Question:

A certain nuclear reaction results in the formation of an alpha particle ($\alpha$) and an unknown nuclear species $X$. The alpha particle has a mass of $6.646 \times 10^{-27}$ kg and a charge of $2e$, where $e$ is the elementary charge. The unknown nuclear species $X$ has a mass of $4.014 \times 10^{-3}$ kg and a charge of $3e$.

If this reaction follows the general nuclear equation $a + A \rightarrow b + X$, where $a$ represents an alpha particle, $A$ represents the parent nucleus, $b$ represents the daughter nucleus, and $X$ represents the unknown nuclear species, determine the following:

a) The charge of the daughter $b$ nucleus. b) The mass of the daughter $b$ nucleus. c) The energy released during this nuclear reaction. d) Whether this nuclear reaction is an exothermic or endothermic process.

Given:

• Elementary charge $e = 1.602 \times 10^{-19}$ C
• Speed of light $c = 2.998 \times 10^8$ m/s

Answer:

a) The charge of the daughter $b$ nucleus can be determined using the principle of conservation of charge. In the reaction, the alpha particle ($\alpha$) has a charge of $2e$ and the unknown nuclear species ($X$) has a charge of $3e$. Therefore, the total initial charge must be equal to the total final charge.

Initial charge = Charge of $\alpha$ particle + Charge of $A$ nucleus
Final charge = Charge of $X$ species + Charge of $b$ nucleus

Since the alpha particle ($\alpha$) and the unknown nuclear species ($X$) represent the initial and final charges respectively, we can write the equation as:

$2e + Charge\ of\ A = 3e + Charge\ of\ b$

Rearranging the equation, we find:

Charge of $b$ nucleus = Charge of $A$ nucleus - $e$

Substituting the values, we get:

Charge of $b$ nucleus = $3e - e$
Charge of $b$ nucleus = $2e$

The charge of the daughter $b$ nucleus is $2e$.

b) The mass of the daughter $b$ nucleus can be determined using the principle of conservation of mass. In the reaction, the alpha particle ($\alpha$) has a mass of $6.646 \times 10^{-27}$ kg and the unknown nuclear species ($X$) has a mass of $4.014 \times 10^{-3}$ kg. Therefore, the total initial mass must be equal to the total final mass.

Initial mass = Mass of $\alpha$ particle + Mass of $A$ nucleus
Final mass = Mass of $X$ species + Mass of $b$ nucleus

Since the alpha particle ($\alpha$) and the unknown nuclear species ($X$) represent the initial and final masses respectively, we can write the equation as:

$6.646 \times 10^{-27}$ kg + Mass of $A$ = $4.014 \times 10^{-3}$ kg + Mass of $b$

Rearranging the equation, we find:

Mass of $b$ nucleus = Mass of $A$ nucleus + $4.014 \times 10^{-3}$ kg - $6.646 \times 10^{-27}$ kg

Substituting the values, we get:

Mass of $b$ nucleus = $4.014 \times 10^{-3}$ kg - $6.646 \times 10^{-27}$ kg
Mass of $b$ nucleus $\approx 4.014 \times 10^{-3}$ kg

The mass of the daughter $b$ nucleus is approximately $4.014 \times 10^{-3}$ kg.

c) The energy released during this nuclear reaction can be determined using Einstein's mass-energy equivalence principle, $E = mc^2$, where $E$ represents the energy released, $m$ represents the mass change, and $c$ represents the speed of light.

The mass change can be calculated as:

Mass change = Mass of $\alpha$ particle + Mass of $A$ nucleus - Mass of $X$ species - Mass of $b$ nucleus

Substituting the given values, we get:

Mass change = $6.646 \times 10^{-27}$ kg + Mass of $A$ nucleus - $4.014 \times 10^{-3}$ kg - $4.014 \times 10^{-3}$ kg

Simplifying the equation, we find:

Mass change = $6.646 \times 10^{-27}$ kg + Mass of $A$ nucleus - $8.028 \times 10^{-3}$ kg

By considering the mass-energy equivalence principle, the energy released can be calculated as:

Energy released = Mass change $\times$ Speed of light$^2$

Substituting the values, we get:

Energy released = (Mass change) $\times$ $(2.998 \times 10^8)^2$ J

However, since the mass of $A$ nucleus and the specific reaction are not provided, we cannot determine the exact energy released without those values.

d) To determine whether this nuclear reaction is exothermic or endothermic, we need to compare the final energy to the initial energy. If the final energy is lower than the initial energy, it is an exothermic process. If the final energy is higher than the initial energy, it is an endothermic process. However, without the specific values for the energy released from part c, we cannot make a definitive determination.