Could you please supply an example of the second law of thermodynamics not applying. I have never seen one.

Note that the vaporization heat calculated by scientists using the second law of thermodynamics (Clapeyron equation) is labeled as an "experimental value", which conforms to the rules of using the second law of thermodynamics. However, subsequent experiments have shown that these calculated values do not match the experiment, indicating that the more advanced the experimental technology, the more inaccurate the prediction of the second law of thermodynamics becomes.

First of all it has to be stated that the formula above is not the second law of thermodynamics, it is just the consequence of it. To add, it is obtained at some rude approximations, namely, assuming that everything goes from equilibrium to equilibrium state. This is entirely wrong for any practical system. On the contrary, the essence of 2nd law is pretty simple - when two bodies with diffrent temperatures are put in contact the heat flow will start from the one that is hotter to the one that is colder. If you observe in any correct experiment the opposite, immediately publish in Nature, Nobel prize will be guaranteed. Of course, you may face some problems with the authorities (as many scientists in the past), but some times afterwards scientific community will recognize your astonishing achievment, because nobody can argue against correctly made experiment.

Being serious I suggest you reading modern papers on thermodynamics that are much more rigorous and postulate up to 16 independent laws (see doi.org/10.1016/S0370-1573(98)00082-9), so that to correctly derive every mathematical expression without the omissions and flaws of the classical picture.

Deviation rate of the second law of thermodynamics: 20-30%, packaged by scientists and diluted to 0.1-0.3%, meeting the requirements for publishing papers

Please refer to the attached diagram and instructions for details

Notes:

• The actual calculation of the second law of thermodynamics: E1=f(P1) or P1=g(E1)
• δ1
  0 votes 0 thanks

First of all, it has to be stated that the relative error of vaporization heat computed with the Clayperon equation again does not imply the 2nd law is wrong. These data are measured in non-equilibrium regime and obviously cannot fully satisfy the criteria of equilibrium thermodynamics. The author of this discussion is maybe familiar with the fact that all practical calculations involving heat and mass transfer in industrial applications always account for the heat loss that is out of the scope of equilibrium thermodynamics.

Second, it is strange to hear that the error which equals 20-30 % is significant. There are multiple areas of physics where the error is much-much worse and still this is acceptable as nobody can offer any reasonable alternative. Considering the relative mathematical simplicity of equations presented the error of 30 % can be seen as a miracle, that guides us all to the happy future.

Third, I cannot understand the claims about the 2nd law being written in the "plot theory" fashion. The problematics of 2nd law's inconsistency has been studied for decades: there exist dozens of claims 2nd law is not working well and people organize conferences on that topic. I strongly recommend to read this book: "Challenges to the Second Law of Thermodynamics Theory and Experiment" by Vladislav Cápek and Daniel P. Sheehan, which reviews a vast amount of the related literature.

Finally, my main argument always relies on the principle of balance between simplicity and efficiency: if some theory allows to explain (at least on a qualitative level) the certain set of data and, moreover, to make some predictions, being at the same time relatively easy to derive, it can be considered valid. The claims like "The inaccuracy of the second law of thermodynamics is revealed" don't make it entirely wrong. If the arguments raised above are true and the error is indeed 30 % one should be really happy with that: such simple theory producing so low error for so complicated systems is an astonishing achievement (maybe only classical mechanics is better in the respective area of applicability). Real science always deals with external corrections, explicit or implicit, but that's the reasonable price. Still, we're living in the age of arising AI which, perhaps, can propose more efficient algorithms that will be able to handle all the inaccuracies mentioned and provide more precise model. But until than it is highly unrealistic something else will be derived that can outperform 2nd law of thermodynamics in terms of the simplicity/efficiency ratio. Too many data should be reconsidered than and, also, too many organizational efforts are to be put in to prove the new model is "better" than the old one.

Thank you for recommending this book. I completely agree with the arguments presented in the book. The experiments in the book are relatively 'peripheral' (not very common). But can a basic physics theory be correct if it is wrong at point A? The basic physics theory is a winner takes all: either all are right or all are wrong. From this perspective, this book lists so many errors. In other places, the correctness of the second law of thermodynamics is just an illusion. I just want to tell everyone that the E and P of water vapor do not satisfy the second law of thermodynamics. This is being examined from the perspective of natural science on the primitive ground of the second law of thermodynamics. This kind of examination is neutral and objective.

