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

### Step 1: Understand the velocity of the electron in Bohr's model
According to Bohr's model of the atom, the velocity \( v \) of an electron in the nth orbit is given by the formula:
\[ v = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number (orbit number).

### Step 2: Set the conditions for the first orbit
For the first orbit, \( n = 1 \). Therefore, the velocity of the electron in the first orbit becomes:
\[ v = 2.18 \times 10^6 Z \text{ m/s} \]

### Step 3: Set the velocity condition against the speed of light
The problem states that this velocity must be less than the speed of light \( c \), which is approximately \( 3 \times 10^8 \text{ m/s} \). Thus, we can write:
\[ 2.18 \times 10^6 Z < 3 \times 10^8 \]

### Step 4: Solve for \( Z \)
To find the maximum value of \( Z \), we can rearrange the inequality:
\[ Z < \frac{3 \times 10^8}{2.18 \times 10^6} \]

### Step 5: Calculate the value of \( Z \)
Now, we perform the division:
\[ Z < \frac{3 \times 10^8}{2.18 \times 10^6} \approx 137.61 \]

### Step 6: 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
Thus, the highest atomic number of an atom that can exist without exceeding the speed of light is:
\[ \boxed{137} \]

---