Number of waves made by a Bohr electron in one complete in its fourth orbit is

Explanation

To find the number of waves made by a Bohr electron in one complete cycle in its fourth orbit, we can follow these steps:

### Step 1: Identify the parameters
We know that:
- For a hydrogen atom, \( Z = 1 \) (since it's a single electron).
- The principal quantum number for the fourth orbit is \( n = 4 \).

### Step 2: Calculate the radius of the fourth orbit
The radius of the nth orbit in a hydrogen-like atom is given by the formula:
\[ r_n = r_0 \cdot \frac{n^2}{Z^2} \]
where \( r_0 = 0.529 \, \text{Å} = 0.529 \times 10^{-10} \, \text{m} \).

Substituting the values:
\[ r_4 = 0.529 \times 10^{-10} \cdot \frac{4^2}{1^2} = 0.529 \times 10^{-10} \cdot 16 = 8.464 \times 10^{-10} \, \text{m} \]

### Step 3: Calculate the velocity of the electron in the fourth orbit
The velocity of the electron in the nth orbit is given by:
\[ v_n = v_0 \cdot \frac{Z}{n} \]
where \( v_0 = 2.16 \times 10^6 \, \text{m/s} \).

Substituting the values:
\[ v_4 = 2.16 \times 10^6 \cdot \frac{1}{4} = 5.4 \times 10^5 \, \text{m/s} \]

### Step 4: Calculate the wavelength of the electron's motion
The wavelength \( \lambda \) can be calculated using the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
where \( h = 6.626 \times 10^{-34} \, \text{Js} \) and \( m = 9.1 \times 10^{-31} \, \text{kg} \).

Substituting the values:
\[ \lambda = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-31} \cdot 5.4 \times 10^5} \]

Calculating the denominator:
\[ 9.1 \times 10^{-31} \cdot 5.4 \times 10^5 = 4.914 \times 10^{-25} \]

Now substituting back:
\[ \lambda = \frac{6.626 \times 10^{-34}}{4.914 \times 10^{-25}} \approx 1.35 \times 10^{-9} \, \text{m} \]

### Step 5: Calculate the number of waves in one complete cycle
The number of waves \( n \) made by the electron in one complete cycle can be calculated as:
\[ n = \frac{2 \pi r}{\lambda} \]

Substituting the values:
\[ n = \frac{2 \pi \cdot 8.464 \times 10^{-10}}{1.35 \times 10^{-9}} \approx \frac{5.316 \times 10^{-9}}{1.35 \times 10^{-9}} \approx 3.94 \]

Since the number of waves must be a whole number, we round this to the nearest whole number:
\[ n \approx 4 \]

### Final Answer:
The number of waves made by a Bohr electron in one complete cycle in its fourth orbit is **4**.

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