Calculate the number of waves made by a Bohr electron in one complete revolution in nth orbit of `He^(+)` ion, if ratio of de-Broglie wavelength associated with electron moving in `n^(th)` orbit and `2^(nd)` orbit is 2.0 :-

Answer

To solve the problem of calculating the number of waves made by a Bohr electron in one complete revolution in the nth orbit of the He⁺ ion, we can follow these steps:

### Step 1: Understand the relationship between de Broglie wavelength and angular momentum
The de Broglie wavelength (λ) of an electron is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \(p\) is the momentum of the electron. For an electron in a circular orbit, \(p = mv\), so:
\[ \lambda = \frac{h}{mv} \]

### Step 2: Use the Bohr model for angular momentum
According to the Bohr model, the angular momentum (L) of an electron in the nth orbit is quantized and given by:
\[ L = n \frac{h}{2\pi} \]
This implies:
\[ mvr = n \frac{h}{2\pi} \]

### Step 3: Calculate the angular momentum for the second orbit
For the second orbit (n=2):
\[ L_2 = 2 \frac{h}{2\pi} = \frac{h}{\pi} \]

### Step 4: Relate the de Broglie wavelengths of the nth and second orbits
We are given that the ratio of the de Broglie wavelengths is:
\[ \frac{\lambda_n}{\lambda_2} = 2 \]
This means:
\[ \lambda_n = 2\lambda_2 \]

### Step 5: Calculate the number of waves in the second orbit
The number of waves (N) made by the electron in one complete revolution in the second orbit can be calculated using the circumference of the orbit divided by the wavelength:
\[ N_2 = \frac{2\pi r_2}{\lambda_2} \]
Since the number of waves in the second orbit corresponds to the orbit number (n=2):
\[ N_2 = 2 \]

### Step 6: Calculate the number of waves in the nth orbit
For the nth orbit, using the relationship derived in Step 4:
\[ N_n = \frac{2\pi r_n}{\lambda_n} = \frac{2\pi r_n}{2\lambda_2} = \frac{2\pi r_n}{\lambda_2} \]
Since we know \(N_2 = 2\):
\[ N_n = 2 \times 2 = 4 \]

### Conclusion
The number of waves made by a Bohr electron in one complete revolution in the nth orbit of the He⁺ ion is:
\[ \boxed{4} \]