Answer :
To calculate nuclear binding energy using the mass defect, we need to follow Einstein's mass-energy equivalence principle, given by the equation [tex]\( E = mc^2 \)[/tex]. Here's a step-by-step explanation of how to find the correct setup:
1. Identify the mass defect: The mass defect is given as [tex]\(0.09564 \, \text{amu}\)[/tex] (atomic mass units).
2. Convert the mass defect to kilograms: Since the binding energy is typically calculated in joules, we need to convert the mass defect from atomic mass units (amu) to kilograms (kg). The conversion factor is [tex]\( 1 \, \text{amu} = 1.6606 \times 10^{-27} \, \text{kg} \)[/tex].
3. Calculate the equivalent mass in kilograms: Multiply the mass defect by the conversion factor:
[tex]\[ 0.09564 \, \text{amu} \times 1.6606 \times 10^{-27} \, \text{kg/amu} \][/tex]
4. Include the speed of light: The speed of light [tex]\( c \)[/tex] is [tex]\( 3.0 \times 10^8 \, \text{m/s} \)[/tex].
5. Apply the mass-energy equivalence formula: Substitute the mass in kilograms and the speed of light into Einstein's formula, [tex]\( E = mc^2 \)[/tex]:
[tex]\[ E = (0.09564 \, \text{amu} \times 1.6606 \times 10^{-27} \, \text{kg/amu}) \times (3.0 \times 10^8 \, \text{m/s})^2 \][/tex]
Now, let's compare this with the given setups:
1. [tex]\( 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This matches the correct application of [tex]\( E = mc^2 \)[/tex] using the appropriate units and conversion factors.
2. [tex]\( 0.09564 \, \text{amu} \times 1 \, \text{amu} / \left(1.6606 \times 10^{-27} \, \text{kg}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This setup is incorrect because it reverses the conversion factor, which would lead to an incorrect mass in kilograms.
3. [tex]\( 0.09564 \, \text{amu} \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This setup is incorrect because it does not convert the mass defect from atomic mass units to kilograms.
4. [tex]\( 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right) \)[/tex]
This setup is incorrect because it does not square the speed of light.
Therefore, the correct setup to calculate the nuclear binding energy is:
[tex]\[ 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \][/tex]
The energy calculated using this setup is approximately [tex]\( 1.429378056 \times 10^{-11} \, \text{J} \)[/tex].
1. Identify the mass defect: The mass defect is given as [tex]\(0.09564 \, \text{amu}\)[/tex] (atomic mass units).
2. Convert the mass defect to kilograms: Since the binding energy is typically calculated in joules, we need to convert the mass defect from atomic mass units (amu) to kilograms (kg). The conversion factor is [tex]\( 1 \, \text{amu} = 1.6606 \times 10^{-27} \, \text{kg} \)[/tex].
3. Calculate the equivalent mass in kilograms: Multiply the mass defect by the conversion factor:
[tex]\[ 0.09564 \, \text{amu} \times 1.6606 \times 10^{-27} \, \text{kg/amu} \][/tex]
4. Include the speed of light: The speed of light [tex]\( c \)[/tex] is [tex]\( 3.0 \times 10^8 \, \text{m/s} \)[/tex].
5. Apply the mass-energy equivalence formula: Substitute the mass in kilograms and the speed of light into Einstein's formula, [tex]\( E = mc^2 \)[/tex]:
[tex]\[ E = (0.09564 \, \text{amu} \times 1.6606 \times 10^{-27} \, \text{kg/amu}) \times (3.0 \times 10^8 \, \text{m/s})^2 \][/tex]
Now, let's compare this with the given setups:
1. [tex]\( 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This matches the correct application of [tex]\( E = mc^2 \)[/tex] using the appropriate units and conversion factors.
2. [tex]\( 0.09564 \, \text{amu} \times 1 \, \text{amu} / \left(1.6606 \times 10^{-27} \, \text{kg}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This setup is incorrect because it reverses the conversion factor, which would lead to an incorrect mass in kilograms.
3. [tex]\( 0.09564 \, \text{amu} \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \)[/tex]
This setup is incorrect because it does not convert the mass defect from atomic mass units to kilograms.
4. [tex]\( 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right) \)[/tex]
This setup is incorrect because it does not square the speed of light.
Therefore, the correct setup to calculate the nuclear binding energy is:
[tex]\[ 0.09564 \, \text{amu} \times \left(1.6606 \times 10^{-27} \, \text{kg/amu}\right) \times \left(3.0 \times 10^8 \, \text{m/s}\right)^2 \][/tex]
The energy calculated using this setup is approximately [tex]\( 1.429378056 \times 10^{-11} \, \text{J} \)[/tex].
