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 solve the problem of finding the ratio of the speed of an electron in the first orbit of a hydrogen atom to that in the first orbit of helium, we can follow these steps:

### Step 1: Understand the Bohr Model
The speed of an electron in a hydrogen atom (or any hydrogen-like atom) can be derived from the Bohr model of the atom. According to the Bohr model, the centripetal force required to keep the electron in a circular orbit is provided by the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron.

### Step 2: Write the Equation for Centripetal Force
The centripetal force can be expressed as:
\[ \frac{mv^2}{r} = \frac{kZe^2}{r^2} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the speed of the electron,
- \( r \) is the radius of the orbit,
- \( k \) is Coulomb's constant,
- \( Z \) is the atomic number (1 for hydrogen, 2 for helium),
- \( e \) is the charge of the electron.

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ mv^2 = \frac{kZe^2}{r} \]
From the Bohr model, we also know that:
\[ Mvr = n\frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 4: Express Velocity
From the above equations, we can express the velocity \( v \) as:
\[ v \propto \frac{Z}{n} \]
This means that the velocity of the electron is directly proportional to the atomic number \( Z \) and inversely proportional to the principal quantum number \( n \).

### Step 5: Calculate the Ratio of Velocities
For hydrogen (H):
- \( Z_H = 1 \)
- \( n_H = 1 \)

For helium (He):
- \( Z_{He} = 2 \)
- \( n_{He} = 1 \)

Now, we can find the ratio of the velocities:
\[ \frac{v_H}{v_{He}} = \frac{Z_H/n_H}{Z_{He}/n_{He}} = \frac{1/1}{2/1} = \frac{1}{2} \]

### Step 6: Conclusion
Thus, the ratio of the speed of an electron in the first orbit of hydrogen to that in the first orbit of helium is:
\[ \frac{v_H}{v_{He}} = \frac{1}{2} \]

### Final Answer
The ratio of the speed of an electron in the first orbit of hydrogen atom to that in the first orbit of helium is \( 1:2 \).

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