Sorry Yannick I meant "is related to" not "equals"!

Could you please explain more, what is it that you want to know?

Are you familiar with how entropy is related to the number of available states (S=k*Ln(omega)) ?

Hi Mahdi, it was a general thermodynamics lecture but in my own studies I'm interested in applying it to [protein] biopolymers, which can be treated as thermodynamic systems (albeit not necessarily ones at thermodynamic equilibrium). There are types of proteins called "intrinsically disordered proteins" hence my interest.

I'm cracking open some thermo textbooks now, ω was shown on a graph against N to explain micro/macrostates, I asked a researcher here and he told me that "we don't tend to look at volume changes in biology", but we do look at thermodynamics...

Like I say, I'm interested in understanding what this dV term means in the context of physical chemistry/biophysics - probably not gaseous.

To my understanding this (first law of thermodynamics) should apply to any thermodynamic system though, so not having the ideal gas law shouldn't be an issue and the volume change should still have a meaning

→ http://physics.stackexchange.com/questions/88358/

Hi Louis, well, in that case you really need to study statistical thermodynamics. in our field of study we don't deal with volume change directly but instead we relate the change in the statistical end-to-end distance of a macromolecule to the force/free energy of the molecule. see for example: http://en.wikipedia.org/wiki/Rubber_elasticity

proteins act almost the same way as synthetic polymers. Thermodynamic laws should be applied specifically for each system of study in this level. Each assumption should be validated by experiments or each system, and you might finally need to study non-equilibrium thermodynamics. proteins and synthetic polymers are analogous in many ways. The equations developed describing polymers, in most cases can be applied to proteins as well.

Anyway, so far I haven't seen a direct use for the equation you mentioned in the field of polymers, but I am sure if you search for it, you might find some.

Here are some keywords that might help: Polymer dynamics/physics+proteins, conformation of proteins/polymers, reptation dynamics of proteins/polymers, theory of rubber elasticity,..

From what I can work out the equation only holds if pV = a constant so it depends on the equation of state.. pV = only constant for ideal gas and as before I'm unsure if this specific concept can be extended to polymers where intermolecular correlations play a role unlike those in gases.. I'm sure end-to-end distance works as a proxy for length, thus provides the volume for a polymer of a single monomer, but with multiple monomers for biopolymers such as proteins, the "volume" is a function of the amino acid composition (whose structures vary at the R group)... I'm still uncertain for now.

I'll take a look at those things, from what I can see they're covered in the textbooks I have. I also found some quite dense polymer dynamics notes at http://cbp.tnw.utwente.nl/PolymeerDictaat/ which I'll look through. Reptation seems cool too, cheers Mahdi

Hi Louis, yes the volume of each monomer could be different and still you can use the specified equations using the concept of coarse graining, which means lumping the specific data about a monomer to an equivalent monomer for which the statistical laws hold. This equivalent monomer is also called Kuhn monomer and is widely used in polymer physics.

Of course using coarse graining, we might lose some details about interactions in atomic level but it still gives some useful information about the behavior of the macromolecules.

Molecular dynamics simulation is a very useful tool for studying macromolecules by taking into account, more details concerning the mentioned interactions, but it also has its own limits.

The more you study it (polymer physics) the more you'll become interested.

Cheers. :-)

One of the problems with understanding thermodynamics is that most books use the ideal gas as a model system, so in the end students often think everything is an ideal gas.

For gases combination of the first and second law gives the fundamental relation:

dU = TdS-pdV

the second term refers to the work done on the system by changing its volume. For condensed systems this term is almost always completely irrelevant, unless one goes to extremely high pressures. The above relation can be inverted, and written as

dS = (1/T)dU + (p/T) dV

Only for ideal gases does U not depend on the volume, just on T, so we can write

dS = (C_V/T)dT + (p/T) dV

where C_V is the specific heat at constant volume (3/2 nR for an atomic gas)

Only for ideal gases p = nRT/V so that:

dS = (C_V/T)dT + nRdV/V

If the temperature is kept constant dT = 0 and we get

dS = nRdV/V

the desired result.

