If the atomic mass unit is designated by u, where 1 u is exactly one 12th of the mass of the 12C atom, equal to 1.661 1027 kg, then, calculate the energy equivalent to a conventional mass equal to 1u. Calculate the energy as using MeV as units.
The Correct Answer and Explanation is :
To calculate the energy equivalent of a mass of 1 atomic mass unit (1 u), we can use Einstein’s mass-energy equivalence formula:
[
E = mc^2
]
Where:
- (E) is the energy equivalent,
- (m) is the mass,
- (c) is the speed of light in a vacuum, approximately (3.00 \times 10^8 \, \text{m/s}).
Step-by-Step Calculation:
- Mass (m): The mass of 1 atomic mass unit is given as (1.661 \times 10^{-27} \, \text{kg}).
- Speed of Light (c): (c = 3.00 \times 10^8 \, \text{m/s}).
Now, substitute these values into the formula:
[
E = (1.661 \times 10^{-27} \, \text{kg}) \times (3.00 \times 10^8 \, \text{m/s})^2
]
First, calculate (c^2):
[
c^2 = (3.00 \times 10^8 \, \text{m/s})^2 = 9.00 \times 10^{16} \, \text{m}^2/\text{s}^2
]
Now calculate the energy:
[
E = (1.661 \times 10^{-27}) \times (9.00 \times 10^{16}) = 1.495 \times 10^{-10} \, \text{Joules}
]
- Convert to MeV: Since 1 electronvolt (eV) is (1.602 \times 10^{-19} \, \text{J}), we can convert the energy from Joules to mega-electronvolts (MeV):
[
1 \, \text{MeV} = 1 \times 10^6 \, \text{eV} = 1.602 \times 10^{-13} \, \text{J}
]
Now convert the energy:
[
E = \frac{1.495 \times 10^{-10} \, \text{J}}{1.602 \times 10^{-13} \, \text{J/MeV}} = 931.5 \, \text{MeV}
]
Final Answer:
The energy equivalent of 1 atomic mass unit (1 u) is approximately 931.5 MeV.
Explanation:
This calculation is a direct application of Einstein’s theory of relativity, which shows that mass and energy are interchangeable. The mass-energy equivalence (E = mc^2) implies that even a very small amount of mass can be converted into a substantial amount of energy, as evidenced by the result of 931.5 MeV. This is particularly relevant in nuclear physics, where small changes in mass can result in large amounts of energy, such as in nuclear reactions (e.g., fission and fusion).
In practical terms, 931.5 MeV is the amount of energy released or absorbed when an atomic mass unit undergoes a transformation, such as in nuclear decay or reactions.