Assuming that the velcity greater than velocity of light is not possible find out the value of the highest atomic number of an atom which can exist given that the velocity of electron in the first orbit of bohr hydrogen atom is `2.18 xx 10^(6) ms^(-1)`

Options

A. 137
B. 238
C. 157
D. 257

Answer

To find the highest atomic number of an atom that can exist without exceeding the speed of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Electron Velocity**:
According to Bohr's theory, the velocity \( v \) of an electron in the nth orbit of a hydrogen-like atom is given by the formula:
\[ v = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number (orbit number).

2. **Set the Condition for Speed of Light**:
We know that the speed of light \( c \) is approximately:
\[ c = 3 \times 10^8 \text{ m/s} \]
According to the problem, the velocity of the electron must not exceed the speed of light:
\[ v < c \]

3. **Substituting for the First Orbit**:
For the first orbit, \( n = 1 \). Therefore, we can substitute \( n \) into the velocity equation:
\[ v = 2.18 \times 10^6 \times Z \]

4. **Setting Up the Inequality**:
We need to ensure that this velocity is less than the speed of light:
\[ 2.18 \times 10^6 \times Z < 3 \times 10^8 \]

5. **Solving for Z**:
To find the maximum value of \( Z \), we can rearrange the inequality:
\[ Z < \frac{3 \times 10^8}{2.18 \times 10^6} \]

6. **Calculating the Right Side**:
Now, we perform the division:
\[ Z < \frac{3 \times 10^8}{2.18 \times 10^6} \approx 137.61 \]

7. **Determine the Highest Integer Value of Z**:
Since \( Z \) must be an integer, the highest possible value of \( Z \) that satisfies this condition is:
\[ Z = 137 \]

### Conclusion:
The highest atomic number of an atom that can exist without exceeding the speed of light is **137**.

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