Two nucleon reactions are given below:

He₂³ → He₂⁴ + 20 MeV

He₂⁴ → He₂⁵ − 0.9 MeV

Find stability order of He₂³, He₂⁴, He₂⁵.

To determine the stability order of the helium isotopes (${\text{He}}_2^3$, ${\text{He}}_2^4$, and ${\text{He}}_2^5$), we need to analyze their binding energy per nucleon. The stability of a nucleus is directly proportional to its binding energy per nucleon ($E_{\mathrm{bn}}$).

Let $B\left({\text{He}}_2^A\right)$ represent the total binding energy of the isotope with mass number $A$.

1. First Reaction:
${\text{He}}_2^3\to {\text{He}}_2^4+20\text{MeV}$
This reaction represents the addition of a neutron to ${\text{He}}_2^3$ to form ${\text{He}}_2^4$, releasing $20\text{MeV}$ of energy. Therefore, the binding energy of ${\text{He}}_2^4$ is greater than that of ${\text{He}}_2^3$ by $20\text{MeV}$:
$B\left({\text{He}}_2^4\right)=B\left({\text{He}}_2^3\right)+20\text{MeV}$

2. Second Reaction:
${\text{He}}_2^4\to {\text{He}}_2^5-0.9\text{MeV}$
This reaction indicates that the addition of a neutron to ${\text{He}}_2^4$ to form ${\text{He}}_2^5$ requires an input of $0.9\text{MeV}$ (or absorbs energy). Thus, the binding energy of ${\text{He}}_2^5$ is less than that of ${\text{He}}_2^4$ by $0.9\text{MeV}$:
$B\left({\text{He}}_2^5\right)=B\left({\text{He}}_2^4\right)-0.9\text{MeV}=B\left({\text{He}}_2^3\right)+19.1\text{MeV}$

3. Calculating Binding Energy per Nucleon:
The binding energy of Helium-3 (${\text{He}}_2^3$) is approximately $7.7\text{MeV}$. Let us use this value to find the binding energy per nucleon ($E_{\mathrm{bn}}=\frac{B}{A}$) for each isotope:
• For ${\text{He}}_2^3$ (nucleons $A=3$):
$E_{\mathrm{bn}}\left({\text{He}}_2^3\right)=\frac{7.7}{3}\approx 2.57\text{MeV/nucleon}$
• For ${\text{He}}_2^4$ (nucleons $A=4$):
$E_{\mathrm{bn}}\left({\text{He}}_2^4\right)=\frac{7.7+20}{4}=\frac{27.7}{4}\approx 6.93\text{MeV/nucleon}$
• For ${\text{He}}_2^5$ (nucleons $A=5$):
$E_{\mathrm{bn}}\left({\text{He}}_2^5\right)=\frac{27.7-0.9}{5}=\frac{26.8}{5}=5.36\text{MeV/nucleon}$

Comparing these values, we get:
$2.57\text{MeV/nucleon}<5.36\text{MeV/nucleon}<6.93\text{MeV/nucleon}$

Therefore, the stability order is:
${\text{He}}_2^3<{\text{He}}_2^5<{\text{He}}_2^4$