If the speed of electron in the first bohr orbit of hydrogen atom is x then the speed of the electron in the third Bohr orbit of hydrogen is

Options

A. x/9
B. x/3
C. 3x
D. 9x

Answer

To solve the problem of finding the speed of the electron in the third Bohr orbit of a hydrogen atom, given that the speed in the first orbit is \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Bohr's Model**: According to Bohr's model of the hydrogen atom, the speed of an electron in the nth orbit can be expressed as:
\[ v_n = \frac{2.18 \times 10^6 \times Z}{n} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)) and \( n \) is the principal quantum number (the orbit number).

2. **Speed in the First Orbit**: For the first orbit (n=1), the speed of the electron is given as:
\[ v_1 = 2.18 \times 10^6 \times \frac{1}{1} = 2.18 \times 10^6 \text{ m/s} \]
However, we are given that this speed is \( x \).

3. **Speed in the Third Orbit**: For the third orbit (n=3), the speed of the electron can be calculated as:
\[ v_3 = 2.18 \times 10^6 \times \frac{1}{3} \]

4. **Relationship Between Speeds**: From the equation, we can see that the speed in the nth orbit is inversely proportional to \( n \). Thus, we can write the ratio of the speeds in the third and first orbits as:
\[ \frac{v_3}{v_1} = \frac{1}{3} \]

5. **Substituting Known Values**: Since \( v_1 = x \), we can substitute this into the equation:
\[ v_3 = \frac{1}{3} v_1 = \frac{1}{3} x \]

6. **Final Answer**: Therefore, the speed of the electron in the third Bohr orbit of the hydrogen atom is:
\[ v_3 = \frac{x}{3} \]

### Conclusion:
The speed of the electron in the third Bohr orbit of the hydrogen atom is \( \frac{x}{3} \).

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