The binding energy per nucleon of deuteron (^2_1H) and helium nucleus (^4_2He) is 1.1 MeV and 7 MeV

Q: Why do we multiply the given values by the number of nucleons?

The values provided in the question are 'binding energy PER nucleon'. To find the total energy holding the entire nucleus together, you must multiply that value by the total number of protons and neutrons (the mass number, A) in that specific nucleus.

Q: Why is energy released when the binding energy increases?

It sounds counterintuitive, but higher binding energy means the nucleus is in a lower, more stable energy state. When nucleons bind more tightly together, they shed their excess potential energy into the surroundings, usually as kinetic energy or gamma-ray photons.

Q: Can I just subtract the B.E. per nucleon values directly (7 - 1.1)?

No! That is a very common trap. You must always convert the values to TOTAL binding energy first. The energy released depends on the total energy state of the entire system, not just the energy per individual particle.

Q: What exactly does the Q-value represent?

The Q-value is the net amount of energy absorbed or released in a nuclear reaction. A positive Q-value means the reaction is exothermic (it releases energy), while a negative Q-value means the reaction is endothermic (it requires an input of energy to occur).

Q: Why do two deuterons fuse instead of breaking apart?

Light nuclei like deuterium have relatively low binding energy per nucleon. Fusing them creates a heavier nucleus (like Helium) with a much higher binding energy per nucleon. Nature always favors moving towards a more stable, lower-energy state, making this fusion a highly favorable, energy-releasing process.

Question

The binding energy per nucleon of deuteron (${}^2_1	ext{H}$) and helium nucleus (${}^4_2	ext{He}$) is $1.1 	ext{ MeV}$ and $7 	ext{ MeV}$ respectively. If two deuteron nuclei react to form a single helium nucleus, then the energy released is

Options:

(A) $13.9 	ext{ MeV}$

(B) $26.9 	ext{ MeV}$

(C) $23.6 	ext{ MeV}$

(D) $19.2 	ext{ MeV}$

Sigma Percentile · Accepted Answer

### The Power of Fusion: Calculating Energy Release in Nuclear Reactions

Have you ever wondered what powers the Sun and the stars? The answer lies in the incredible process of **nuclear fusion**, where light atomic nuclei combine to form heavier, more stable ones. In this problem, we are going to explore a miniature version of this cosmic event: the fusion of two deuteron nuclei to form a single helium nucleus.

Our goal is to calculate the exact amount of energy released during this reaction. To do this, we need to understand the concept of **binding energy**.

### Understanding Binding Energy

Binding energy is the energy required to completely disassemble a nucleus into its constituent protons and neutrons. Conversely, it is also the energy released when those nucleons bind together to form the nucleus.

A crucial metric is the **binding energy per nucleon**. This tells us how tightly bound each individual proton or neutron is within the nucleus. The higher the binding energy per nucleon, the more stable the nucleus.

When a nuclear reaction occurs, the energy released (often called the **Q-value**) is simply the difference between the total binding energy of the final products and the total binding energy of the initial reactants:

$$Q = 	ext{Total B.E. of Products} - 	ext{Total B.E. of Reactants}$$

Let's break down our specific reaction step-by-step.

### Analyzing the Reactants: Deuterium

Our reactants are two deuteron nuclei (${}^2_1	ext{H}$). A deuteron is an isotope of hydrogen containing one proton and one neutron, giving it a total of $2$ nucleons.

The problem states that the binding energy per nucleon for a deuteron is $1.1 	ext{ MeV}$.

To find the total binding energy of a single deuteron, we multiply the binding energy per nucleon by its total number of nucleons:

$$	ext{B.E. of one } {}^2_1	ext{H} = 2 	imes 1.1 	ext{ MeV} = 2.2 	ext{ MeV}$$

Since our reaction involves **two** deuterons fusing together, we must account for both of them. Therefore, the total binding energy of our reactants is:

$$	ext{Total B.E. of Reactants} = 2 	imes 2.2 	ext{ MeV} = 4.4 	ext{ MeV}$$

### Analyzing the Product: Helium

When the two deuterons fuse, they form a single helium nucleus (${}^4_2	ext{He}$), also known as an alpha particle. This nucleus contains two protons and two neutrons, making a total of $4$ nucleons.

Helium is a remarkably stable nucleus. The problem gives its binding energy per nucleon as $7 	ext{ MeV}$.

Just like we did for the deuterons, we calculate the total binding energy of the helium nucleus by multiplying its binding energy per nucleon by its total number of nucleons:

$$	ext{Total B.E. of Product} = 4 	imes 7 	ext{ MeV} = 28 	ext{ MeV}$$

### The Final Calculation: Energy Released

Now we have all the pieces of the puzzle. We know the total binding energy of what we started with (the reactants) and what we ended up with (the product).

To find the energy released ($Q$), we subtract the reactants' binding energy from the product's binding energy:

$$Q = 28 	ext{ MeV} - 4.4 	ext{ MeV}$$

$$Q = 23.6 	ext{ MeV}$$

And there we have it! The fusion of two deuterons into a helium nucleus releases a massive **$23.6 	ext{ MeV}$** of energy.

This result perfectly illustrates a fundamental principle of nuclear physics: whenever light nuclei fuse to form a nucleus with a higher binding energy per nucleon, the system transitions to a lower, more stable energy state, and the "excess" energy is radiated away into the universe.

Animated Solution for Physics - Atoms and Nuclei: The binding energy per nucleon of deuteron ( $_{1}^{2} H$ ) and helium nucleus ( $_{2}^{4} He$ ) is $1.1 MeV$ and $7 MeV$ respectively. If two deuteron nuclei react to form a single helium nucleus, then the energy released is

Select Answer:

Visualized Solution

$N u c l e a r F u s i o n R e a c t i o n$

$E n er g y R e l e a se d (Q)$

$B . E . o f R e a c t an t s$

$B . E . o f P r o d u c t s$

$C a l c u l a t in g Q - v a l u e$

$C o n c l u s i o n$

The Sigma Insight: Nucleus and Nuclear Reaction

The Power of Fusion

Understanding Binding Energy

Analyzing the Reactants

Analyzing the Product

The Final Calculation

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Comprehension Passage

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