Find the mass defect and binding energy for the helium nucleus? (Answer: 0.03038 amu, 28.3 MeV)
Mass of the helium Nucleus = m = 6.6447 × 10⁻²⁷ kg
Atomic mass of helium = A = 4
Atomic number of helium = z = 2
Mass of Proton = `\m_p` = 1.67262 × 10⁻²⁷ kg
Mass of Neutron = `\m_n`= 1.67493 × 10⁻²⁷ kg
To Find:
mass defect = Δ m =?
Binding energy = B.E =?
Solution:
The mass defect can be calculated by the following formula
Δ m = ( Z `\m_p` + (A - Z)`\m_n` ) - m
by putting the corresponding values
Δ m = ( 2 ｘ 1.67262 × 10⁻²⁷ kg + (4 - 2) ｘ1.67493 × 10⁻²⁷ kg ) - 4.00603 U
Δ m = 3.3452 × 10⁻²⁷ kg + 3.3498 × 10⁻²⁷ kg - 6.6447 × 10⁻²⁷ kg
Now converting in amu (atomic mass unit) we have
1.660 × 10⁻²⁷ kg = 1 amu
1 kg = (1/1.660) × 10²⁷ amu
1 kg = 0.602409 × 10²⁷ amu
0.0503 × 10⁻²⁷ kg = 0.0503 × 10⁻²⁷ × 0.602409 × 10²⁷ amu
Δm = 0.0303 amu ---------------Ans.1
To Find the Binding energy = B.E =?
1st Method:
We know that 1 amu = 931.5 MeV so,
Binding Energy = B.E = Mass deficit (in amu) × 931.5 MeV
Binding Energy = B.E = 0.0303 × 931.5 MeV
B.E = 28.225 MeV -----------------------Ans.2
2nd Method:
The formula for binding energy is
B.E. = Δmc², Where Δm should be in kg unit so,
B.E. = 0.0503 × 10⁻²⁷ kg × (3 × 10⁸ m s⁻¹)²
B.E. = 0.0503 × 10⁻²⁷ kg × 9 × 10¹⁶ m² s⁻²
B.E. = 0.4527 × 10⁻¹¹ J
Now to convert into eV we have 1 eV = 1.60218 × 10⁻¹⁹ J Thus,
B.E. = `frac {0.4527 × 10⁻¹¹ J}{1.60218 × 10⁻¹⁹J} eV
B.E. = 0.28255 × 10⁸ eV
or
B.E. = 28.3 × 10⁶ eV
or
B.E. = 28.3 MeV --------------------Ans. 2
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