The force between two `alpha`-particles separated by a distance .r. is F. In order to have same force F, the distance between singly ionized chlorine atoms separated by a distance of

Options

A. 2r
B. 4r
C. r/2
D. r/4

Answer

To solve the problem, we need to compare the electrostatic forces between two alpha particles and two singly ionized chlorine atoms.

### Step-by-Step Solution:

1. **Identify the Charge of Alpha Particles:**
An alpha particle consists of 2 protons and 2 neutrons, making its charge \( Q_{\alpha} = 2e \) (where \( e \) is the elementary charge).

2. **Calculate the Force Between Alpha Particles:**
The force \( F \) between two alpha particles separated by a distance \( r \) can be expressed using Coulomb's law:
\[ F = k \frac{Q_{\alpha}^2}{r^2} \]
Substituting \( Q_{\alpha} = 2e \):
\[ F = k \frac{(2e)^2}{r^2} = k \frac{4e^2}{r^2} \]

3. **Identify the Charge of Singly Ionized Chlorine Atoms:**
A singly ionized chlorine atom has lost one electron, giving it a charge of \( Q_{Cl} = e \).

4. **Set Up the Force Equation for Chlorine Atoms:**
Let the distance between the singly ionized chlorine atoms be \( x \). The force \( F \) between them can be expressed as:
\[ F = k \frac{Q_{Cl}^2}{x^2} \]
Substituting \( Q_{Cl} = e \):
\[ F = k \frac{e^2}{x^2} \]

5. **Equate the Two Forces:**
Since we want the force between the chlorine atoms to be the same as that between the alpha particles, we set the two equations equal:
\[ k \frac{4e^2}{r^2} = k \frac{e^2}{x^2} \]

6. **Cancel Out Common Terms:**
We can cancel \( k \) and \( e^2 \) from both sides:
\[ \frac{4}{r^2} = \frac{1}{x^2} \]

7. **Cross-Multiply to Solve for \( x^2 \):**
Cross-multiplying gives:
\[ 4x^2 = r^2 \]

8. **Solve for \( x \):**
Dividing both sides by 4:
\[ x^2 = \frac{r^2}{4} \]
Taking the square root:
\[ x = \frac{r}{2} \]

### Final Answer:
The distance between the singly ionized chlorine atoms should be \( \frac{r}{2} \).

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