Question

\(\alpha\) -particles are projected towards the nucleus of following metals, with the same kinetic energy. Towards which metal, the distance of closest approach will be minimum? (a) \(\operatorname{Cu}(Z=29)\) (b) \(\operatorname{Ag}(Z=47)\) (c) \(\mathrm{Au}(Z=79)\) (d) \(\mathrm{Ca}(Z=20)\)

Short answer

The distance of closest approach will be minimum for \(\mathrm{Au}(Z=79)\) because it has the highest nuclear charge among the given options.

Step-by-step solution

Understanding the Distance of Closest Approach

The distance of closest approach in a scattering experiment is inversely proportional to the charge of the nucleus that the \(\alpha\)-particles are projected towards. This relationship is derived from the conservation of energy, where the kinetic energy of the \(\alpha\)-particle is converted entirely into electric potential energy at the closest point of approach. The electric potential energy is given by \(\frac{kQ_1Q_2}{r}\), where \(k\) is the Coulomb's constant, \(Q_1\) and \(Q_2\) are the charges of the \(\alpha\)-particle and nucleus respectively, and \(r\) is the distance of closest approach.

Considering the Charge of the \(\alpha\)-Particle

Since all \(\alpha\)-particles have the same charge (which is twice the charge of a proton), the factor \(Q_1\) remains constant for all experiments with different metals. Therefore, the charge of the nucleus \(Z\) will be the deciding factor, as the metallic nucleus with which \(\alpha\)-particles interact will have a charge \(Q_2 = Ze\), where \(e\) is the elementary charge.

Comparing the Nuclear Charge of Each Metal

The distance of closest approach will be minimum for the nucleus with the highest charge. By comparing the given atomic numbers (\(Z\)) of the metals, we can determine that gold (\(\mathrm{Au}\)) with \(Z=79\) has the highest atomic number among the given options, indicating the highest nuclear charge.

Key concepts

Nuclear Charge

The concept of nuclear charge, often symbolized as Z, is critical when examining the behavior of particles as they approach atomic nuclei. Nuclear charge refers to the total charge of the protons within a nucleus. Since each proton carries a positive charge, the total nuclear charge could be calculated by multiplying the number of protons (the atomic number) by the fundamental charge of a proton. This is essential to understand because nuclear charge directly influences the interaction strength between charged particles and the nucleus. In the given exercise, \( \alpha \) -particles, which themselves are positively charged, are influenced by the nuclear charges of different metal atoms. Higher the nuclear charge, stronger the electrostatic forces that repel the approaching \( \alpha \) -particles, leading to a smaller distance of closest approach.

Coulomb's Law

Coulomb's law is instrumental to understanding why particles behave the way they do in electromagnetic fields. This fundamental law quantifies the amount of force between two charged objects. According to Coulomb's law, the force (\( F \) is directly proportional to the product of the charges (\( Q_1 \) and \( Q_2 \) and inversely proportional to the square of the distance (\( r^2 \) between them, that is \( F = k \frac{Q_1Q_2}{r^2} \) where \( k \) is Coulomb's constant. In the scattering experiment scenario, this law explains why \( \alpha \) -particles with the same kinetic energy will have different distances of closest approach to different nuclei: the force of repulsion increases with the increased nuclear charge, thus reducing the distance at which the particle can approach before its kinetic energy has been fully converted into electric potential energy.

Scattering Experiment

A scattering experiment is a method used in physics to probe the structure and properties of atomic nuclei. In these experiments, particles such as \( \alpha \) -particles are fired at a target nucleus, and the paths they take after encountering the electric field of the nucleus offer clues about the nuclear size, charge distribution, and even the forces at work within the atom. In the context of the textbook exercise, the distance of closest approach in such an experiment is equivalent to the point at which the entire kinetic energy of the \( \alpha \) -particles has been converted to electric potential energy without the particle actually colliding with the nucleus, illustrating the non-contact nature of electromagnetic force.

Electric Potential Energy

In the scenario of electric potential energy, this is the energy stored due to the position of a charged object in an electric field. The formula for electric potential energy (\( U \) between two point charges is \( U = k \frac{Q_1Q_2}{r} \), closely related to Coulomb's law but focusing on energy instead of force. It's important to note that potential energy is maximum at the distance of closest approach since it's the point where kinetic energy of the approaching \( \alpha \) -particle is fully transferred to potential energy. Through examining this energy transformation, we can infer the relationship between the nuclear charge and the kinetic energy of the \( \alpha \) -particles to find the distance where the particle's kinetic energy is entirely converted, illuminating the distance of closest approach.