Latent Heat and Kinetic Energy

Question

The kinetic theory says that temperature is the measure of the average kinetic energy. That would mean then: The temperature rises with the rise in kinetic energy. We know that when matter changes state 'latent heat' helps in 'breaking the bonds' at the transition phase and that this 'extra heat' is not noticed on a thermometer. But energy can neither be made nor destroyed. Taking that into consideration latent heat has to convert into some form of energy. Since temperature is not rising, it's not kinetic energy it's getting converted to. What energy does it get converted to?

Note: I already know why the temperature does not change. What I want to know is what the latent heat energy gets converted to.

Answer 1

Latent heat isn't heat; it is energy released or absorbed during a constant temperature phase change.

The kinetic theory of gases has little to say about phase changes. Phase changes, involve liquids or solids in general and most often don't involve gases.

It's an error of comprehension (or articulation) to claim a phase change releases latent heat. If I slowly accelerate a tank of gas, its kinetic energy increases with no increase in temperature. Temperature is a measure of kinetic energy, not the only measure of kinetic energy. (We can notice this from the fact that most solids at their melting point do not increase in temperature when heat energy is added to them.)

In a more general sense, latent energy is the energy difference between the two phases. So where it goes to, or comes from, is the conversion of one phase to the other. The assumption here is that one phase is lower in potential energy compared to the other. You may know that entropy is defined in terms of heat and temperature. In classical thermodynamics energy and entropy are two fundamental concepts, they are on equal footing. But in statistical thermodynamics, energy is fundamental, while entropy can be derived from the ensemble of possible states that a system can be in. Whoops! Too much!

In the case of ice melting, the intermolecular bonds (mostly hydrogen bonding) are 'broken' (which means there are a lot fewer of them and they are (on average) weaker). The energy that was in the form of potential energy of the bonds, now becomes "kinetic" energy as chemists understand the term in this context.

In this context (as contrasted with the kinetic energy of a bullet or a ballistic electron), the kinetic energy includes all of the energy of translation, rotation and vibration: the rotation of a molecule doesn't change its speed, nor do the movements of the two hydrogen atoms with respect to the oxygen atom - their bond to the central O atom can be looked at as a (coil) spring and they can stretch and bend that spring. Faster and greater spring movement occurs during melting. (The rotation of the H atoms also occurs but the energy required to spin a H atom about its bond is very small as bond energies go, so is generally neglected with little loss of accuracy; this wouldn't be true for bonds between larger groups, where rotation around the bond could require quite a bit of energy). So, in the isothermal melting of ice, the place where the energy is going is in these intramolecular movements. (intra = inside). And in the isothermal freezing of water, the place where the energy is coming from is these same intramolecular motions.

There is no one unique definition of kinetic energy (K.E.) and another for potential energy (P.E.). In a group of atoms or molecules, the 'potential' energy is 'internal' and kinetic is 'external' and yet if one focuses in on a single molecule, some of its potential energy is due to its movement (or movements of its parts), so what is and what is not kinetic energy or p potential energy may change with the context. In fact, the basic laws of energy conservation may be K.E. + P.E = constant or K.E. = P.E. (which is often expressed as K.E. - P.E. = 0. These two approaches are embodied in Hamiltonian and LaGrangian Mechanics which are the two related but complimentary ways modern physics is expressed. The LaGrangian way is L = T - V (where T is K.E. and V is P.E.) and the Hamiltonian way is H = T + V. These are used in both classical Physics and in quantum Physics.

Answer 2

Experimentally heat is absorbed to melt a solid or to vaporise a liquid and as the temperature remains constant at the melting or boiling temperature, the entropy must be increasing. If the entropy change from one phase to the other is $\Delta S_{1,2} = S_2 - S_1 $ and the heat absorbed (i.e. enthalpy or latent* heat) is positive $\Delta H_{1,2}$ in transforming phase 1 to phase 2, the entropy change is positive, $\Delta S_{1,2} = \Delta H_{1,2}/T$.

Any solid or liquid is held together by intermolecular potential energy, in general called van-der-Waals potentials (ion-dipole, dipole - dipole etc). Starting from a temperature just below the melting point as energy is added the molecules absorb this both as kinetic and potential energy (vibration and rotation quantum numbers increase) both internally (normal mode vibrations & rotations) and also as motion in the intermolecular potential. The potential interaction between molecules is unchanged as heat is added, but the molecules start to gain enough energy to overcome the well formed by the potential and that holds them in place.

There is now an increase in entropy as there are more energy levels that the molecules can occupy and so more ways of occupying them, i.e the number of possible configurations $\Omega$ (or complexions) increases. Entropy is proportional to the number of these as $S=k_B\ln(\Omega)$.

Another form of the increase in entropy is that of position, which is strictly fixed in a solid, but exchange of a molecule’s position occurs all the time in a liquid. Positional exchange largely accounts for the entropy increase in atomic solid-liquid transitions, $\approx 8~ \pu{J K^{-1}mol^{-1}}$ e.g. solid - liquid argon.

Eventually there is enough energy that molecules start to escape from the intermolecular potential and so the phase changes.

Generally the process of initiating a phase change, say, freezing relies on nucleation, then growth of the nucleus. If this does not occur supercooling or superheating is observed. In supercooling, which is common, water for example, can still be liquid many degrees below normal freezing point. Part of the effect of overcoming the intermolecular potential is to form a nucleation/seed site, say a small region of vapour inside a liquid or a region of crystal inside a liquid. Next this region will either grow and form some bigger region or shrink to nothing in size.

The nucleus/seed will not grow spontaneously due to the relatively large surface energy which is relatively larger when a nuclei is small, however, a fluctuation in the structure due to some random accumulation of thermal energy may increase its size and so it now becomes stable. The competition is between the negative free energy gain from the volume of the new phase 2 being lower than that of phase 1, and the loss in free energy due to the positive surface energy. As the radius of the nucleus/seed increases the volume term wins out and growth is now spontaneous.

Finally, the average translational kinetic energy of a molecule is $3k_BT/2$ irrespective of whether the molecule is in the gas, liquid or solid phase. In the liquid or the solid the motion is merely restricted to a narrower range around the minimum of the intermolecular potential, than it is in the gas phase. Thus the notion that it is the kinetic energy that molecules acquire is the reason for molecules to go into the vapour phase is incorrect.

• The term ‘latent heat’ of vaporisation or fusion is rarely used nowadays and instead enthalpy of vaporisation or fusion are used. ‘Fusion’ is an old -fashioned word for melting.