Q.What will be the value of neutron multiplication factor for controlled chain reactions?

Mass number and atomic number already balance between reactants and product, so only energy (a gamma photon) is released — no material particle.

The reaction is:

$_3^7\text{Li} + \ _1^1\text{p} \rightarrow \ _4^8\text{Be} + X$

Check conservation of mass number (A) and atomic number (Z):

• Mass number: $7 + 1 = 8$, and $_4^8\text{Be}$ already has mass number 8 — balanced.
• Atomic number: $3 + 1 = 4$, and $_4^8\text{Be}$ already has atomic number 4 — balanced. …

Conserve mass number and atomic number across the nuclear reaction $_4Be^9 + \,_2He^4 \rightarrow X + \,_0n^1$.

Mass number: $9+4 = 13 = A_X + 1 \Rightarrow A_X = 12$

Atomic number: $4+2 = 6 = Z_X + 0 \Rightarrow Z_X = 6$

…

Correct α-decay: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He.

In α-decay a nucleus emits a ⁴₂He nucleus, so the mass number decreases by 4 and the atomic number by 2. Applying conservation:

• Mass number: 238 = 234 + 4 ✓
• Atomic number: 92 = 90 + 2 ✓ …

Both reactions are radioactive beta decays. Reaction (a) is $\beta^-$ decay (a neutron converts to a proton, emitting an electron and an antineutrino); reaction (b) is $\beta^+$ decay (a proton converts to a neutron, emitting a positron and a neutrino).

(a) ${}^{32}_{15}\text{P}\rightarrow{}^{32}_{16}\text{S}+\_\_\_\_+\bar\nu$

Mass number stays $32\to32$ (unchanged), but atomic number increases $15\to16$. An increase in $Z$ by 1 with no change in $A$, accompanied by an antineutrino, is the signature of $\beta^-$ (electron) decay:

$n \rightarrow p + e^- + \bar\nu$

So:

${}^{32}_{15}\text{P} \rightarrow {}^{32}_{16}\text{S} + {}^{0}_{-1}e + \bar\nu$

(b) ${}^{22}_{11}\text{Na}\rightarrow{}^{22}_{10}\text{Ne}+\_\_\_\_+\nu$

…

$Z = 2$ gives 2 protons; $A - Z = 4 - 2 = 2$ gives 2 neutrons: option (B).

Nucleons are the particles inside a nucleus — protons and neutrons (electrons are NOT nucleons).

For helium $_2He^4$:

• Atomic number $Z = 2$ → number of protons $= 2$. …

This is a deuterium-deuterium fusion reaction; balancing mass and atomic numbers shows a free neutron is released.

Check conservation of mass number (A) and atomic number (Z):

Left side: ${}_1H^2 + {}_1H^2$ → $A=2+2=4$, $Z=1+1=2$

Right side: ${}_2He^3$ → $A=3$, $Z=2$ …

Balancing mass number and atomic number across the nuclear reaction shows the emitted particle must be a proton.

The reaction is:

7N14 + 0n1 -> 6C14 + X

Conserve mass number (A): 14 + 1 = 14 + A_X, so A_X = 1.

Conserve atomic number (Z): 7 + 0 = 6 + Z_X, so Z_X = 1.

…

Conserving mass number and charge, W has A = 1 and Z = 0 → a neutron: option (c).

In a nuclear reaction, both the mass number (superscript) and the atomic number/charge (subscript) must be conserved.

Reaction: ₂He⁴ + ₓX^A → ₍Z₊₂₎Y^(A+3) + W.

Mass number balance:

4 + A = (A + 3) + (mass number of W)

⇒ mass number of W = 4 + A − A − 3 = 1.

Charge (atomic number) balance:

2 + Z = (Z + 2) + (charge of W) …

Neutrons = mass number − atomic number = A − Z.

The atomic number Z is the number of protons. The mass number A counts protons plus neutrons (the nucleons):

…

Nuclear reactions conserve charge, linear momentum and angular momentum (and energy, nucleon number).

The fundamental conservation laws hold in every nuclear reaction:

• Total electric charge is conserved.
• Total linear momentum is conserved.
• Total angular momentum is conserved. …

A controlled chain reaction requires the multiplication factor K = 1.

The neutron multiplication factor K is the ratio of the number of neutrons (or fissions) in one generation to that in the preceding generation:

$K = \frac{\text{neutrons produced in a generation}}{\text{neutrons in the preceding generation}}$

• If $K > 1$, the number of fissions grows each generation → the reaction is uncontrolled (as in a bomb).
• If $K < 1$, the reaction dies out. …