Animated Solution for Chemistry - Chemical Kinetics: In the decay sequence : ${}_{92}^{238}\text{U}\stackrel{-x_1}{\to }{}_{90}^{234}\text{Th}\stackrel{-x_2}{\to }{}_{91}^{234}\text{Pa}\stackrel{-x_3}{\to }{}^{234}\text{Z}\stackrel{-x_4}{\to }{}_{90}^{230}\text{Th}$ $x_1$, $x_2$, $x_3$ and $x_4$ are particles/ radiation emitted by the respective isotopes. The correct option(s) is/are-

Nuclear decay problems might look like a cryptic puzzle of numbers and letters, but they are actually governed by one of the most beautiful and unbreakable rules in physics: Conservation Laws.

In any nuclear reaction, two fundamental quantities must always balance out perfectly: 1. Total Mass Number ($A$): The superscripts on both sides of the reaction arrow must sum to the same value. 2. Total Atomic Number ($Z$): The subscripts on both sides must also sum to the same value.

Let's use these simple rules to decode the entire sequence step by step.

The First Decay ($x_1$)

We start with Uranium-238 decaying into Thorium-234:

Look at the mass numbers. We go from $238$ down to $234$. That's a loss of $4$ mass units. Now look at the atomic numbers. We drop from $92$ to $90$, a loss of $2$ protons.

What particle has a mass of $4$ and a charge of $+2$? It's an alpha particle ($\alpha$), which is essentially a Helium nucleus (${}_2^4\text{He}$). Because it carries a positive charge, if it passes through an electric field, it will naturally deflect towards the negatively charged plate. This makes Option (C) absolutely correct.

The Second Decay ($x_2$)

Next, Thorium-234 decays into Protactinium-234:

Notice something interesting? The mass number stays exactly the same at $234$. However, the atomic number increases by $1$, from $90$ to $91$. How can a nucleus gain a proton without gaining mass? This happens when a neutron inside the nucleus spontaneously converts into a proton and ejects an electron.

This ejected electron is called a beta-minus particle (${\beta }^{-}$), represented as ${}_{-1}^{0}e$. This confirms that Option (B) is correct.

Finding the Mystery Element Z

To figure out what element Z is, it's actually easier to look at its decay rather than its formation. Let's look at the final step:

The mass number drops from $234$ to $230$, a decrease of $4$. Just like our first step, this means $x_4$ must be an alpha particle (${}_2^4\text{He}$).

Since the alpha particle takes away $2$ protons, the atomic number of Z must be $2$ greater than Thorium's $90$.

Any element with an atomic number of $92$ is, by definition, Uranium. Therefore, Z is an isotope of Uranium (specifically, U-234). This makes Option (A) correct.

The Third Decay ($x_3$)

Now that we know Z is ${}_{92}^{234}\text{U}$, let's look at the third step:

Once again, the mass number is unchanged ($234\to 234$), but the atomic number increases by $1$ ($91\to 92$). Just like the second step, this is a classic beta-minus decay.

Since $x_3$ is a beta particle (an electron), it is definitely not a gamma ray (which is a massless, chargeless photon). Therefore, Option (D) is incorrect.

By systematically applying conservation laws, we've completely decoded the sequence. The correct statements are (A), (B), and (C).