The ratio of velocity of the electron in the third and fifth orbit of `Li^(2+)` would be :

Options

A. `3:5`
B. `5:3`
C. `25:9`
D. `9:25`

Answer

To find the ratio of the velocity of the electron in the third and fifth orbits of the lithium ion \( Li^{2+} \), we can use the formula derived from Bohr's model of the atom. The velocity \( V \) of an electron in a given orbit can be expressed as:

\[ V = C \cdot \frac{Z}{n} \]

Where:
- \( C \) is a constant,
- \( Z \) is the atomic number of the element,
- \( n \) is the principal quantum number (the orbit number).

### Step 1: Identify the atomic number and the orbits
For lithium \( (Li) \), the atomic number \( Z \) is 3. We need to find the velocities for the third orbit \( (n=3) \) and the fifth orbit \( (n=5) \).

### Step 2: Calculate the velocity for the third orbit
Using the formula for the third orbit:
\[ V_3 = C \cdot \frac{Z}{n} = C \cdot \frac{3}{3} = C \]

### Step 3: Calculate the velocity for the fifth orbit
Now, using the formula for the fifth orbit:
\[ V_5 = C \cdot \frac{Z}{n} = C \cdot \frac{3}{5} = \frac{3C}{5} \]

### Step 4: Calculate the ratio of velocities
Now, we can find the ratio of the velocity in the third orbit to that in the fifth orbit:
\[ \text{Ratio} = \frac{V_3}{V_5} = \frac{C}{\frac{3C}{5}} = \frac{C \cdot 5}{3C} = \frac{5}{3} \]

Thus, the ratio of the velocity of the electron in the third orbit to that in the fifth orbit is:
\[ \text{Ratio} = \frac{5}{3} \]

### Final Answer
The ratio of the velocity of the electron in the third and fifth orbit of \( Li^{2+} \) is \( 5:3 \).

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