The ratio of the speed of an electron in the first orbit of hydrogen atom to that in the first orbit of He is

Options

A. `1:2` .
B. `2:1` .
C. `1:4` .
D. `4:1` .

Answer

To find the ratio of the speed of an electron in the first orbit of a hydrogen atom to that in the first orbit of a helium atom, we can use Bohr's model of the atom. Here’s a step-by-step solution:

### Step 1: Understand Bohr's Model
Bohr's model states that the speed of an electron in the nth orbit of a hydrogen-like atom can be expressed as:
\[ v_n = \frac{Z e^2}{2 n h \epsilon_0} \]
where:
- \( Z \) is the atomic number,
- \( e \) is the charge of the electron,
- \( h \) is Planck's constant,
- \( \epsilon_0 \) is the permittivity of free space,
- \( n \) is the principal quantum number (orbit number).

### Step 2: Calculate Speed for Hydrogen
For hydrogen (H), which has \( Z = 1 \) and for the first orbit \( n = 1 \):
\[ v_H = \frac{1 \cdot e^2}{2 \cdot 1 \cdot h \epsilon_0} = \frac{e^2}{2 h \epsilon_0} \]

### Step 3: Calculate Speed for Helium
For helium (He), which has \( Z = 2 \) and for the first orbit \( n = 1 \):
\[ v_{He} = \frac{2 \cdot e^2}{2 \cdot 1 \cdot h \epsilon_0} = \frac{2 e^2}{2 h \epsilon_0} = \frac{e^2}{h \epsilon_0} \]

### Step 4: Find the Ratio of Speeds
Now, we need to find the ratio of the speed of the electron in hydrogen to that in helium:
\[ \frac{v_H}{v_{He}} = \frac{\frac{e^2}{2 h \epsilon_0}}{\frac{e^2}{h \epsilon_0}} \]

### Step 5: Simplify the Ratio
When we simplify the ratio:
\[ \frac{v_H}{v_{He}} = \frac{e^2}{2 h \epsilon_0} \cdot \frac{h \epsilon_0}{e^2} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the speed of an electron in the first orbit of hydrogen atom to that in the first orbit of helium is:
\[ \frac{v_H}{v_{He}} = \frac{1}{2} \]