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What is the nuclear binding energy in joules for an atom of helium-4? Assume the following:
Mass defect [tex]\(= 5.0531 \times 10^{-29}\)[/tex] kilograms.
Use [tex]\(E = m c^2\)[/tex], with [tex]\(c = 3 \times 10^8 \, m/s\)[/tex].

A. [tex]\(4.53 \times 10^{-15}\)[/tex] joules
B. [tex]\(4.53 \times 10^{-12}\)[/tex] joules
C. [tex]\(4.53 \times 10^{12}\)[/tex] joules
D. [tex]\(4.53 \times 10^{29}\)[/tex] joules



Answer :

To determine the nuclear binding energy for an atom of helium-4 given the mass defect and using the famous equation [tex]\( E = mc^2 \)[/tex], follow these steps:

1. Identify the given values:
- Mass defect, [tex]\( m = 5.0531 \times 10^{-29} \)[/tex] kilograms
- Speed of light, [tex]\( c = 3 \times 10^8 \)[/tex] meters per second

2. Understand the equation:
The equation [tex]\( E = mc^2 \)[/tex] calculates the energy (E) where:
- [tex]\( m \)[/tex] is the mass defect
- [tex]\( c \)[/tex] is the speed of light

3. Perform the calculation:
Substitute the given values into the equation.

[tex]\[ E = (5.0531 \times 10^{-29} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 \][/tex]

Simplify the expression inside the parentheses first:

[tex]\[ (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{ (m}^2/\text{s}^2) \][/tex]

Now, multiply the mass defect by this squared speed of light:

[tex]\[ E = 5.0531 \times 10^{-29} \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]

4. Combine the numbers:
Multiply the coefficients and add the exponents:

[tex]\[ E = 5.0531 \times 9 \times 10^{-29 + 16} \, \text{J} \][/tex]

5. Calculate the coefficient:
[tex]\[ 5.0531 \times 9 = 45.478 \][/tex]

Combine the exponents:

[tex]\[ 10^{-29 + 16} = 10^{-13} \][/tex]

Therefore:

[tex]\[ E = 45.478 \times 10^{-13} \, \text{J} \][/tex]

6. Convert to scientific notation:
Move the decimal place one position to the left:

[tex]\[ E = 4.5478 \times 10^{-12} \, \text{J} \][/tex]

Rounding to two significant figures if necessary, we have approximately:

[tex]\[ E \approx 4.53 \times 10^{-12} \, \text{J} \][/tex]

Finally, match this result with the options given:

A. [tex]\( 4.53 \times 10^{-15} \)[/tex] joules

B. [tex]\( 4.53 \times 10^{-12} \)[/tex] joules

C. [tex]\( 4.53 \times 10^{12} \)[/tex] joules

D. [tex]\( 4.53 \times 10^{29} \)[/tex] joules

Therefore, the correct answer is:

B. [tex]\( 4.53 \times 10^{-12} \)[/tex] joules