Mass of Water Increasing with Heating

[Ebi Rogha]

TL;DR Summary

Mass-energy relation

Suppose we heat up a sealed container of water (no vapour escape), will it mass increase according to E=m.c^2 ?

 

[PeterDonis]

Suppose we heat up a sealed container of water (no vapour escape), will it mass increase according to E=m.c^2 ?

Yes.

 

[DaveE]

If by mass you mean weight due to gravity, then yes.

E=mc² is really referring to the rest mass of whatever, water molecules in your case. The more general version is E² = m²c⁴ + p²c², where p is momentum. When you heat water, you increase the momentum of the molecules, which in turn increases the energy content, which creates more gravitational force. But in this version the mass doesn't change.

This is mostly a semantic issue, I think. Older texts will describe the mass as increasing with velocity (relativistic mass).

 

[PeterDonis]

If by mass you mean weight due to gravity, then yes.

It is true that the water's weight will increase, but that's not a separate thing from the increase in its invariant mass. See below.

When you heat water, you increase the momentum of the molecules, which in turn increases the energy content

No, this is not correct. The correct statement is that heating the water increases the average kinetic energy of the molecules and their average momentum (in the center of mass frame). One does not cause the other; they are both part of the same thing.

The increase in the kinetic energy of the molecules, in the center of mass frame, increases the invariant mass of the water as an overall system.

which creates more gravitational force.

Gravity is not a force in relativity. The weight of the water increases (assuming it is at rest in a constant gravitational field) because its invariant mass increases. If we want to talk about the effect on the water as a source of gravity, that's a whole other issue that I don't think is part of the OP's question and probably deserves a separate thread of its own.

But in this version the mass doesn't change.

This is not correct. See above.

 

[PeterDonis]

E=mc2 is really referring to the rest mass of whatever, water molecules in your case.

It can also refer to the invariant mass of the water as an overall system (and to be strictly correct we would also include the container in the overall system). That is what I took the OP to be asking about. Invariant mass is not additive, so the invariant mass of the overall system is not the same as the sum of invariant masses of all the individual constituents (the molecules).

 

[Ebi Rogha]

Invariant mass is not additive, so the invariant mass of the overall system is not the same as the sum of invariant masses of all the individual constituents (the molecules).

Could you pleass explain this?

 

[PeroK]

Could you pleass explain this?

Imagine a system of two particles of equal mass $m$. If the particles are relatively at rest, then the mass of the system is $2m$.

If, however, the particles have equal and opposite non-zero momenta, then the mass of the system is greater than $2m$.

Are you able to do the calculation?

 

[vanhees71]

Let's stick to special relativity and look at the water in global thermal equilibrium. Then there's a reference frame, where the water is at rest as a whole and it has an energy content dependent on the temperature. Heating up the water means the energy of the water in this refernce frame changes by $\Delta q$ (where $\Delta q$ is the added heat energy). Thus the invariant mass of the water changes by $\Delta m=\Delta q/c^2$ since by definition the invariant of a system is the total energy of this system in its center-of-momentum (NOT center-of-mass!) frame of reference.

 

[Ebi Rogha]

Are you able to do the calculation?

Thanks for the explanation.
I am not sure how to the calculation, if you could please help, that will also help to understand the concept.

 

[PeroK]

Thanks for the explanation.
I am not sure how to the calculation, if you could please help, that will also help to understand the concept.

If you are serious about this you should post questions like this under homework.

 

[Ibix]

Thanks for the explanation.
I am not sure how to the calculation, if you could please help, that will also help to understand the concept.

The thing to look up is the four momentum, which is a four vector comprised of the energy and momentum of the particle. The modulus of the four momentum is the mass of the particle. You can, of course, add vectors, so if you have a system of two or more particles you can write the total four momentum of the system as the sum of the four momenta of the particles. The modulus of that sum of vectors is the mass of the system - but the modulus of a sum of vectors is not in general equal to the sum of their moduli. So the mass of a system is not the sum of the masses of its components in relativity.