Why do bound systems have less rest mass than the sum of its parts?

[dimwatt]

Hi PF. This a fact well aware to just about anyone that has had even basic chemistry, but I'm having a hard time coming to an understanding as to why this must be true. So why?

Also, if I knew that some box contained, say, a proton and an electron, could I ever know whether or not, inside the box, the proton and the electron are in a bound state (hydrogen, as opposed to just two free particles) without puncturing or tampering with the boundaries of the box?

 

[dauto]

When the parts come together energy is released. That energy is called binding energy. To pull them apart the same amount of energy must now be provided. If the energy is not provided, the parts can't move apart - they are bound. By Einstein's relation E=mc² the energy lost in the binding process corresponds to an amount of lost mass - called the mass deficit. Therefore the bound system contains less mass than the sum of its parts.

 

[dauto]

Also, if I knew that some box contained, say, a proton and an electron, could I ever know whether or not, inside the box, the proton and the electron are in a bound state (hydrogen, as opposed to just two free particles) without puncturing or tampering with the boundaries of the box?

Yes, you could put the box on a scale and measure its mass. The hydrogen is lighter than the sum of the masses of the proton and electron.

 

[dimwatt]

Yes, you could put the box on a scale and measure its mass. The hydrogen is lighter than the sum of the masses of the proton and electron.

True but whether or not the particles "came together" no energy escaped the box, so it would have the same mass. One way we could tell is by opening the box and finding out if it weighs less than it did before opening, in which case the binding energy escaped in the form of heat or radiation or something like that. But I'm asking if there is anyway to discern between hydrogen and proton+electron (free) without disturbing the closed system.

 

[Nugatory]

True but whether or not the particles "came together" no energy escaped the box, so it would have the same mass. One way we could tell is by opening the box and finding out if it weighs less than it did before opening, in which case the binding energy escaped in the form of heat or radiation or something like that. But I'm asking if there is anyway to discern between hydrogen and proton+electron (free) without disturbing the closed system.

It's really hard to construct such an ideally closed system, but if you could then you are right: the mass of the box plus its contents would be the same whether the particles are bound (there's a hydrogen atom and some energy in the form of light/heat in the box) or unbound (there's a proton and an electron and the same energy in the form of an electrical field between them in the box).

There are other ways, in principle, of distinguishing the two cases; for example, the temperature of the box will be different; the electrical field outside the box will be different. However, they all depend, one way or another, on the box not being a perfect ideally closed system.

 

[dauto]

True but whether or not the particles "came together" no energy escaped the box, so it would have the same mass. One way we could tell is by opening the box and finding out if it weighs less than it did before opening, in which case the binding energy escaped in the form of heat or radiation or something like that. But I'm asking if there is anyway to discern between hydrogen and proton+electron (free) without disturbing the closed system.

You could look for the recoil of the box every time some particle inside if bounces off the walls and infer what kind of particles are inside