Theoretical limits on enthalpy of phase change

Question

As an Amazon Associate, I earn from qualifying purchases. For more practice problems like this, see Schaum’s Outline of College Physics, 12th Edition.

1. What the question is really asking

A phase change is a transformation of a material in which the temperature stays constant while a certain amount of energy per unit mass (the latent heat or enthalpy of transition) is absorbed or released.

The student wonders:

Is there a universal upper bound on the amount of energy that can be taken up per kilogram of material during any possible phase change – even exotic ones that we have never made?
If such a bound exists, what is it and where does it come from (e.g. the speed‑of‑light limit, the rest‑mass energy, etc.)?

In other words we have to find the maximum possible specific latent heat (L) (units J kg(^{-1})) that a system of mass (M) can absorb while staying at a single temperature, before the temperature must start to rise again.

2. Step‑by‑step reasoning

2.1 What “latent heat’’ really measures

For a transition between two equilibrium states A and B at the same temperature (T),

[ L \equiv \frac{Q_{\text{latent}}}{M} = \frac{\Delta H}{M} = \frac{U_{B}+pV_{B} - (U_{A}+pV_{A})}{M}, ]

where (U) is the internal energy, (p) the pressure and (V) the volume.
Because the temperature is the same on both sides, the change in internal energy is not due to a change in kinetic energy of the microscopic degrees of freedom (which would raise the temperature).
Instead it is the difference in binding energy (or, more generally, the difference in the rest‑mass energy of the constituents) between the two phases.

Hence the latent heat per unit mass is essentially the change in the rest‑energy per kilogram caused by the rearrangement of the constituents.

2.2 The ultimate relativistic ceiling

Special relativity tells us that any isolated system of rest mass (M) has a total energy

[ E_{\text{total}} = M c^{2}, ]

and no process can change the total rest‑mass energy by more than this amount without either creating or destroying mass.

If we insist that the total mass of the system stays the same during the phase change (the usual definition of a phase transition – the material does not disappear), then the maximum possible change in internal energy per kilogram is bounded by the total rest‑energy that could, in principle, be released:

[ \boxed{L_{\max}^{\;(\text{absolute})}=c^{2}\approx 9.0\times10^{16}\ {\rm J\;kg^{-1}} } . ]

This bound would be approached only if the transition could convert all of the rest‑mass of the material into other forms of energy (e.g. matter–antimatter annihilation). In that case the “phase’’ is really a reaction rather than a conventional phase change, but the bound still applies.

2.3 More realistic bound: binding‑energy limits

In ordinary condensed‑matter phases the particles are electrons and nuclei bound by electromagnetic forces.
The characteristic energy scale of an electromagnetic bond is the Rydberg energy ((\sim 13.6) eV per electron).
Per kilogram this gives

[ L_{\text{chem}} \sim \frac{13.6\ {\rm eV}\times N_A}{\text{kg of material}} \approx 10^{5}\text{–}10^{6}\ {\rm J\;kg^{-1}} . ]

So chemical phase changes (melting, vaporisation, etc.) are many orders of magnitude below the relativistic ceiling.

If we go to nuclear forces, the binding energy per nucleon is at most ≈ 9 MeV (the iron‑peak maximum).
A kilogram of nucleons contains

[ \frac{N_A}{1\ \text{g/mol}} \times 10^{3}\ {\rm g} = 6.02\times10^{26}\ \text{nucleons}. ]

Thus the largest nuclear latent heat that could be achieved by a transition between two nuclear phases (e.g. ordinary nuclear matter ↔ quark‑gluon plasma) is

[ L_{\text{nuc}} \lesssim 9\ {\rm MeV}\times 6.0\times10^{26} \approx 9\times10^{13}\ {\rm J\;kg^{-1}} . ]

That is roughly (10^{-3}\,c^{2}). It is still far below the absolute (c^{2}) limit, but it is the largest value we know how to obtain in a mass‑conserving phase transition.

2.4 Why we cannot exceed the binding‑energy limit

The binding energy of a bound system is the amount by which its rest mass is reduced relative to the sum of the rest masses of its constituents taken far apart.
If a phase transition changed the internal configuration, the change in binding energy cannot be larger than the total binding energy that the system already possesses, because:

Energy conservation – the extra energy must come from somewhere; the only reservoir is the rest‑mass of the constituents.
Stability – if a configuration could release more than its total binding energy, the system would be unstable and would spontaneously disintegrate (i.e. it would not be a distinct phase).

