Question

In the disintegration series ${^{238}_{92}}U{ \to ^\alpha }X{ \to ^{{\beta ^{ - 1}}}}{^A_Z}Y$
The values of $Z$ and $A$ are

A) $92,326$
B) $88,230$
C) $90,234$
D) $91,234$

Answer 1

Emission of an alpha particle causes a loss of mass of $4amu$ and a polarity loss of $+ 2$. The emission of a beta particle causes no loss of mass but a polarity loss of $- 1$. By balancing the mass and charge we will find the atomic weight and mass of the final particle.

Formulae used: Loss of mass and a polarity loss due to alpha particle $\alpha$ emission: $\Delta M = 4(2n + 2p),\Delta Z = 2$.
Where $\Delta M$ is the loss of mass, $n$ is the mass of a neutron particle and $p$ is the mass of proton particles.

Loss of mass and a polarity loss due to beta particle $\beta$ emission: $\Delta M = 0,\Delta Z = - 1$

Step by step solution:
For atomic mass calculation:
When an alpha particle is emitted, the body from which it is emitted loses a mass equivalent to $2n$ and $2p$. Applying these we get the relation $\Delta M = (2n + 2p)$ for mass change. We know that the mass of a neutron and proton is approximately $1amu$ each. Therefore, the loss in mass is equivalent to $\Delta M = (2n + 2p) = (2 + 2) = 4amu$.
The change in mass due to beta emission is $\Delta M = 0$ as they have negligible mass.
The atomic mass of the given particle is $238amu$.
Therefore, total change in mass due to alpha emission is determined by substituting these values:
$A = 238 - 4 = 234$
For atomic charge calculation:
When an alpha particle is emitted, the body from which it is emitted from losses charge equivalent to $+ 2$. Applying these we get the relation $\Delta Z = 2$ for charge loss.
The change in charge due to beta emission is $\Delta Z = - 1$.
The atomic mass of the given particle is $92$.
Therefore, total change in mass due to alpha emission is determined by substituting these values:
$Z = 92 - ( + 2 - 1) = 92 - 2 + 1 = 91$
Therefore, the final atomic mass $A$ and atomic charge $Z$ of the particle $Y$ are $234$ and $91$ respectively.

In conclusion, the correct option is D.

Note: An alpha particle is a helium nuclei that has been stripped of its electrons. Therefore, it only consists of two proton and neutron particles. A beta particle is nothing but a high speed electron. Therefore it has negligible mass and a unit negative charge, just like an electron.