Find the fractional increase in inertial mass when an ice cube melts

[MaestroBach]

Summary: Apparently an ice cube gains mass when it melts

So I'm asked to "Find the fractional increase in inertial mass when an ice cube melts ".

All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is added, so the energy of the liquid is Q + mc^2. I think, technically, what my professor wants me to use is mass dilation, so the liquid and mass will have the same energy I guess? (Which is funny because I remember him saying we wouldn't be using mass dilation, saying "we just don't do that anymore").

I'm not sure how to change what I have in terms of energies into an expression relating masses. I know there's a mass dilation equation, but it's not like the liquid is moving... and I assumed the net average velocity of all the water molecules was 0, maybe I'm wrong and it has something to do with that.

Any help would be appreciated!

 

[Pencilvester]

The molecules of a melted (liquid) ice cube at 0 degrees C have more random kinetic energy than a frozen ice cube at 0 degrees C. If you divide this amount of added energy by the rest mass of the frozen ice cube then you get the fractional increase that you’re looking for.

 

[PeterDonis]

If you divide this amount of added energy by the rest mass of the frozen ice cube then you get the fractional increase that you’re looking for.

There's a caveat to this, though. The rest masses quoted for various substances are effectively assuming zero absolute temperature. A block of ice at melting point is not at 0 K, though; it's at 273 K. There is some added heat, as compared to zero temperature, in that block of ice. That added heat has to be counted in the total energy of the ice which forms the "baseline" for the fractional increase. The added heat is many, many orders of magnitude smaller than $mc^2$ for the ice, of course--but so is the latent heat that gets added during the melting process.

 

[Orodruin]

Summary: Apparently an ice cube gains mass when it melts

All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is added, so the energy of the liquid is Q + mc^2. I think, technically, what my professor wants me to use is mass dilation, so the liquid and mass will have the same energy I guess? (Which is funny because I remember him saying we wouldn't be using mass dilation, saying "we just don't do that anymore").

We do not like to talk about ”relativistic mass” (it is somewhat misleading). This does not mean that the increase in the invariant mass of a heated system does not increase. It is fundamentally different.

 

[Ibix]

The rest masses quoted for various substances are effectively assuming zero absolute temperature.

Wouldn't this depend on how the rest mass is measured? Surely if I do something like put an ice cube on a frictionless plane, then tap it with a hammer and measure its resulting speed I'm including the vibrational energy of the atoms in the mass.

 

[Herman Trivilino]

All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is added, so the energy of the liquid is Q + mc^2.

The fractional increase in rest energy is equal to the fractional increase in mass. Note that the heat of fusion and $c^2$ have the same units, the SI unit being joules per kilogram.

 

[MaestroBach]

The fractional increase in rest energy is equal to the fractional increase in mass. Note that the heat of fusion and $c^2$ have the same units, the SI unit being joules per kilogram.

Would this mean that all I need to do to calculate the fractional increase in mass is to say the fractional increase in energy = Q (which equals m * heat of fusion) / mc^2? And then that is equal to the fractional increase in mass?

 

[Orodruin]

Would this mean that all I need to do to calculate the fractional increase in mass is to say the fractional increase in energy = Q (which equals m * heat of fusion) / mc^2? And then that is equal to the fractional increase in mass?

Yes.

 

[MaestroBach]

Yes.

Thank you very much!

 

[PeterDonis]

Wouldn't this depend on how the rest mass is measured?

For actual measurements, yes, it would depend on what temperature the object was at when you made the measurement. I was thinking more of calculating the rest mass based on, for example, an Avogadro's number of water molecules in the ice cube, times the ground state mass of a water molecule.

 

[Herman Trivilino]

Would this mean that all I need to do to calculate the fractional increase in mass is to say the fractional increase in energy = Q (which equals m * heat of fusion) / mc^2? And then that is equal to the fractional increase in mass?

Yes, but note that when you do that you can show that the result has a dimensionless value that does not depend on the mass $m$.