Question

Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

Hint:

Rutherford scattering key idea:

Most particles undeflected → empty space
Few large-angle deflections → tiny dense nucleus

Answer 1

Concept: This refers to Rutherford’s alpha-particle scattering experiment. The graph shows how the number of scattered alpha particles varies with scattering angle.

Graph description: Plot: 

X-axis → Scattering angle (\( \theta \)) 
Y-axis → Number of scattered particles 
Shape of graph: 

Very large number of particles at small angles (near \( 0^\circ \)) 
Rapid decrease as angle increases 
Very few particles scattered at large angles 
Extremely small number scattered backward (near \( 180^\circ \)) 
So, the curve starts high at small angles and falls sharply with increasing angle. Conclusion 1: Atom is mostly empty space. Since most alpha particles pass through with little or no deflection: 

Positive charge and mass are not uniformly spread. 
Most of the atom is empty. 
Conclusion 2: Presence of a small, dense nucleus. A very small fraction of particles are deflected through large angles: 

Indicates strong repulsive force. 
Positive charge is concentrated in a tiny central region (nucleus). 
Additional inference (optional): 

Nucleus is positively charged and very small compared to atom size. 
 

Answer 2

The shape of the scattered-particle-versus-angle plot in Rutherford's alpha-scattering experiment can be understood through the idea of impact parameter, the perpendicular distance between the original straight-line path of an incoming alpha particle and the nucleus it passes near.

Relating impact parameter to scattering angle:
An alpha particle that passes far from any nucleus (large impact parameter) feels only a weak repulsive force and is deflected by a tiny angle. An alpha particle heading almost directly at a nucleus (very small impact parameter) feels a strong repulsive force at closest approach and can be deflected through a large angle, occasionally even bounced back the way it came. Since the nucleus occupies only a minuscule fraction of the atom's cross-sectional area, only a tiny fraction of the incoming alpha particles have a small enough impact parameter to be strongly deflected.

Describing the plot:
Plotting the number of scattered particles (y-axis) against scattering angle \( \theta \) (x-axis), the curve starts very high near \( \theta = 0^\circ \), since most alpha particles have large impact parameters and pass through with negligible deflection. It then falls off sharply as \( \theta \) increases; the number of particles scattered beyond a given angle drops roughly as \( 1/\sin^4(\theta/2) \), a very steep fall. By the time \( \theta \) approaches \( 180^\circ \), the count is extremely small, only a tiny fraction of alpha particles get deflected by more than \( 90^\circ \).

Conclusion 1: the atom is mostly empty space.
Because the overwhelming majority of alpha particles pass through with little or no deflection, most of the space inside an atom must be empty, with nothing dense enough in most of that volume to noticeably push a fast, heavy alpha particle off course.

Conclusion 2: nearly all of the atom's mass and positive charge is concentrated in a tiny, dense nucleus.
The rare but real large-angle and near-\( 180^\circ \) deflections can only be produced by a very strong, concentrated repulsive force acting over a very small region, meaning the positive charge, and essentially all the mass, of the atom is packed into a nucleus far smaller than the atom itself, exactly the region that gives a small enough impact parameter to produce those large-angle events.