The approximate size of the nucleus can be calculated by using energy conservation theorem in Ruther

Question

The approximate size of the nucleus can be calculated by using energy conservation theorem in Rutherford's $\alpha$-scattering experiment. If an $\alpha$-particle is projected from infinity with speed $v$, towards the nucleus having $Z$ protons, then the $\alpha$-particle which is reflected back or which is deflected by $180^\circ$ must have approached closest to the nucleus. It can be approximated that $\alpha$-particle collides with the nucleus and gets back. Now, if we apply the energy conservation equation at initial point and collision point, then :

$(	ext{Total Energy})_{	ext{initial}} = (	ext{Total Energy})_{	ext{final}}$
$(	ext{K.E.})_i + (	ext{P.E.})_i = (	ext{K.E.})_f + (	ext{P.E.})_f$

$(	ext{P.E.})_i = 0$, since P.E. of two charge system separated by infinite distance is zero. Finally the particle stops and then starts coming back.

$\frac{1}{2}m_{\alpha} v_{\alpha}^2 + 0 = \frac{K q_1 q_2}{R}$
$\Rightarrow \quad \frac{1}{2}m_{\alpha} v_{\alpha}^2 = K \frac{2e 	imes Ze}{R}$
$\Rightarrow \quad R = \frac{4KZe^2}{m_{\alpha} v_{\alpha}^2}$

Thus the radius of nucleus can be calculated using above equation. The nucleus is so small a particle that we can't define a sharp boundary for it.

Professor PC Thomas Classes · Answer

See detailed solution for explanation.