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What is the nuclear binding energy of an atom that has a mass defect of [tex]\(1.643 \times 10^{-28} \, \text{kg}\)[/tex]? Use [tex]\(E = mc^2\)[/tex]. (The speed of light is approximately [tex]\(3.00 \times 10^8 \, \text{m/s}\)[/tex]).

A. [tex]\(1.83 \times 10^{-48} \, \text{J}\)[/tex]
B. [tex]\(4.93 \times 10^{-20} \, \text{J}\)[/tex]
C. [tex]\(1.48 \times 10^{-11} \, \text{J}\)[/tex]
D. [tex]\(5.48 \times 10^{45} \, \text{J}\)[/tex]



Answer :

GinnyAnswer
To determine the nuclear binding energy of an atom with a given mass defect, we use Einstein’s famous equation [tex]\( E = mc^2 \)[/tex]. Here’s a step-by-step breakdown of the calculation:

1. Identify the given data:
- Mass defect ([tex]\( m \)[/tex]) = [tex]\( 1.643 \times 10^{-28} \)[/tex] kg
- Speed of light ([tex]\( c \)[/tex]) = [tex]\( 3.00 \times 10^8 \)[/tex] m/s

2. Write down the formula:
The equation for energy is [tex]\( E = mc^2 \)[/tex].

3. Square the speed of light:
[tex]\[ c^2 = (3.00 \times 10^8)^2 = 9.00 \times 10^{16} \, (\text{m}^2/\text{s}^2) \][/tex]

4. Multiply the mass defect by the squared speed of light to find the binding energy:
[tex]\[ E = (1.643 \times 10^{-28}) \times (9.00 \times 10^{16}) \][/tex]

5. Perform the multiplication:
- Multiply the numerical coefficients:
[tex]\[ 1.643 \times 9.00 = 14.787 \][/tex]
- Combine the powers of ten:
[tex]\[ 10^{-28} \times 10^{16} = 10^{-28 + 16} = 10^{-12} \][/tex]
- Putting it all together:
[tex]\[ E = 14.787 \times 10^{-12} \, \text{J} \][/tex]

6. Convert the result into scientific notation:
[tex]\[ E = 1.4787 \times 10^{-11} \, \text{J} \][/tex]

7. Compare the result with the provided choices:
The choices are:
- [tex]\( 1.83 \times 10^{-48} \)[/tex] J
- [tex]\( 4.93 \times 10^{-20} \)[/tex] J
- [tex]\( 1.48 \times 10^{-11} \)[/tex] J
- [tex]\( 5.48 \times 10^{45} \)[/tex] J

The value [tex]\( 1.4787 \times 10^{-11} \)[/tex] J is very close to the choice [tex]\( 1.48 \times 10^{-11} \)[/tex] J.

8. Conclude the correct answer:
The nuclear binding energy of the atom with a mass defect of [tex]\( 1.643 \times 10^{-28} \, \text{kg} \)[/tex] is [tex]\(\boxed{1.48 \times 10^{-11} \, \text{J}}\)[/tex].