of `n//z` is :

Options

A. 1.5
B. 0.75
C. `1.0`
D. `0.40`

Explanation

To solve the problem, we need to find the value of \( \frac{n}{z} \) given that the de Broglie wavelength of the electron in the \( n^{th} \) Bohr orbit of a hydrogenic atom is equal to \( 1.5 \pi a_0 \), where \( a_0 \) is the Bohr radius.

### Step-by-Step Solution:

1. **Understanding the de Broglie Wavelength**:
The de Broglie wavelength \( \lambda \) of an electron in a Bohr orbit can be expressed in terms of the orbit's circumference and the number of wavelengths that fit into that circumference. The formula is:
\[ n \lambda = 2 \pi r \]
where \( n \) is the principal quantum number, \( \lambda \) is the de Broglie wavelength, and \( r \) is the radius of the orbit.

2. **Substituting the Given Wavelength**:
We know from the problem that:
\[ \lambda = 1.5 \pi a_0 \]
Therefore, substituting this into the equation gives:
\[ n (1.5 \pi a_0) = 2 \pi r \]

3. **Expressing the Radius**:
The radius \( r \) of the \( n^{th} \) Bohr orbit in a hydrogenic atom is given by:
\[ r = \frac{n^2 a_0}{z} \]
where \( z \) is the atomic number of the hydrogenic atom.

4. **Substituting the Radius into the Equation**:
Now substitute \( r \) into the previous equation:
\[ n (1.5 \pi a_0) = 2 \pi \left(\frac{n^2 a_0}{z}\right) \]

5. **Simplifying the Equation**:
Cancel \( \pi a_0 \) from both sides:
\[ n (1.5) = 2 \frac{n^2}{z} \]
Rearranging gives:
\[ 1.5nz = 2n^2 \]

6. **Dividing by \( n \)**:
Assuming \( n \neq 0 \), we can divide both sides by \( n \):
\[ 1.5z = 2n \]

7. **Finding \( \frac{n}{z} \)**:
Rearranging this equation gives:
\[ \frac{n}{z} = \frac{1.5}{2} = 0.75 \]

### Final Answer:
Thus, the value of \( \frac{n}{z} \) is:
\[ \frac{n}{z} = 0.75 \]