Question

An alpha particle $\left({}_2{}^4\mathrm{He}\right)$ has a mass of $4.00300amu$. A proton has mass of $1.00783amu$ and a neutron has mass of $1.00867amu$ respectively. The binding energy of alpha particles estimated from these data is the closest to:

• A. $20.4MeV$
• B. $32.0MeV$
• C. $24.3MeV$
• D. $27.9MeV$

Answer 1

The correct option is D

$27.9MeV$

Step 1: Given

Atomic mass: $A=4$

Atomic number: $Z=2$

Mass of proton: $m_p=1.00783amu$

Mass of neutron: $m_n=1.00867amu$

Mass of helium atom: $m_{He}=4.00300amu$

Step 2: Formula Used

$∆M=Z{m}_p+\left(A-Z\right)m_n-m_{atom}$

$B.E.=∆M{c}^2$

Step 3: Calculate mass defect using the formula

$$\begin{gathered} ∆M=2\times 1.00783+\left(4-2\right)\times 1.00867-4.00300 \\ =2\times 1.00783+2\times 1.00867-4.00300 \\ =2.01566+2.01734-4.00300 \\ =0.03000amu \end{gathered}$$

Step 4: Calculate the binding energy of the alpha particle

$$\begin{gathered} B.E.=∆M{c}^2 \\ =0.3000amu\times 931.5 \\ =27.9MeV \end{gathered}$$

Hence, the binding energy is $27.9MeV$. Hence, option D is correct.