Converting Energy to Mass: Understanding the Relationship Between MeV and kg

[Saibot]

Homework Statement

Convert 1672 MeV/c^2 to kg

Relevant Equations

E=mc^2

(1672 MeV/c^2) * c^2 = 1.505 * 10^20 MeV = 1.505 * 10^26 eV = 2.41 * 10^7 J

Since E = mc^2, m = E/c^2

Therefore, m = 2.41 * 10^7 / (3 * 10^8)^2 = 2.68 * 10^-10 kg

But the answer is 2.97 * 10^-27 kg

Help! What is wrong with my logic?

 

[kuruman]

$1~\text{eV} = 1.609 \times 10^{-19}~\text{J}$
eV is a unit of energy so eV/c² is a unit of mass. Thus 1 J/c² is kilograms. Conversely, if you multiply kilograms by c² in (m/s)², you get Joules.
Take it from there.

 

[Saibot]

Indeed, but if I directly convert 1672 MeV/c^2 using the fundamental charge:

(1672 * 10^6 eV/c^2) * 1.609 J/eV, I get 2.68 * 10^-10 J/c^2. This is kilograms.

Same wrong answer. I'm missing something here. I have to divide again by c^2 and I have no idea why.

 

[Steve4Physics]

(1672 MeV/c^2) * c^2 = 1.505 * 10^20 MeV

No. You haven't handled the conversion/units correctly. The 'c²'s (on the 'top' and the 'bottom') cancel, so the energy (in MeV) is 1672MeV.

 

[Saibot]

OK, so it was incorrect to replace the "unit" c with the actual "value" of c (3*10^8). Got it, thanks mate.

 

[kuruman]

(1672 * 10^6 eV/c^2) * 1.609 J/eV, I get 2.68 * 10^-10 J/c^2. This is kilograms.

Mind your units.

If $~1~\text{eV} = 1.609 \times 10^{-19}~\text{J}$, $~1~\text{J} = \frac{1}{1.609 \times 10^{-19}}~\text{eV}\implies 1~ \rm{J/eV}=6.21\times 10^{18}.$

 

[DrClaude]

Therefore, m = 2.41 * 10^7 / (3 * 10^8)^2 = 2.68 * 10^-10 kg

But the answer is 2.97 * 10^-27 kg

You are off by a factor of $\approx 9 \times 10^{16}$. This should give you a clue as to what you are doing wrong.

 

[Steve4Physics]

OK, so it was incorrect to replace the "unit" c with the actual "value" of c (3*10^8). Got it, thanks mate.

It's ok to replace the "unit" c with its actual value. But you didn't do it everwhere. So you should have done this:

$1672 MeV/c^2 \times c^2$

$= 1672 \times \frac {MeV}{(3 \times 10^8 m/s)^2} \times (3 \times 10^8 m/s)^2$

$= 1672MeV$

Of course, all that work is unnecesary once you understand that a mass of $X~ MeV/c^2$ is equivalent (using $E=mc^2$) to an energy of $X ~MeV$.

 

[Saibot]

Understood. Thanks so much :)