Maybe I disappoint someone, but EVERY physical theory is wrong somewhere, this is not a discovery. Such kind of things are typical for any sort of knowledge that tries to make things systematic. Most times the inconsistencies are hidden so that to make an impression everything is fine. Still, this is, as I said in previous comments, a reasonable fee for the simplicity provided. If you make an equation that will be able to accurately reproduce water vaporization properties without extra parameters required, but will be computationally inefficient, why using it? We work with those concepts that can be handled and understood. In this sense, new theories are extremely hard to be imagined.

To make this discussion useful, I suggest you to ask yourself, how the flaw you've noticed can be improved without radical reformulation of thermodynamics (this is fairly possible, I think). The disagreement you refer to sounds not too high to be crucial (at least, the slope is predicted, right?), hence it can be corrected. If I'm correct this is usually done by Antoine's equation. Accordingly, maybe another possibility can be constructed.

Notes:

A： Figure 1 is from the authoritative book "The Properties of Gases and Liquids", and the enthalpy change H1 is calculated from the equation of state. From Figure 1A, it can be seen that the accuracy of the state equation is high, but the deviation rate of enthalpy change derived is large. The theoretical basis for the derivation is the differential equation of the second law of thermodynamics.

B， Plot the deviation rate of the state equation (

I cannot understand the purpose of last figure. It states that the real EOS is better than the ideal gas? I mean, those enthalpies derived in the scope of Peng-Robinson, RK and others are obtained by similar relation between internal energy and volume (which is claimed as "wrong"). The procedure is much more complicated (as one needs to deal with fugacities and numerical integration), but in the end the dependence of pressure on temperature is established (the enthalpy change is hidden within but cannot be straight-forwardly extracted). Comparing this with simple equation that originates from ideal gas approxiamation does not imply the 2nd law is wrong here. Again, all calculations involving EOS rely on the general E=f(V) dependence that is derived because of the 2nd law. So, I cannot consider the discrepance mentioned above as a "proof" 2nd law is incorrect.

Strictly speaking, there is a direct test for 2nd law which is hidden in the 2nd derivatives of entropy: the heat capacity of any fluid presented at phase equilibrium in thermodynamically closed system at const T,P should be higher than 0, as well as its isothermal compressibility should be higher than 0 (i.e. under pressure increase, volume of any fluid should decrease). The disproval of these will indeed indicate 2nd law is incorrect (at least partially). Otherwise, everything seems to be fine.

IAPS84: High precision equation of state for water vapor, with a deviation rate of 15-20% for specific heat and sound speed obtained according to the second law of thermodynamics

Notes:（See image for details） 1. Deviation rate between water vapor thermal properties table IAPS84 and experiment 2. The accuracy of the state equation is very high, with specific heat and sound velocity obtained according to the second law of thermodynamics (Formula 1) ranging from 15% to 20%. 3. In Formula 1, "dP/dT * T" is the contribution of the second law of thermodynamics, which separates the "dP/dT * T" in sound velocity and specific heat and estimates a deviation rate of 40-50% through experimental comparison. 4. The water vapor thermal properties table IAPWS97 has made great progress, which is due to the combination of data.

I think you refer to quite old-fashioned data. For instance, in the reformulation of water EoS made in 1995 (i.e. producing IAWPS-95) the precision of the resulting equation improved drastically (see full report at [Wagner, W., & Pruß, A. (2002). The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of physical and chemical reference data, 31(2), 387-535.]).

For instance, here are the screens for speed of sound and heat capacity (I had to add also, that unlike speed of sound, heat capacity is much related to caloric equation of state which is not given by 2nd law; rather, it originates from statistical mechanics or is purely empirical). As seen, the accuracy is much better than 15-20 %.

IAPS84: High precision equation of state for water vapor, with a deviation rate of 15-20% for specific heat and sound speed obtained according to the second law of thermodynamics

Notes:（See image for details）

1. Deviation rate between water vapor thermal properties table IAPS84 and experiment

2. The accuracy of the state equation is very high, with specific heat and sound velocity obtained according to the second law of thermodynamics (Formula 1) ranging from 15% to 20%.

