# Change in internal energy when ice melts to water at 0C

If we melt ice at 0'C to water at 0C, what is the change in the internal energy of the ice-water system?
As per the first law of thermodynamics, ΔU = ΔQ +ΔW
where they are increase in internal energy, heat flow to the system and work done on the system respectively.
If we melt ice at 0C, we have to supply heat to the system, so Q is positive
My initial thought on change in internal energy would have been that, as we melt a solid to liquid, there is increase in intermolecular distance resulting in increase in potential energy of the system BUT ice is less dense than water. So,the intermolecular spacing actually decreases when ice melts to water resulting in decrease in potential energy.
As U= K.E. + P.E.
and as K.E. is constant due to the constant temperature, U has to decrease due to decrease in potential energy

Further, the ice contracts on melting (due to the greater density of water), so the atmosphere actually does work on ice when it melts to a smaller volume of water. So, work done on the system is positive.

Therefore, we have a decrease in potential energy, increase in heat energy and increase in work done on the system.
So, what is the net effect on the internal energy of the system?
I believe my reasoning on work done and decrease in potential energy could be flawed as well. So, please help me out with this thought process.

Andrew Mason
Homework Helper
As U= K.E. + P.E.
and as K.E. is constant due to the constant temperature,
This is not correct. Temperature measures only translational kinetic energy. Total kinetic energy depends on vibrational, rotational and translational energy. The specific heat of ice is much lower than the specific heat of liquid water. What does that tell you about the degrees of freedom of water molecules in liquid form compared to that of ice? How does this relate to internal energy?

This is just a matter of applying the first law. Since dQ = dU + dW, if you add a great deal of heat to ice to melt it and there is very little work done (either on or by the ice) the internal energy must increase a great deal.

AM

This is not correct. Temperature measures only translational kinetic energy. Total kinetic energy depends on vibrational, rotational and translational energy. The specific heat of ice is much lower than the specific heat of liquid water. What does that tell you about the degrees of freedom of water molecules in liquid form compared to that of ice? How does this relate to internal energy?

This is just a matter of applying the first law. Since dQ = dU + dW, if you add a great deal of heat to ice to melt it and there is very little work done (either on or by the ice) the internal energy must increase a great deal.

AM

I do no know what "degrees of freedom of water molecule" refers to. Could you please give the answer to my question in simple terms.

Good morning Pranav Jha,

You have correctly worked out that a substantial amount of heat is absorbed by the system during melting, without significant work taking place.

You have further correctly deduced that this heat must therefore result in an increase in internal energy.

Well done.

Now you are enquiring what happened to the heat.

I don't know if you have yet met the Second Law of Thermodynamics but this is a case where there is a massive increase in entropy.

The entropy of fusion is about 22.5 Joules per deg per mole.

Andrew Mason
Homework Helper
I do no know what "degrees of freedom of water molecule" refers to. Could you please give the answer to my question in simple terms.
My comment was with respect to your statement that the KE of the liquid water molecules at 0C and ice molecules at 0C are the same. The translational KE is the same. But not the vibrational and rotational kinetic energies of the water molecules. Water molecules in liquid form have greater vibrational and rotational energy than the molecules in ice. They have more freedom to move.

Comment re: PE.

Most of the heat that is absorbed by the water molecules in melting is used in breaking the bonds between the molecules. So, although the density of water in liquid form is lower than in ice (maximum density is at 4C) this does not mean that the potential energy of water is less than in ice. Molecules that are bound have much less potential energy than unbound molecules. It takes energy to break those bonds to form liquid water.

AM

Comment re: PE.

Most of the heat that is absorbed by the water molecules in melting is used in breaking the bonds between the molecules. So, although the density of water in liquid form is lower than in ice (maximum density is at 4C) this does not mean that the potential energy of water is less than in ice. Molecules that are bound have much less potential energy than unbound molecules. It takes energy to break those bonds to form liquid water.

AM

I got the part relating to the increased rotational kinetic energy.
However, my question says that, ice is less dense than water at 0C. So, on melting at 0C, shouldn't the volume of the system decrease? And that would translate into decreased intermolecular spacing and hence decreased potential energy. So, could you please explain the second part using this information!

why hasn't the decreased intermolecular distance, decreased the potential energy in water?

You haven't said if you have heard of entropy?

Andrew Mason
Homework Helper
why hasn't the decreased intermolecular distance, decreased the potential energy in water?
Decreased intermolecular spacing does not imply a lower potential energy. It depends on the forces between the molecules, not the spacing.

Although on average the molecules are closer together they are not bound to each other the way they are in ice. They have broken their molecular bonds. They have higher potential energy when those bonds are broken.

The physics of water is rather complicated. Water is a polar molecule (one side is slightly positive and the other slightly negative). The bonds between molecules depend not only on the average distance between molecules but also on the orientation of the molecules. Bonds form between molecules when those negative and positive regions get close. But when the molecules are rotating and vibrating, the bonds cannot form even when separation decreases.

AM

nasu
Gold Member
This is not correct. Temperature measures only translational kinetic energy. Total kinetic energy depends on vibrational, rotational and translational energy.
AM
Not really related to the main discussion, but I don't know if you mean this for the phase transition or in general.
And what do you mean by it? In what sense "measures"? You mean that a temperature increase cannot affect the kinetic energy of the vibrations or rotations?
How about the temperature in a solid? What translational kinetic energy is measured by temperature?

