Alpha-particles are projectied towards the nuclei of the following metals with the same kinetic energy. Towards which metal, the distance of closest approach is minimum?

Options

A. `Cu(Z=29)`
B. `Ag(Z=47)`
C. `Au(Z=79)`
D. `Pd(Z=46)`

Explanation

To solve the problem of determining towards which metal the distance of closest approach for alpha particles is minimum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Closest Approach**:
The distance of closest approach (R) occurs when the kinetic energy (KE) of the alpha particle is equal to the potential energy (PE) due to electrostatic interaction with the nucleus of the metal.

2. **Equating Kinetic and Potential Energy**:
The kinetic energy of the alpha particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
The potential energy at the distance of closest approach is given by:
\[ PE = \frac{k \cdot Q_1 \cdot Q_2}{R} \]
where:
- \( k \) is Coulomb's constant,
- \( Q_1 \) is the charge of the alpha particle (which is +2e),
- \( Q_2 \) is the charge of the nucleus of the metal,
- \( R \) is the distance of closest approach.

3. **Setting Up the Equation**:
At the closest approach, we set KE equal to PE:
\[ \frac{1}{2} mv^2 = \frac{k \cdot Q_1 \cdot Q_2}{R} \]

4. **Rearranging for R**:
Rearranging this equation gives us:
\[ R = \frac{k \cdot Q_1 \cdot Q_2}{KE} \]
From this equation, we can see that the distance of closest approach \( R \) is directly proportional to \( Q_2 \) (the charge of the nucleus).

5. **Analyzing the Charge of the Nucleus**:
Since \( R \) is proportional to \( Q_2 \), the distance of closest approach will be minimum when \( Q_2 \) is minimum. Therefore, we need to identify which metal has the smallest nuclear charge.

6. **Comparing the Metals**:
We need to compare the nuclear charges of the given metals. For example, if we have metals like copper (Cu), lead (Pb), and others, we look at their atomic numbers:
- Copper (Cu) has an atomic number of 29.
- Lead (Pb) has an atomic number of 82.
- Other metals will have varying atomic numbers.

7. **Conclusion**:
The metal with the smallest nuclear charge will result in the minimum distance of closest approach. In this case, copper has the smallest nuclear charge among the options provided, leading us to conclude that the distance of closest approach is minimum for copper.

### Final Answer:
The distance of closest approach is minimum towards **Copper (Cu)**.