B)1/m....I.e.,kinetic energy=potential energy....1/2mv^2=2Ze^2/4πE• r....from this we get "r" is inversely proportional to 'm'...

The distance of closest approach (R) of an alpha particle to a heavy nucleus can be determined using the Rutherford scattering formula. This formula relates the impact parameter (b) to the mass (m), velocity (v), and charge (Ze) of the alpha particle, as well as the Coulomb constant (k) and the energy (E) of the alpha particle.

The Rutherford scattering formula is given by:

R = 1 / (4πε₀) * [(2Ze² / mv²) * (1 / (1 + √(1 + (2EZe / mv²))))]
where ε₀ is the permittivity of free space.

The dependence of R on the mass (m) can be analyzed by examining the term (2Ze² / mv²) in the formula.

1. Direct dependence on mass (m):
The term (2Ze² / mv²) in the formula suggests that the distance of closest approach is inversely proportional to the square of the alpha particle's mass (m). This means that as the mass of the alpha particle increases, the distance of closest approach decreases.

2. Inverse dependence on mass (m):
However, this interpretation is not entirely accurate. The term (1 / (1 + √(1 + (2EZe / mv²)))) also affects the distance of closest approach. This term involves the square root of (1 + (2EZe / mv²)).

As the mass (m) of the alpha particle increases, the denominator of the term (2EZe / mv²) becomes larger, resulting in a smaller value for the square root term. This, in turn, increases the overall value of the term (1 / (1 + √(1 + (2EZe / mv²)))), leading to a larger distance of closest approach.

Based on these considerations, we can conclude that the distance of closest approach (R) depends on the mass (m) of the alpha particle, but not in a simple linear or inverse linear manner. Therefore, none of the given options (A) m, (B) 1/m, (C) 1/√m, or (D) 1/m² accurately describe the dependence of R on m.