Andrew Mason
Homework Helper
Not really related to the main discussion, but I don't know if you mean this for the phase transition or in general.
And what do you mean by it? In what sense "measures"? You mean that a temperature increase cannot affect the kinetic energy of the vibrations or rotations?
How about the temperature in a solid? What translational kinetic energy is measured by temperature?
Temperature is a statistical measurement of the energy of motion of the centre of mass of molecules relative to the centre of mass of the macroscopic object. Rotational energy of a molecule does not contribute to temperature. Vibration of a molecule about its centre of mass does not contribute to temperature. But intermolecular vibrational energy does contribute to temperature. In solids, this is the only kind of translational kinetic energy that exists.

AM

You haven't said if you have heard of entropy?

Heard of entropy but never understood it

Decreased intermolecular spacing does not imply a lower potential energy. It depends on the forces between the molecules, not the spacing.

Although on average the molecules are closer together they are not bound to each other the way they are in ice. They have broken their molecular bonds. They have higher potential energy when those bonds are broken.

The physics of water is rather complicated. Water is a polar molecule (one side is slightly positive and the other slightly negative). The bonds between molecules depend not only on the average distance between molecules but also on the orientation of the molecules. Bonds form between molecules when those negative and positive regions get close. But when the molecules are rotating and vibrating, the bonds cannot form even when separation decreases.

AM

Thank you, that makes so much more sense.
A further question: Does the contraction of an object necessarily mean that some work has been done by the atmosphere on the object?

Heard of entropy but never understood it
In post 1 you quoted the First Law.

Well there is another expression (in fact there are lots more)

$$\Delta$$U = T$$\Delta$$S - P$$\Delta$$V

T is temperature, S is entropy, the rest you already know.

You have already established that the P$$\Delta$$V is insignificant. So this expression shows where the heat input goes when a solid melts.

It goes into a large increase in entropy.

So what does this mean?

Well, the water molecule forms crystals held together by hydrogen bonds. The crystals have a regular tetrahedral symmetry, which means that every water molecule has a hydrogen bond to 4 surrounding water molecules arranged at the vertices of a tetrahedron around it.

The point is that this is a regular long range structure.
We say that it has long range order, because to align the water molecules like this requires each molecule to orient itself in a particular way.

When a solid changes to a liquid this regular array structure is lost. No only are the molecules no longer arranged in rows, but they have also rotated relative to each other. The molecules have not moved further apart, or in the case of water the rotation has allowed the molecules to actually close up a bit.
The closing up happens because unlike many molecules water is a very chunky three dimensional molecule, also tetraheral in shape.
In solid ice the molecules fit together vertex to vertex. Inputting some energy allows them to fit side to side or face to face or eventually just randomly aligned. Vertex to vertex is not a very efficient molecular packing so it should be no suprise that a small realignment increases the density.

Back to entropy.
One explanation of entropy is that it is a measure of the disorder of a system.
The more the disorder the higher the entropy

The solid is very well ordered, but as we allow disorder to creep in by melting we increase the entropy.

Sorry if this is rather rambling.

Andrew Mason
Homework Helper
A further question: Does the contraction of an object necessarily mean that some work has been done by the atmosphere on the object?
Work is done by the surroundings on the ice. But the work done is small compared to the heat of fusion. The collapse of the structure of ice when it turns to liquid water results in the density increasing even under low pressure.

Water is a very, very interesting molecule. It is not simple.

AM

Work is done by the surroundings on the ice. But the work done is small compared to the heat of fusion. The collapse of the structure of ice when it turns to liquid water results in the density increasing even under low pressure.

Water is a very, very interesting molecule. It is not simple.

AM

However, if you were to give it a (+, -, 0) , which sign would you give to the work done by the atmosphere on the melting ice?

In post 1 you quoted the First Law.

Well there is another expression (in fact there are lots more)

$$\Delta$$U = T$$\Delta$$S - P$$\Delta$$V

T is temperature, S is entropy, the rest you already know.

You have already established that the P$$\Delta$$V is insignificant. So this expression shows where the heat input goes when a solid melts.

It goes into a large increase in entropy.

So what does this mean?

Well, the water molecule forms crystals held together by hydrogen bonds. The crystals have a regular tetrahedral symmetry, which means that every water molecule has a hydrogen bond to 4 surrounding water molecules arranged at the vertices of a tetrahedron around it.

The point is that this is a regular long range structure.
We say that it has long range order, because to align the water molecules like this requires each molecule to orient itself in a particular way.

When a solid changes to a liquid this regular array structure is lost. No only are the molecules no longer arranged in rows, but they have also rotated relative to each other. The molecules have not moved further apart, or in the case of water the rotation has allowed the molecules to actually close up a bit.
The closing up happens because unlike many molecules water is a very chunky three dimensional molecule, also tetraheral in shape.
In solid ice the molecules fit together vertex to vertex. Inputting some energy allows them to fit side to side or face to face or eventually just randomly aligned. Vertex to vertex is not a very efficient molecular packing so it should be no suprise that a small realignment increases the density.

Back to entropy.
One explanation of entropy is that it is a measure of the disorder of a system.
The more the disorder the higher the entropy

The solid is very well ordered, but as we allow disorder to creep in by melting we increase the entropy.

Sorry if this is rather rambling.

You just inspired me to force my self into studying entropy!

Well Physics Forum is a good place to study Entropy, there is lots of discussion here as any search will reveal.

You should bear in mind that there are two routes to the matter.

The first and original route leads to equations such as the one I posted. This is a mathematical/mechanical approach.
Entropy was originally introduced to pair with temperature on what is known as an indicator diagram. See post#11 of this thread