Masses of two isobars 64_29 Cu and 64_30 Zn are 63.9298 u and 63.9292 u respectively. It can be conc

Q: Why can't Zn decay into Cu?

Because Zn has a smaller mass than Cu. A spontaneous decay requires the parent to be heavier than the daughter to release energy.

Q: What happens to the missing mass?

The missing mass (mass defect) is converted into kinetic energy of the emitted particles (beta particle and antineutrino) according to Einstein's equation $E = mc^2$.

Q: Why is it a beta decay and not alpha?

In alpha decay, the mass number decreases by 4. Here, the mass number remains 64, which is characteristic of beta or gamma decay. Since the atomic number changes, it must be beta decay.

Q: Are atomic masses or nuclear masses given?

Usually, atomic masses are given. In $\beta^-$ decay, the condition $M_{\text{atomic}}(\text{parent}) > M_{\text{atomic}}(\text{daughter})$ automatically accounts for the mass of the emitted electron.

Q: What is an isobar?

Isobars are atoms of different chemical elements that have the same number of nucleons (same mass number) but different atomic numbers.

Question

Masses of two isobars $_{29}^{64}	ext{Cu}$ and $_{30}^{64}	ext{Zn}$ are $63.9298	ext{ u}$ and $63.9292	ext{ u}$ respectively. It can be concluded from these data that

Options:

(A) both the isobars are stable

(B) $	ext{Zn}^{64}$ is radioactive, decaying to $	ext{Cu}^{64}$ through $\beta$-decay

(C) $	ext{Cu}^{64}$ is radioactive, decaying to $	ext{Zn}^{64}$ through $\gamma$-decay

(D) $	ext{Cu}^{64}$ is radioactive, decaying to $	ext{Zn}^{64}$ through $\beta$-decay

Sigma Percentile · Accepted Answer

### The Tale of Two Isobars

Imagine you have two atoms sitting side by side: Copper-64 ($_{29}^{64}	ext{Cu}$) and Zinc-64 ($_{30}^{64}	ext{Zn}$). They are **isobars**, meaning they share the exact same mass number ($A = 64$) but have different atomic numbers. At first glance, they might seem like peaceful neighbors, but the universe is governed by a strict rule: **everything seeks the lowest possible energy state**.

To understand their fate, we must look at their masses. The mass of Copper-64 is $63.9298	ext{ u}$, while the mass of Zinc-64 is slightly less, at $63.9292	ext{ u}$. This tiny difference is the key to unlocking the mystery of their stability.

### The Rule of Spontaneous Decay

For a nucleus to spontaneously decay into another, the process must release energy. According to Einstein's famous equation, $E = mc^2$, energy and mass are interchangeable. Therefore, for a decay to occur, the mass of the parent nucleus must be strictly greater than the mass of the daughter products.

Let's calculate the mass defect ($\Delta m$) if Copper were to decay into Zinc:

$$\Delta m = M(_{29}^{64}	ext{Cu}) - M(_{30}^{64}	ext{Zn})$$

$$\Delta m = 63.9298	ext{ u} - 63.9292	ext{ u} = 0.0006	ext{ u}$$

Since $\Delta m > 0$, the decay of Copper-64 into Zinc-64 is energetically favorable. Copper is sitting at a higher energy state and wants to roll down the hill to become the more stable Zinc.

### Unmasking the Decay Process

Now that we know Copper decays into Zinc, we need to figure out *how*. Let's write down the nuclear reaction and balance the atomic and mass numbers:

$$_{29}^{64}	ext{Cu} \longrightarrow \ _{30}^{64}	ext{Zn} + 	ext{Unknown Particle}$$

**1. Conserving Mass Number ($A$):**
The parent has $A = 64$, and the daughter has $A = 64$. 
$$64 = 64 + A_{	ext{particle}} \implies A_{	ext{particle}} = 0$$

**2. Conserving Atomic Number ($Z$):**
The parent has $Z = 29$, and the daughter has $Z = 30$.
$$29 = 30 + Z_{	ext{particle}} \implies Z_{	ext{particle}} = -1$$

A particle with a mass number of $0$ and a charge of $-1$ is an electron ($_{-1}^{0}e$). In the realm of nuclear physics, the emission of an electron from the nucleus is known as **$\beta^-$ decay**.

### The Final Verdict

By simply comparing the masses and applying the laws of conservation, we have deduced that Copper-64 is radioactive and will spontaneously transform into Zinc-64 by emitting a beta particle. The universe, once again, finds its balance.

Animated Solution for Physics - Atoms and Nuclei: Masses of two isobars $_{29}^{64} Cu$ and $_{30}^{64} Zn$ are $63.9298 u$ and $63.9292 u$ respectively. It can be concluded from these data that

Select Answer:

Visualized Solution

Analyzing the Given Masses

Condition for Spontaneous Decay

Calculating Mass Defect

Identifying the Decay Type

Conclusion

The Sigma Insight: Radioactivity

The Tale of Two Isobars

The Rule of Spontaneous Decay

Unmasking the Decay Process

The Final Verdict

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