Consider the nuclear fission Ne20 - 2 He4 + C12. Given that the binding energy/nucleon of Ne20, He4

Q: Why do we multiply the given binding energies by the mass numbers?

The problem provides the binding energy *per nucleon*. To find the total energy holding the entire nucleus together, you must multiply this value by the total number of nucleons (protons + neutrons), which is the mass number.

Q: Why is the Q-value formula `Products - Reactants` for binding energy, but `Reactants - Products` for mass?

Binding energy is the energy *released* when a nucleus is formed from its constituent nucleons. Higher binding energy means a more stable, lower-energy state. Therefore, if products have more binding energy than reactants, the system has moved to a lower energy state, releasing the difference. Hence, $Q = BE_{\text{products}} - BE_{\text{reactants}}$. Conversely, mass defect is related to this released energy by $E=mc^2$, so $Q = (M_{\text{reactants}} - M_{\text{products}})c^2$.

Q: What does a negative Q-value signify physically?

A negative Q-value means the products have *less* total binding energy than the reactants. They are less tightly bound overall. To force the tightly bound reactant to break into less tightly bound products, you must *supply* energy to the system.

Q: Why are there two Helium nuclei in the products?

The balanced nuclear equation is $^{20}\text{Ne} \longrightarrow 2^{4}\text{He} + ^{12}\text{C}$. The coefficient '2' indicates that two alpha particles (Helium-4 nuclei) are produced to conserve the total number of nucleons ($20 = 2 \times 4 + 12$).

Q: What if none of the options match my calculated answer?

In competitive exams like JEE, occasionally a question may have incorrect options due to a typographical error in the paper. If you are confident in your concepts and calculations, trust your result. In this specific historical question, none of the given options were correct, and it was likely treated as a bonus question.

Question

Consider the nuclear fission
$^{20}	ext{Ne} \longrightarrow 2^{4}	ext{He} + ^{12}	ext{C}$
Given that the binding energy/nucleon of $^{20}	ext{Ne}$, $^{4}	ext{He}$ and $^{12}	ext{C}$ are respectively, $8.03 	ext{ MeV}$, $7.07 	ext{ MeV}$ and $7.86 	ext{ MeV}$, identify the correct statement.

Options:

(A) Energy of $3.6 	ext{ MeV}$ will be released.

(B) Energy of $12.4 	ext{ MeV}$ will be supplied.

(C) $8.3 	ext{ MeV}$ energy will be released.

(D) Energy of $11.9 	ext{ MeV}$ has to be supplied.

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine you are observing a nuclear fission event. A heavy nucleus, in this case, Neon-20 ($^{20}	ext{Ne}$), is splitting apart into smaller fragments. Specifically, it breaks down into two Helium-4 nuclei ($^{4}	ext{He}$, also known as alpha particles) and one Carbon-12 nucleus ($^{12}	ext{C}$).

Whenever a nuclear reaction occurs, there is an exchange of energy. This energy change is quantified by the **Q-value** of the reaction. To determine whether energy is released (exothermic) or absorbed (endothermic), we need to compare the total binding energy of the products with the total binding energy of the reactants.

### The Master Equation
The fundamental equation governing the energetics of this reaction is:

$$\Delta Q = (	ext{B.E.})_{	ext{products}} - (	ext{B.E.})_{	ext{reactants}}$$

However, the problem provides us with the binding energy *per nucleon*, not the total binding energy. To find the total binding energy of any nucleus, we must multiply its binding energy per nucleon by its mass number ($A$), which represents the total number of protons and neutrons it contains.

### Calculating the Binding Energies
Let's start by calculating the total binding energy of our products. We have two $^{4}	ext{He}$ nuclei and one $^{12}	ext{C}$ nucleus.

$$(	ext{B.E.})_{	ext{products}} = 2 	imes (4 	imes 7.07) + (12 	imes 7.86)$$

Computing the values inside the brackets:

$$(	ext{B.E.})_{	ext{products}} = 2 	imes 28.28 + 94.32 = 56.56 + 94.32 = 150.88 	ext{ MeV}$$

Next, we calculate the total binding energy of our single reactant, the $^{20}	ext{Ne}$ nucleus:

$$(	ext{B.E.})_{	ext{reactants}} = 20 	imes 8.03 = 160.6 	ext{ MeV}$$

### Final Calculation and Interpretation
Now, we substitute these total binding energies back into our Q-value equation:

$$\Delta Q = 150.88 - 160.6 = -9.72 	ext{ MeV}$$

We arrive at a **negative Q-value**. What does this signify physically? A negative Q-value indicates that the products are less tightly bound overall than the reactants. Therefore, the reaction cannot happen spontaneously; energy must be **supplied** (or absorbed) to force the Neon-20 nucleus to break apart.

Looking at the given options, none of them state that $9.72 	ext{ MeV}$ of energy must be supplied. In competitive exams, such discrepancies occasionally occur, and this question would likely be treated as a bonus.

Animated Solution for Physics - Atoms and Nuclei: Consider the nuclear fission $^{20} Ne ⟶ 2^{4} He +^{12} C$ Given that the binding energy/nucleon of $^{20} Ne$ , $^{4} He$ and $^{12} C$ are respectively, $8.03 MeV$ , $7.07 MeV$ and $7.86 MeV$ , identify the correct statement.

Select Answer:

Visualized Solution

Visualizing the Reaction

Formula for $Δ Q$

Total Binding Energy

Setup for $(B.E.)_{products}$

Computing $(B.E.)_{products}$

Setup for $(B.E.)_{reactants}$

Computing $(B.E.)_{reactants}$

Calculating $Δ Q$

Final Conclusion

The Sigma Insight: Nucleus and Nuclear Reaction

Analyzing the Setup Imagine you are observing a nuclear fission event

The Master Equation

Calculating the Binding Energies Let's start by calculating the total binding energy of our products

Final Calculation and Interpretation

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