Q.Find the energy equivalent of one atomic mass unit, first in Joules and then in MeV. Using this, express the mass defect of $^{16}_{8}\text{O}$ in $\text{MeV}/c^2$.

Use $E = mc^2$ with $1\ \text{u} = 1.6605\times 10^{-27}\ \text{kg}$.

Energy in joules:

In MeV (using $1\ \text{MeV} = 1.602\times 10^{-13}\ \text{J}$):

Mass defect of $^{16}_{8}\text{O}$ ($Z=N=8$), with $m(^1\text{H}) = 1.007825\ \text{u}$, $m_n = 1.008665\ \text{u}$, $m(^{16}\text{O}) = 15.994915\ \text{u}$: …

Energy equivalent of one atomic mass unit

Mass-energy equivalence, $E = mc^2$, converts any mass into an energy. One atomic mass unit is $1\ \text{u} = 1.6605\times 10^{-27}\ \text{kg}$, so

Convert to MeV using $1\ \text{MeV} = 1.602\times 10^{-13}\ \text{J}$:

Therefore

Mass defect of $^{16}_{8}\text{O}$

Oxygen-16 has $Z = 8$ protons and $N = 8$ neutrons. The mass defect is the difference between the total mass of the free constituents and the actual atomic mass. Using atomic masses (so the proton is represented by the $^1\text{H}$ atom, which balances the 8 electrons):

$m(^1\text{H}) = 1.007825\ \text{u},\quad m_n = 1.008665\ \text{u},\quad m(^{16}_{8}\text{O}) = 15.994915\ \text{u}.$ …

Method: Direct Application of $E = mc^2$ Using the Unified Mass Unit

The core idea is simple: one atomic mass unit (u) is defined as $1/12$ the mass of a carbon-12 atom. Its energy equivalent comes straight from Einstein's relation — multiply the mass (in kg) by $c^2$ to get Joules, then convert Joules to MeV using the known conversion factor.

Step 1 — Energy equivalent of 1 u in Joules

First, recall the value of 1 u in kilograms:

Speed of light: $c = 2.99792458 \times 10^8 \text{ m/s}$

Now apply $E = mc^2$:

Square $c$ first:

Multiply:

Step 2 — Convert to MeV

We need the conversion: $1 \text{ eV} = 1.602176634 \times 10^{-19} \text{ J}$

So $1 \text{ MeV} = 1.602176634 \times 10^{-13} \text{ J}$

Divide the energy in Joules by the energy of 1 MeV:

Step 3 — Mass defect of oxygen-16 in MeV/$c^2$

Oxygen-16 has $Z=8$ protons and $N=8$ neutrons. A key bookkeeping trick: use atomic

masses throughout (not bare nuclear masses), because atomic masses already include their

own orbital electrons — comparing $Z$ hydrogen ATOMS (each carrying 1 electron) against

the O-16 ATOM (carrying $Z=8$ electrons) makes the electron masses cancel automatically,

so you never need to add or subtract electron mass separately.

Atomic masses: $m(^1\text{H}) = 1.007825\text{ u}$ (proton + its own electron),

$m_n = 1.008665\text{ u}$ (neutrons have no electron either way),

$m(^{16}\text{O}) = 15.994915\text{ u}$ (the actual atomic mass, 8 electrons included).

• Mass of 8 hydrogen atoms: $8 \times 1.007825 \text{ u} = 8.062600 \text{ u}$
• Mass of 8 neutrons: $8 \times 1.008665 \text{ u} = 8.069320 \text{ u}$ …

Common Mistakes: Mass–Energy Equivalence

Mistake 1: Using the wrong value of $c$

Students often take $c = 3 \times 10^8 \text{ m/s}$ for convenience — and that’s fine for an estimate. But for the energy equivalent of 1 u, the exact value matters. The standard value is $c = 2.99792458 \times 10^8 \text{ m/s}$. Using the rounded value gives $E \approx 9 \times 10^{16} \text{ J per kg}$, which when multiplied by $1.66 \times 10^{-27} \text{ kg}$ yields about $1.49 \times 10^{-10} \text{ J}$ — close, but not the accepted $1.492 \times 10^{-10} \text{ J}$.

How to avoid: Use $c = 3.00 \times 10^8 \text{ m/s}$ only if the problem explicitly allows approximation. For board exams, stick to $c = 3 \times 10^8$ is usually acceptable, but for precise work (like binding energy calculations), use the exact value.

Mistake 2: Forgetting to convert atomic mass unit to kilograms

One atomic mass unit is $1 \text{ u} = 1.660539 \times 10^{-27} \text{ kg}$. A common error is to plug in 1 directly into $E = mc^2$ as if $m$ were in kg.

How to avoid: Always write the conversion explicitly:

Then:

Mistake 3: Confusing Joules with MeV in the conversion

The conversion $1 \text{ MeV} = 1.602 \times 10^{-13} \text{ J}$ is often misremembered as $1.6 \times 10^{-19}$ (which is the charge of an electron in coulombs). That error throws the MeV value off by a factor of a million.

How to avoid: Memorise the pair:

• $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$
• $1 \text{ MeV} = 1.602 \times 10^{-13} \text{ J}$

So to convert Joules to MeV, divide by $1.602 \times 10^{-13}$.

Mistake 4: Writing the final answer in $\text{MeV}$ instead of $\text{MeV}/c^2$

The mass defect is a mass, not an energy. When the problem asks for it in $\text{MeV}/c^2$, students often just give the energy equivalent in MeV and stop.

How to avoid: Remember: $E = mc^2$ means $m = E/c^2$. So if you compute the energy equivalent of the mass defect in MeV, the mass in $\text{MeV}/c^2$ is numerically the same number. For example, if the mass defect corresponds to 127.5 MeV of energy, then the mass defect is $127.5 \text{ MeV}/c^2$. The unit tells you it's mass, not energy.

Mistake 5: Using the mass of the nucleus instead of the mass defect

For $^{16}_{8}\text{O}$, the mass defect is:

Students sometimes plug in the atomic mass (which includes electrons) or forget to subtract the nuclear mass. …