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I swear my stomach and thighs get bigger by every bite!
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So there's a little bit of a loss of energy there. So, that means you're going to lose a little bit of mass.
I think not.
Digestion is not a nuclear reaction!
You are converting the potential energy of e.g. sugar bonds, into kinetic energy.
During this process energy is converted to heat and that heat is radiated out of the body in various forms thus decreasing the mass of your body.
No nuclei are transformed by 3nO2 + 2(CH2)n → 2nCO2 + 2nH2O + heat
so the final mass is the same as the starting mass plus the mass of oxygen you inhaled to oxidise your food.
Were it not so, the universe would have disappeared into a burst of photons a long time ago.
The more common association of massenergy equivalence with nuclear processes derives from the fact that the large amounts of energy released in such reactions may exhibit enough mass that the mass loss (which is called the mass defect) may be measured, when the released energy (and its mass) have been removed from the system; while the energy released in chemical processes is smaller by roughly six orders of magnitude, and so the resulting mass defect is much more difficult to measure.
You are losing weight all the time (ignoring the mass of air being breathed in and out which causes fluctuations up and down if you include it), but if you eat the burger you will lose weight a bit faster as you munch and swallow it, even before you start digesting it. Some of the energy will be lost as sound and other vibrations (in addition to heat), but all of that lost energy starts as energy released from the rearrangement of bonds between atoms where a process similar to combustion is going on.