In a Rutherford scattering experiment, alpha particles...

In a Rutherford scattering experiment, alpha particles having kinetic energy of 7.70 $\mathrm{MeV}$ are fired toward a gold nucleus. (a) Use energy conservation to determine the distance of closest approach between the alpha particle the and gold nucleus. Assume the nucleus remains at rest. (b) What If? Calculate the de Broglic wavelength for the 7.70 -MeV alpha particle and compare it with the distance obtained in part (a). (c) Based on this comparison, why is it proper to treat the alpha particle as a particle and not as a wave in the Rutherford scattering experiment?

In a Rutherford scattering experiment, alpha particles having kinetic energy of 7.70 $\mathrm{MeV}$ are fired toward a gold nucleus. (a) Use energy conservation to determine the distance of closest approach between the alpha particle the and gold nucleus. Assume the nucleus remains at rest. (b) What If? Calculate the de Broglic wavelength for the 7.70 -MeV alpha particle and compare it with the distance
obtained in part (a). (c) Based on this comparison, why is it proper to treat the alpha particle as a particle and not as a wave in the Rutherford scattering experiment?

Key Concepts

Distance of Closest Approach

This is the minimum separation achieved by a projectile and a target during a scattering interaction. It is determined by setting the initial kinetic energy equal to the electrostatic potential energy at the closest point, under the assumption that the target remains stationary. This concept is crucial in experiments involving particle collisions as it gives insight into the spatial scale of the interaction and the underlying force dynamics.

Energy Conservation

This concept involves the principle that in an isolated system the total energy remains constant over time. In scattering experiments, the initial kinetic energy of a projectile is converted into potential energy as it interacts with another particle, which is used to determine quantities like the distance of closest approach. The energy conservation equation allows one to equate the kinetic energy with the potential energy at the point where the projectile’s motion momentarily stops before it is repelled.

Electrostatic Potential Energy

Electrostatic potential energy is the energy stored in a system of charged particles due to their electric charges and separations. In contexts like scattering experiments, the interaction between positively charged particles is described by Coulomb’s law, and the potential energy is given by the product of the charges divided by the distance separating them. This energy plays a pivotal role in determining how far a charged particle can approach another charged target before being repelled.

de Broglie Wavelength

The de Broglie wavelength is a measure of the wave-like behavior of matter, establishing that every particle exhibits both wave and particle characteristics. It is inversely proportional to the momentum of the particle. The concept is central in quantum mechanics as it explains phenomena such as diffraction and interference, and it aids in assessing whether classical or quantum mechanical descriptions are more appropriate for a given system.

Wave-Particle Duality

Wave-particle duality is the principle that particles such as electrons, photons, and even composite particles like alpha particles, can exhibit properties of both waves and particles. However, when the de Broglie wavelength is significantly smaller than the physical dimensions involved in an experiment, the particle behavior dominates, justifying the use of classical mechanics in the analysis. This comparison helps in deciding the appropriate theoretical framework for describing the behavior of the particles in scattering experiments.

Transcript

00:01 In this question, we are given alpha particles, having kinetic energy of 7 .7 mvp being fired towards gold particle, right, or gold nucleus.

00:17 We want to use energy conservation to find what is the distance of closest approach.

00:22 So the initial total energy is just the kinetic energy of our alpha particle.

00:32 The final energy is only the potential energy of the system.

00:47 No kinetic energy.

00:48 That's the point where they are closest to each other, and the alpha particle is not moving.

00:57 Both the alpha particle and the particles are not moving.

01:00 So we just have to equate the kinetic energy to the potential energy, which in this case is the electric potential energy.

01:08 Due to the quorum interaction.

01:11 So this equals to ke times q1, q2 over r, where r is the distance between the nuclei at this particular point, which is our distance of closest approach.

01:30 Q1 and q2 is the charge of the alpha particle and the charge of the gold nuclei respectively.

01:38 So for the gold nuclei, this is given as 79e, because there are 79 protons, for the alpha particle is 2e.

01:57 Creating this to the kinetic energy, 7 .7, m .v.

02:01 We will have to convert the mep into joules to work in sir units.

02:09 So multiply by 1 .6, then power of 13.

02:12 So give us in terms of juice...

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