Question

the energy required to seperate the 3 protons and 4 neutrons in the nucleus of a lithium atom is 39.3 megaelectronvolts determine the mass equivalent of this energy in universal mass units

Answer 1

To find the mass equivalent of the energy required to separate the protons and neutrons in a lithium nucleus, we can use Einstein's famous equation:

\[
E = mc^2
\]

Where:
- \( E \) is energy,
- \( m \) is mass,
- \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).

We need to convert the energy from megaelectronvolts (MeV) to joules first. The conversion factor is:

\[
1 \text{ MeV} = 1.602 \times 10^{-13} \text{ joules}
\]

Given the energy \( E = 39.3 \) MeV, we convert it to joules:

\[
E = 39.3 \text{ MeV} \times 1.602 \times 10^{-13} \text{ J/MeV} \approx 6.302 \times 10^{-12} \text{ J}
\]

Next, we can solve for the mass \( m \) using \( c = 3 \times 10^8 \text{ m/s} \):

\[
m = \frac{E}{c^2}
\]

Substituting in the values:

\[
m = \frac{6.302 \times 10^{-12} \text{ J}}{(3 \times 10^8 \text{ m/s})^2}
\]

Calculating \( c^2 \):

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

Now substitute this into the equation:

\[
m = \frac{6.302 \times 10^{-12}}{9 \times 10^{16}} \approx 6.992 \times 10^{-29} \text{ kg}
\]

To convert this mass to universal mass units (u), we use the conversion:

\[
1 \text{ u} \approx 1.660539 \times 10^{-27} \text{ kg}
\]

Now, converting the mass from kg to u:

\[
m \approx \frac{6.992 \times 10^{-29} \text{ kg}}{1.660539 \times 10^{-27} \text{ kg/u}} \approx 0.0421 \text{ u}
\]

Thus, the mass equivalent of the energy required to separate the 3 protons and 4 neutrons in the nucleus of a lithium atom is approximately:

\[
\boxed{0.0421 \text{ u}}
\]