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I know that in a closed system mass+energy is always conserved. For example, in an exothermic reaction, some mass is converted to energy. Now, I'm looking for a situation where mass and energy are separately conserved.

Given below is a hypothetical situation I've come up with:

Consider a closed system of 2 identical balls, one moving (ball 1) and the other stationary (ball 2). Ball 1 collides with ball 2 and transfers its kinetic energy to ball 2. As ball 1 lost its KE in the process, its mass was also decreased. Similarly, as ball 2 gained KE, its mass was increased. In this case, both mass and energy are separately conserved.

Does this make sense? I'm not sure because I'm fairly new to the concept of mass-energy equivalence.

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  • $\begingroup$ Aside from the relativistic consideration explained by the answer(s) below, I think you could easily model an open system where energy is conserved but the number of particle is not or viceversa. $\endgroup$
    – Mauricio
    Commented Sep 29, 2021 at 15:15

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I know that in a closed system mass+energy is always conserved.

Unfortunately, it seems that you have been prey to some poor teaching of the kind often found in pop-sci treatments of physics.

For any isolated system the energy is conserved, and for any isolated system the momentum is conserved. The mass is related to energy and momentum by $m^2 c^2 = E^2/c^2 - p^2$. Since $E$ and $p$ are conserved the conservation of $m$ is implied.

Mass is conserved in any isolated system, but as you can see from the expression above, mass is not additive. It is non-linear. This means that the mass of a system is always larger than the sum of the masses of the parts of the system.

Using units where $c=1$, let’s work an example. Suppose we have an electron and a positron at rest next to each other. Each have energy $E= 511 \text{ keV}$ and momentum $p= 0 \text{ keV}$. So the total system energy is $1022 \text{ keV}$ and the total system momentum is $0 \text{ keV}$ so the system mass is $1022 \text{ keV}$. Now, suppose they anhilate and produce two photons, each with $E=511 \text{ keV}$ energy and one with $p=511 \text{ keV}$ and the other with $p= -511 \text{ keV}$. Note that each photon has 0 mass. The system of two photons together again has total system energy $1022 \text{ keV}$ and the total system momentum is again $0 \text{ keV}$ so the system mass is again $1022 \text{ keV}$

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  • $\begingroup$ In an exothermic reaction, the mass of the products is less than the mass of the reactants. Mass is not conserved there, right? In what kind of system is this true? $\endgroup$
    – Sasikuttan
    Commented Sep 29, 2021 at 15:26
  • $\begingroup$ The total system mass is always conserved for all isolated systems. Mass is not additive so the sum of the masses of the products is not equal to the mass of the system. It is the mass of the whole system which is always conserved. $\endgroup$
    – Dale
    Commented Sep 29, 2021 at 15:42
  • $\begingroup$ But what about closed systems, where energy is allowed to leave, but matter is not? Is mass conserved there? $\endgroup$
    – Sasikuttan
    Commented Sep 29, 2021 at 15:47
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    $\begingroup$ If energy is allowed to leave then mass is not conserved. The conservation of mass requires conservation of energy and conservation of momentum. So if either of those are violated then so is conservation of mass $\endgroup$
    – Dale
    Commented Sep 29, 2021 at 15:50
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    $\begingroup$ Possibly, but then mass+energy is not conserved either $\endgroup$
    – Dale
    Commented Sep 29, 2021 at 16:39

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