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

• MaestroBach
In summary: For actual measurements, yes, it would depend on what temperature the object was at when you made the measurement.
MaestroBach
Moved from a technical forum, so homework template missing.
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!

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.

Pencilvester said:
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.

Pencilvester
MaestroBach said:
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.

hutchphd
PeterDonis said:
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.

MaestroBach said:
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.

Mister T said:
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?

MaestroBach said:
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.

hutchphd and Bystander
Orodruin said:
Yes.

Thank you very much!

berkeman
Ibix said:
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.

MaestroBach said:
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##.

## What is the definition of inertial mass?

Inertial mass is a measure of an object's resistance to changes in its state of motion. It is often referred to as the "quantity of matter" in an object.

## How is inertial mass related to the melting of an ice cube?

The inertial mass of an object does not change when it undergoes a physical change, such as melting. This means that the inertial mass of an ice cube remains the same before and after it melts.

## What is the fractional increase in inertial mass when an ice cube melts?

As mentioned earlier, the inertial mass of an ice cube does not change when it melts. Therefore, there is no fractional increase in inertial mass during the melting process.

## Is there any change in the mass of an ice cube when it melts?

While the inertial mass of an ice cube remains the same, its physical mass does change when it melts. This is because some of the water molecules in the ice cube become liquid and may evaporate. Overall, the mass of the ice cube decreases as it melts.

## How does the melting of an ice cube impact its density?

The density of an ice cube decreases as it melts. This is because the volume of the ice cube increases as it turns into liquid, while the mass remains the same. Therefore, the density decreases because the same amount of mass now occupies a larger volume.

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