The ratio between total acceleration of the electron in singly ionized helium atom and hydrogen atom (both in ground state) is

Options

A. 1
B. 8
C. 4
D. 16

Answer

To find the ratio between the total acceleration of the electron in a singly ionized helium atom and a hydrogen atom in their ground states, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Acceleration in Atoms**:
The total acceleration \( a \) of an electron in an atom can be expressed in terms of its velocity \( V \) and radius \( r \) of the orbit:
\[ a \propto \frac{V^2}{r} \]

2. **Relating Velocity and Radius**:
For an electron in an atom, the velocity \( V \) is proportional to the atomic number \( Z \) divided by the principal quantum number \( n \):
\[ V \propto \frac{Z}{n} \]
The radius \( r \) of the orbit is proportional to \( n^2 \):
\[ r \propto n^2 \]

3. **Substituting into the Acceleration Formula**:
Substituting the expressions for \( V \) and \( r \) into the acceleration formula gives:
\[ a \propto \frac{\left(\frac{Z}{n}\right)^2}{n^2} = \frac{Z^2}{n^4} \]

4. **Finding the Proportionality for Acceleration**:
Therefore, we can express the acceleration as:
\[ a \propto \frac{Z^2}{n^4} \]

5. **Considering Ground State**:
For the ground state, the principal quantum number \( n = 1 \). Thus, the acceleration becomes:
\[ a \propto Z^2 \]

6. **Calculating the Ratio for Helium and Hydrogen**:
The atomic number \( Z \) for hydrogen is 1 and for singly ionized helium (He\(^+\)) is 2. Therefore, the ratio of the accelerations \( a_{He} \) to \( a_{H} \) is:
\[ \frac{a_{He}}{a_{H}} = \frac{Z_{He}^2}{Z_{H}^2} = \frac{2^2}{1^2} = \frac{4}{1} = 4 \]

### Final Answer:
The ratio between the total acceleration of the electron in singly ionized helium atom and hydrogen atom (both in ground state) is **4**.