This is thus only valid for ideal classical particle gases, and only when the temperature is kept constant.

To give another example: consider the photon gas (relevant in black body radiation). For such a gas U = bT^4V and p = U/3. b is a constant. Performing the same calculation as above I get

dS = 4bT^2V dT + (4/3) bT^3 V dV

so for constant T the result is completely different:

dS = (4/3) bT^3 V dV

Doing this type of calculation for other systems will show you that dS proportional to dV/V really is (almost?) never valid. If you want to give it another try, use the van der Waals gas.

One final remark: S is in fact a dimensionless quantity, basically the number of microscopic realizations of a macroscopic system. Both J and K are energies, but on different scales, so J/K is just a constant.. Boltzmann's constant is a numerical factor that connects the two energy scales.

Sorry, tiny mistake: for the photon gas p= U/3V, so that for constant T:

dS = (4/3) bT^3 dV

This does not change my argument though.

The answer of Gert is correct and complete. However, chemists use a funny notation for non-ideal systems (mostly mixtures) where the notion of fugacity introduced by G. N. Lewis in 1908 is applied. The quantity has been renamed as (relative) "activity", but for gas mixtures, the term fugacity is usually retained. According to this notion, the ideal gas law is formally retained for real gases, with the introduction of a "fugacity coefficient" that one should use to multiply pressure with, and the law is thus extended for real gases and their mixtures. Thus, we can write formally at constant T:

dS = ((phi p)/T) dV

Putting the ideal gas pressure p = nRT/V into this, we get dS = (phi nR) dV / V .

Though this equation holds, we should keep in mind that the fugacity coefficient phi is a state function, i. e. the function of temperature and pressure. Thus, as it has been stated in the original question, dS is RELATED TO dV/V, but is not proportional to it.

Coming to the protein solutions, we can write similar equations for the partial molar entropy, and the partial molar volume. Thus, the volume meant is the partial molar volume of the protein (or any other chemical component) in the particular solution within the cell, and dS_i/dV_i is proportional to it as long as the fugacity coefficient does not change much. (In this case, the fugacity coefficient also refers only to the relevant component.)

The same reasoning also holds if we consider other types of relative activities, related to different reference states, with the exception that the dependence of another activity coefficient on p and T is necessarily different from that of the fugacity coefficient.

I agree colpetely with Louis Madox

Entropy is best understood as a historical concept that has developed and that has acquired different meaning in different scientific disciplines and in engineering (even in cancer research and in psychotherapy). Although science in theory is unified, in practice it has become fragmented. In order to work with entropy one should be aware of these different meanings.

One can relate these meanings to the ideas formulated by different researchers: Entropy started with Clausius. It was further developed by Gibbs and Boltzmann. Engineering, macrophysics, statistical mechanics, physics, physical chemistry all make use of the concept. it has further evolved in computer science. Exergy is also an interesting extension. Related info is easily found on the Internet.

The picture is quite complex, if not chaotic. One easily gets involved in arguments with people that know only a small part of the picture, and think that that small part is or should be applicable to other scientific disciplines. For instance, a physicist who makes use of the definition of entropy made in statistical mechanics, can erroneously claim or assume that the same definition applies to the entropy of biomolecules in water (whereas water itsekf has not yet been succesfully modeled by statistical mechanics).

For a recent discussion of the fallacy of relating disorder to entropy see Lambert.

Again: if you want to understood entropy, follow its historical development. It is elucidating how the concept, which in itself makes much sense, acquired during history a different meaning in different disciplines.

Unfortunately, one often cannot talk about entropy in science in a detached manner; Many pretend that they got it, but it is fake, and they get angry when challenged. It is easier for those in power, who are always too busy with other projects, to ridicule someone who has a different opinion than to do one's homework and sort arguments out. It is worthwhile to cite what Joe Jackson said on Lavoisier:

"Naively placing his faith in reason, he could never see why, for those who glorify power, reason is hated and feared."

So if you got it, it may be safer to keep it for yourself.

