# If $E=mc^2$, then why do different substances have different calorific values?

Today during a classroom discussion, I realised that if we consider the equation $$E=mc^2$$, then we are establishing a relation between energy and mass but we often observe that different substances produce different amount of energy when they are burned. For example: Burning a kilogram of wood will not produce same amount of energy as 1 kilogram of petrol.

Different $$E$$. Burning is a chemical process that releases some amount of chemical energy, which is vastly less that the total rest energy contained in the initial substance. The $$E$$ in $$E = mc^2$$ is how much energy you'd get if you converted all of the mass into pure radiation and left nothing left, which is a totally different process from burning.

One has to keep clear in what frame the statements are made. The $$E=mc^2$$ equation holds in a frame of very large energies and momenta, the special relativity frame. The classical frame, until the discovery of nuclear reactions and other effects that necessitated the frame of special relativity, masses were invariant. That is how our value system developed, on the invariant mass of gold.

The energies and momenta involved in burning wood and petrol are in changing the atomic and molecular bonds, and these are very small, below an electron volt per interaction. The energies and momenta involved in nuclear reactions are of the order of MeV, one million electron volts. There is a very very tiny change in mass with chemical reactions, but it is not measurable.

In a chemical reaction a very, very small amount of the mass of the reagents is converted to energy. For example you mention burning petrol and if you burn a mole of octane (a component of petrol) then it releases about $$5$$ MJ, which is the equivalent of converting about $$56$$ picograms ($$5.6 \times 10^{-11}$$ grams) of the octane to energy. The molar mass of octane is $$114.2$$g so only about $$5 \times 10^{-11}$$ percent of the mass of the octane is converted to energy.

Anyhow, the reason that different reactions produce different amounts of energy is because in those reactions different percentages of the reagents are converted to energy. The equation $$E = mc^2$$ always applies, but in different reactions the value of $$m$$ in the equation is different.

The mass of your reaction products after burning the two substances will slightly differ, indeed, because a different amount of mass has been converted to energy and passed to the surroundings.

If you converted 1 pound of fuel completely into energy it would be the same as converting 1 pound of wood completely into energy.

But when you burn wood, you get leftover mass in two forms: smoke and ashes. When you burn gas, you get leftover mass in exhaust ($$CO_2$$, $$CO$$, etc.)

The energy you get from burning a substance is:

$$E = c^2(m_{substance} - m_{leftovers})$$

But if you convert the whole thing to energy so that there's no mass left then you get

$$E = mc^2$$

Although often used, I would not use the term "conversion to energy". $$E=mc^2$$ says that energy and mass are the same things (in the center of mass frame). If a system at rest has some mass $$m$$, it has, at the same time, a corresponding total energy $$mc^2$$. No conversion is going on. I would rather say that in any reaction (chemical or nuclear) a certain amount of energy is released as radiation and then it is considered lost by the system. As a consequence of such a change of energy of the system, the energy of the reaction products, and therefore their total mass is decreased.

The calorific value of a material corresponds to the difference of energy between products and reactants after the energy corresponding to some interaction has been released in the form of radiation.

From a pound of oil one can get $$2.4~10^7~$$J. From a pound of $$~^{235}$$U, $$3.7~10^{13}$$ J. The corresponding mass difference between the original system and the final atomic/molecular products is $$2.7~10^{-10}$$ kg in the case of oil and $$4~10^{-4}$$ kg in the case of Uranium. Below our ability of measuring in the first case, small, but measurable, in the second one. But in both cases the variation of mass is a small fraction of the total mass of the system.

In order to transform the whole system into radiation, obtaining the same calorific value per unit of mass, one would need to have half of the system made by matter and the other half by anti-matter. Something which is presently possible only in the high energy physics labs, by using beams of particles/antiparticles.

• FWIW, to make this equivalence more clear, we sometimes refer to mass as rest energy. Jun 21, 2021 at 7:04