The velocity of electron in a certain Bohr's orbit of H-atom bears the ratio `1: 275` to the velocity of light. What is the number of orbit ?

Answer

To solve the problem of finding the number of the Bohr orbit for an electron in a hydrogen atom based on its velocity, we can follow these steps:

### Step 1: Determine the velocity of the electron
Given the ratio of the velocity of the electron (v_e) to the velocity of light (c) is \( \frac{1}{275} \), we can express this mathematically as:

\[ v_e = \frac{c}{275} \]

Where the speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s. Therefore:

\[ v_e = \frac{3 \times 10^8 \, \text{m/s}}{275} \]

### Step 2: Calculate the velocity of the electron
Now, we can calculate \( v_e \):

\[ v_e = \frac{3 \times 10^8}{275} \approx 1.09 \times 10^6 \, \text{m/s} \]

### Step 3: Use the Bohr model to relate velocity and orbit number
According to the Bohr model, the velocity of the electron in the nth orbit is given by:

\[ v_e = \frac{2.18 \times 10^6 \, \text{m/s}}{n} \]

Where \( n \) is the principal quantum number (the number of the orbit) and \( 2.18 \times 10^6 \, \text{m/s} \) is a constant for hydrogen.

### Step 4: Set the two expressions for velocity equal to each other
Now we can set the two expressions for \( v_e \) equal to each other:

\[ \frac{3 \times 10^8}{275} = \frac{2.18 \times 10^6}{n} \]

### Step 5: Solve for n
Rearranging the equation to solve for \( n \):

\[ n = \frac{2.18 \times 10^6 \times 275}{3 \times 10^8} \]

### Step 6: Calculate the value of n
Now we can compute \( n \):

\[ n = \frac{2.18 \times 275}{3} \times 10^{-2} \approx \frac{599.5}{3} \approx 199.83 \approx 2 \]

### Conclusion
Thus, the number of the orbit \( n \) is approximately 2.

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