What Anthonie writes is basically O.K. Frank Lambert's webpage (http://entropysite.oxy.edu) also describes fallacies associated to the notion of entropy. However, the cited textbooks on this webpae are neither physics or math books, not surprisingly.

The notion of entropy - though its origin is really in phenomenological thermodynamics - is of STATISTICAL nature. Its definition is also very simple. It is the expectation value of the logarithm of the probability density function, munltiplied by a suitable scaling constant. Scientists who are well trained in maths and/or physics do understand this, and do not have problems to apply the concept in any branches of science. But those who do not understand this statistical definition, often mystify entropy and misuse it.

There remains always a problem: WHAT DISTRIBUTION should be taken to compute the entropy. Reasearchers who understand the field where the problem emerges and the definition of entropy as well, do know the right answer. (Or, to cite the ancient latin saying: sapienti sat.)

@Keszei

Sorry, I disagree. "Phenomenological entropy" applies to all macroscopic systems; according to the second law it would be universally applicable. Not only to gases, but also liquids and whatever composed complex objects.

"Statistical mechanics entropy" can only apply to those systems for which a distribution function can be formulated: mostly systems resembling a gas or a solid. It is certainly NOT universally applicable.

What statistical mechanics shows is that a few cases of phenomenological entropy can be explained as statistical entropy. But statistical mechanics goes only as far as simple solids and liquid noble gases (if I remember it well). Entropies of liquid water and simple molecules cannot be calculated by statistical mechanics.

Moreover, in order to calculate the scaling constant you mention, you need quantum mechanics.

Comparing phenomenological entropy and statistical mechanics entropy therefore resembles comparing apples are oranges: one can compare them, but they are basically distinct. The entropy considered by Shannon is different as well.

'Scientists who are well trained in maths and/or physics do understand this': if such 'well trained' scientists think statistical mechanical entropy is universally applicable, their training is not good enough ;)

Note that the phenomena of the hydrophobic effect and enthalpy-entropy compensation show the importance of entropy for biosystems.

Forgot to mention that the Sackur-Tetrode equation for an ideal gas contains Plank's constant: it depends on quantum mechanics, and is therefore certainly not simple or intuitive, such as the Second Law in phenomenological thermodynamics.

"But statistical mechanics goes only as far as simple solids and liquid noble gases (if I remember it well)."

Well, statistical mechanics (or statistical physics in a broader sense) can be applied to ALL substances, even water or any biologically important substances, whatever complicated they are. The only problem is the CACULABILITY of the relevant distribution functions, or mor precisely, the partition function. (It is the norm of the emerging distribution function for a given ensemble.) Nevertheless, it can be calculated using computer simulations AND some model of the molecular description. Thus, it is model-dependent, but it does not mean ANY theoretical problems but practical calculability only.

Statistical physics has shown that the "second law" entropy IS EQUIVALENT at a molecular level to the expectation value of the logarithm of the probability distribution function in the actual ensemble. Shannon's information theory entropy is exacly the same. The only difference between the two is that the scaling factor in thermodynamics is to arrange for the usual unit (J / K) for one molecule, while in information theory, it arranges for the unit of information in bits. That's why the scaling factor in thermodynamics is Planck's constant, and that in info theory is ln 2.

By the way; to get Planck's constant in the thermodynamic expressions does not follow from quantum mechanics, but from the conversion of units. Planck's constant is simply the universal gas constant (R) divided by the number of particles in one mol substance.

Things are not that much complicated as they seem to be if we understand basic principles underlying areas in several different branches of science.

Sorry, but I have to admit that I mistakenly wrote at two occasions Planck's constant instead of Boltzmann's constant.

To be correct, the scaling factor in thermodynamics to arrange for the usual unit (J / K) for one molecule is the BOLTZMANN constant k MULTIPLIED BY MINUS 1, while in information theory, to arranges for the unit of information in bits, it is -ln 2.

Taken this into account, we can see that the correct interpretation in statiscical physics does not need to consider quantum mechanics. However, quantum-statistical mechanics is a more convenient way to understand the underlying principles of statistical physics.

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity