Question

The energy equivalent of 1g of substance is:

• A. \( 11.2 \times 10^{24} \, \text{MeV} \)
• B. \( 5.6 \times 10^{12} \, \text{MeV} \)
• C. \( 5.6 \, \text{eV} \)
• D. \( 5.6 \times 10^{26} \, \text{MeV} \)

Answer 1

The Correct Option is D

To find the energy equivalent of 1 gram of a substance, we will use Einstein's mass-energy equivalence principle, given by the famous equation:

\(E = mc^2\)

where:

• \(E\) is the energy equivalent.
• \(m\) is the mass of the substance.
• \(c\) is the speed of light, approximately \(3 \times 10^8 \, \text{m/s}\).

For 1 gram of a substance:

• \(m = 1 \, \text{g} = 1 \times 10^{-3} \, \text{kg}\) (since 1 gram is \(0.001\) kilograms)

Substituting these values into the equation:

\(E = (1 \times 10^{-3} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2\)

Calculate the value inside the parentheses:

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

Now, multiply them together:

\(E = (1 \times 10^{-3}) \times (9 \times 10^{16})\)

\(E = 9 \times 10^{13} \, \text{joules}\)

To convert joules to electronvolts (eV), use the conversion factor:

\(1 \, \text{joule} = 6.242 \times 10^{18} \, \text{eV}\)

Therefore,

\(E = 9 \times 10^{13} \, \text{joules} \times 6.242 \times 10^{18} \, \text{eV/joule}\)

\(E = 5.6178 \times 10^{32} \, \text{eV}\)

Now, convert electronvolts to mega-electronvolts (MeV):

\(1 \, \text{MeV} = 10^6 \, \text{eV}\)

\(E = \frac{5.6178 \times 10^{32} \, \text{eV}}{10^6}\)

\(E = 5.6178 \times 10^{26} \, \text{MeV}\)

Thus, the energy equivalent of 1 g of substance is \(5.6 \times 10^{26} \, \text{MeV}\), which matches the given correct option.

Answer 2

The energy equivalent of a mass is given by Einstein’s equation:

\( E = mc^2, \)

where:

• \( m = 1 \, \text{g} = 10^{-3} \, \text{kg}, \)
• \( c = 3 \times 10^8 \, \text{m/s}. \)

Step 1: Substitute values into \( E = mc^2 \)

\( E = (10^{-3}) \times (3 \times 10^8)^2 \, \text{J}. \)

Simplify:

\( E = (10^{-3}) \times (9 \times 10^{16}) \, \text{J}. \)

\( E = 9 \times 10^{13} \, \text{J}. \)

Step 2: Convert energy to electron volts (eV)

Using the conversion \( 1 \, \text{J} = 6.241 \times 10^{18} \, \text{eV}: \)

\( E = (9 \times 10^{13}) \times (6.241 \times 10^{18}) \, \text{eV}. \)

\( E = 56.169 \times 10^{31} \, \text{eV}. \)

Convert to MeV (\( 1 \, \text{MeV} = 10^6 \, \text{eV} \)):

\( E = 56.169 \times 10^{25} \, \text{MeV}. \)

Approximate:

\( E \approx 5.6 \times 10^{26} \, \text{MeV}. \)

Final Answer: \( 5.6 \times 10^{26} \, \text{MeV}. \)