Question

A radio-active nucleus \({ }^{\mathrm{A}} \mathrm{Z} \mathrm{X}\) emits \(3 \alpha\) -particles and 2 Positrons. the ratio of number of neutron to that of Protons in the final nucleus will be (A) \([(\mathrm{A}-\mathrm{Z}-8) /(\mathrm{Z}-8)]\) (B) \([(\mathrm{A}-\mathrm{Z}-12) /(\mathrm{Z}-4)]\) (C) \([(\mathrm{A}-\mathrm{Z}-4) /(\mathrm{Z}-8)]\) (D) \([(A-Z-4) /(Z-2)]\)

Short answer

The short answer for the problem is: The ratio of the number of neutrons to protons in the final nucleus after emitting 3 alpha particles and 2 positrons is given by (C) \(\frac{(A - Z) - 4}{Z - 8}\).

Step-by-step solution

Determine the initial composition

In the beginning, we have a nucleus represented by \({ }^{\mathrm{A}} \mathrm{Z} \mathrm{X}\). Here, A is the mass number, and Z is the atomic number. The mass number is the sum of the number of protons and neutrons, and the atomic number is the number of protons. We can use these definitions to express the initial number of protons and neutrons. Number of initial protons (P_initial) = Z Number of initial Neutrons (N_initial) = A - Z

Analyze the emissions

In the given decay process, the nucleus emits 3 alpha particles and 2 positrons. Let's see how these emissions affect the composition of the nucleus. 1. Emission of an alpha particle: An alpha particle consists of 2 protons and 2 neutrons. Therefore, when an alpha particle is emitted, the nucleus loses 2 protons and 2 neutrons. 3 alpha particles mean there is a loss of 6 protons and 6 neutrons. 2. Emission of a positron: A positron is an anti-particle of an electron, and when it is emitted, a proton changes into a neutron. Therefore, when a positron is emitted, the nucleus loses 1 proton and gains 1 neutron. 2 positrons mean there is a loss of 2 protons and a gain of 2 neutrons.

Calculate the final composition

Now, let's find out the final number of protons and neutrons in the nucleus after all emissions. Number of final protons (P_final) = P_initial - 6 (from alpha particles) - 2 (from positrons) = Z - 8 Number of final neutrons (N_final) = N_initial - 6 (from alpha particles) + 2 (from positrons) = (A - Z) - 4

Find the ratio of neutrons to protons

The ratio of the number of neutrons to protons in the final nucleus can now be determined: Ratio = N_final / P_final = \(\frac{(A - Z) - 4}{Z - 8}\) The correct answer is (C).

Key concepts

Alpha Particle Emission

Alpha particle emissions are a common type of radioactive decay that involves the emission of helium nuclei from the parent atom. An alpha particle is essentially a helium nucleus, consisting of two protons and two neutrons. When a nucleus emits an alpha particle, it loses these four nucleons, resulting in a decrease in its atomic number by 2 and its mass number by 4.

• Each alpha particle emission reduces the number of protons in the nucleus by 2.
• It also reduces the number of neutrons by 2.

In the context of the original exercise, the emission of three alpha particles results in the parent nucleus losing 6 protons and 6 neutrons. Understanding how this type of emission affects the nuclear composition is essential in predicting the new element formed and the change in its nuclear properties.

Positron Emission

Positron emission is a type of beta decay where a proton is converted into a neutron while emitting a positron and a neutrino. A positron is the antimatter counterpart of an electron, carrying the same mass but a positive charge instead of negative.

• When a positron is emitted, the number of protons decreases by 1.
• The number of neutrons increases by 1, due to the proton-neutron conversion.

In the problem exercise, the emission of two positrons alters the nucleus' composition by decreasing the number of protons by 2 and increasing the number of neutrons by 2. This process is significant in changing the neutron-to-proton ratio, which plays a crucial role in determining nuclear stability.

Neutron to Proton Ratio

The neutron to proton ratio is a crucial concept in nuclear physics, as it influences the stability of a nucleus. Generally, a balanced ratio helps maintain nuclear stability, while imbalances can lead to radioactivity.
As observed in the given exercise, after emitting particles, the neutron to proton ratio changes as follows:

• Initial neutrons = A - Z
• Initial protons = Z
• Final neutrons = (A - Z) - 4
• Final protons = Z - 8

The emitted alpha particles and positrons result in a revised ratio of neutrons to protons given as \(\frac{(A - Z) - 4}{Z - 8}\). Identifying this ratio helps in understanding the characteristic changes in a nucleus following radioactive decay events, such as shifts toward a more stable configuration.

Nuclear Composition Change

The nuclear composition is adjusted through emission events that change the number of protons and neutrons within an atom. Both alpha and positron emissions influence the atomic and mass numbers in different ways.
When these emissions occur:

• Alpha emission decreases both protons and neutrons by 6 in this scenario.
• Positron emission results in a net change of minus 2 protons but plus 2 neutrons, due to conversion effects.

The combined changes due to these emissions result in a new nuclear composition characterized by a specific neutron to proton ratio. Understanding these changes allows scientists to predict the behavior of the remaining nucleus, explore its stability, and understand the pathways of nuclear transformation.