Consequently, for any stable phase transition the latent heat per mass is bounded by the largest binding‑energy fraction that a material can have. The strongest known binding is nuclear, ≈ 10 % of the rest‑mass energy, giving the ( \sim 10^{14}\ \text{J kg}^{-1}) figure above.

2.5 Exotic possibilities

Exotic “phase’’	What changes	Approx. latent‑heat per kg
Matter → Antimatter annihilation (mass not conserved)	Rest‑mass converted to photons	( \approx c^{2}=9\times10^{16}\ \text{J kg}^{-1})
Neutron‑star → Black‑hole (gravitational binding)	Gravitational binding energy ≈ 0.2 (c^{2}) released	(\sim 2\times10^{16}\ \text{J kg}^{-1}) (theoretically)
Quark–gluon plasma ↔ Hadronic matter (QCD deconfinement)	QCD binding (≈ 100 MeV per baryon)	(\sim 10^{15}\ \text{J kg}^{-1}) (still < (c^{2}))
Chemical melting / vaporisation	Electromagnetic bonds	(10^{5}\text{–}10^{6}\ \text{J kg}^{-1})

Only when mass is allowed to change (annihilation, collapse to a black hole) can we approach the absolute (c^{2}) ceiling. As long as the system retains the same rest mass, the practical universal bound is set by the strongest known binding, i.e. nuclear, giving

[ \boxed{L_{\text{practical}}\;\lesssim\;10^{14}\ {\rm J\;kg^{-1}} } . ]

3. Final answer

Absolute theoretical limit (mass may change):

[ L_{\max}=c^{2}=9.0\times10^{16}\ {\rm J\;kg^{-1}} . ]

Maximum realistic latent heat for a mass‑conserving phase transition (the strongest binding we know, nuclear):

[ L_{\text{realistic}}\lesssim 10^{14}\ {\rm J\;kg^{-1}}\;( \sim 0.001\,c^{2}) . ]

All ordinary chemical and condensed‑matter phase changes are many orders of magnitude smaller ((10^{5}\text{–}10^{6}\ \text{J kg}^{-1})).

Thus, the only universal bound is the rest‑mass energy (c^{2}); practical limits are set by the strongest binding energy available, which is nuclear and gives a specific latent heat of order (10^{14}\ \text{J kg}^{-1}).

4. Common mistakes

Mistake	Why it’s wrong	How to avoid it
Treating latent heat as a temperature‑independent constant for all materials.	Different phases have different microscopic binding; the latent heat can vary by many orders of magnitude.	Always relate (L) to the difference in internal (binding) energy between the two phases.
Assuming the speed‑of‑sound limit (v_s\le c) directly caps latent heat.	The speed of sound limits how fast a pressure disturbance propagates, not how much energy can be stored in a phase.	Use energy‑conservation arguments (rest‑mass, binding energy) instead of sound‑speed arguments.
Confusing a chemical reaction with a phase change.	A reaction can change composition and mass; a phase change must keep composition (and usually mass) fixed.	Verify that the process being considered conserves the total number of particles (or at least the total rest mass).
Thinking that any “exotic’’ phase could arbitrarily increase (L).	Even exotic QCD or gravitational phases are limited by the fraction of rest‑mass that can be bound (≈ 10 % for nuclear, ≈ 20 % for gravitational).	Compare the binding energy fraction of the proposed phase with the known limits (nuclear, gravitational).
Neglecting relativity when estimating large latent heats.	At energies (>10^{13}) J kg(^{-1}) the particles become relativistic; non‑relativistic formulas break down.	Use relativistic energy–mass relation (E=mc^{2}) when the latent heat approaches (\sim10^{14}) J kg(^{-1}).

Keeping these points in mind prevents over‑optimistic estimates and clarifies why the ultimate ceiling is set by the speed of light through the rest‑mass energy of the material.

Original question: Theoretical limits on enthalpy of phase change on Physics Stack Exchange, licensed CC BY-SA.