3. In Formula 1, "dP/dT * T" is the contribution of the second law of thermodynamics, which separates the "dP/dT * T" in sound velocity and specific heat and estimates a deviation rate of 40-50% through experimental comparison.

4. The water vapor thermal properties table IAPWS97 has made great progress, which is due to the combination of data.

Can you prove your point 4 (The water vapor thermal properties table IAPWS97 has made great progress, which is due to the combination of data)?

The problem is that you don't understand how to prove the correctness of the second law of thermodynamics.

dE/dV=(dp/dT)*T-P（1） The contribution of the second law of thermodynamics is

dq={(dp/dT)*T}dV（2）

The measurement of sound speed and specific heat includes (2), which requires separating the contribution of the second law of thermodynamics in sound speed and specific heat (formula 2). Compare theory and experiment.

IAPWS97 IAPS84 pieced together E and P using formula (1),

Physics requires theoretical predictions and experimental comparisons. See the picture for details.

From 1950 to 1970, the thermal properties of water did not meet the prediction of the second law of thermodynamics. Scientists have already discovered it. Scientists dare not say that the second law of thermodynamics is wrong, they only say that basic theories need to be developed. Kick the ball to someone else.

From 1980 to 1995, with the development of computer technology, Wagner came up with data piecing together.

so no enthalpic losses ? Carnot engine efficiency as maximum is wrong? wow -how? your math has to conform to reality and if not, your equations are wrong.

dEdV=(dPdT)T−P\frac{dE}{dV} = \left( \frac{dP}{dT} \right) T - P

This equation is not a general statement of the second law. It appears to be derived from a thermodynamic identity involving partial derivatives, possibly from a Maxwell relation or a TdS equation. But it’s not the second law itself.

• The second law is more fundamentally expressed as:

dS≥dQTdS \geq \frac{dQ}{T}

for any real process, with equality for reversible ones.

• Your formula might be valid under specific assumptions (e.g. constant entropy or ideal gas behavior), but it’s not universally applicable. If experimental data diverges from it, that doesn’t invalidate the second law—it suggests the formula’s assumptions don’t hold in that regime.

🧊 Critical Points, Hydrogen Bonds, and Fluctuations

You mentioned frustration with “technical justifications” like fluctuations near critical points or hydrogen bonding. But these aren’t excuses—they’re real physical phenomena that affect measurements:

• Near the critical point, thermodynamic quantities like heat capacity and compressibility diverge. This makes precise measurements tricky, but not meaningless.
• Hydrogen bonding in water, for example, leads to anomalous behavior in entropy and heat capacity. Ignoring it would be scientifically irresponsible.

The formula dEdV = (dPdT)T−P\frac{dE}{dV} = \left( \frac{dP}{dT} \right) T - P does not align with experimental results. If this is the case, Carnot's efficiency would no longer be equal to 1-T1/T2, and the second law of thermodynamics would be incorrect. This is the fundamental knowledge of the second law of thermodynamics. I do not wish to see any strange explanations that deviate from basic understanding.

As I said before, each theory is not designed to be perfect; the error of equation discussed is in quite good range considrening the simplicities assumed. Still, the last assumption seems to be totally wrong. Carnot's efficiency has nothing to do with the formula you discuss. Therefore, stating dEdV = (dPdT)T−P\frac{dE}{dV} = \left( \frac{dP}{dT} \right) T - P is wrong does not mean "Carnot's efficiency would no longer be equal to 1-T1/T2". These are different things.

I have a headache. I need a lot of time to correct someone's misunderstandings about the book.

dEdV = (dPdT)T−P\frac{dE}{dV} = \left( \frac{dP}{dT} \right) T - P =

Carnot's efficiency=1-T1/T2

Well, but Carnot's formula can be derived without all of this. In fact, it is extremely straight derivation. And the (dE/dV) term goes from further approximations.

The formula DE/dV = (dP/dT) * T - P determines the existence of thermodynamic entropy and is related to the validity of the second law of thermodynamics. Like the first law of thermodynamics (dE = dq - dW), it is subject to experimental verification. The accuracy of DE/dV = (dP/dT) * T - P needs to reach the level of dE = dq - dW for the second law of thermodynamics to hold. I am very tired of all kinds of excuses.