{"id":1079,"date":"2018-07-03T09:48:32","date_gmt":"2018-07-03T09:48:32","guid":{"rendered":"http:\/\/crediblehulk.org\/?p=1079"},"modified":"2018-07-03T10:32:09","modified_gmt":"2018-07-03T10:32:09","slug":"state-functions-entropy-path-dependence-and-energy-conservation-in-thermodynamic-systems","status":"publish","type":"post","link":"https:\/\/www.crediblehulk.org\/index.php\/2018\/07\/03\/state-functions-entropy-path-dependence-and-energy-conservation-in-thermodynamic-systems\/","title":{"rendered":"State Functions, Entropy, Path Dependence, and Energy Conservation in Thermodynamic Systems"},"content":{"rendered":"

State Functions vs Path Dependent Functions<\/strong><\/h3>\n

In thermodynamics, scientists distinguish between what are called state functions vs path functions<\/a><\/em>. State functions are properties of a system whose values do not depend on how they were arrived at from a prior state of the system. They depend only on the starting and ending states of the system. On the other hand, path functions can take on very different values depending on the path by which the system arrived at a state from its previous state.<\/p>\n

For example, there are probably a few different ways you could get from your house to your friend\u2019s house across town, all of which would result in the same displacement between your home and your friend\u2019s. However, some of those paths would be less direct than others, and would therefore require you to travel a greater path length (and spend more on gas if you\u2019re driving).<\/p>\n

The displacement is therefore a state function, whereas the path length (and gas required) depends on the path taken. Examples of thermodynamic state functions include temperature, pressure, internal energy, density, entropy, and enthalpy. Examples of path dependent thermodynamic variables include heat and work.<\/p>\n

System vs surroundings and the Laws of Thermodynamics<\/strong><\/h3>\n

One of the fundamental concepts of thermodynamics is the idea of dividing the universe into a system + its surroundings. In this way, scientists and engineers can choose to define the boundaries between the environment and the system they are analyzing in a way that makes it convenient to apply the 1st<\/sup> and 2nd<\/sup> Laws of Thermodynamics to them both.<\/p>\n

The 1st<\/sup> Law: Conservation of Energy<\/strong><\/h3>\n

The 1st<\/sup> Law<\/a> is just the law of conservation of energy, which states that the total energy of an isolated system is constant. It is often conveyed by the statement that energy can be neither created nor destroyed in an isolated system, but rather can only change from one form of energy to another.<\/p>\n

\"Image<\/a>

Image source:<\/a><\/em><\/p><\/div>\n

This means that changes in the internal energy of a thermodynamic system are compensated for by the surroundings such that the total energy of the universe is conserved. An increase in a system\u2019s internal energy requires an input of heat and\/or work from<\/em> the surrounding environment, whereas a decrease requires a release of heat from<\/em> the system to<\/em> the environment, and\/or work done by<\/em> the system on<\/em> the environment.<\/p>\n

*As an interesting (albeit admittedly tangential) side-note, energy-momentum conservation is not so well-defined<\/a> for certain cosmological applications of general relativity. See footnote at the end of this article.<\/em><\/p>\n

Entropy and the 2nd<\/sup> Law<\/strong><\/h3>\n

The 2nd<\/sup> Law<\/a> is a little subtler, and can be stated and explained in different (albeit ultimately equivalent<\/a>) ways, either in terms of classical or statistical thermodynamics.<\/p>\n

For example, the Clausius statement<\/a> of the 2nd<\/sup> Law is that<\/p>\n

\n

“No process is possible whose\u00a0sole<\/span><\/em>\u00a0result is the transfer of heat from a cooler to a hotter body,”<\/p>\n<\/blockquote>\n

whereas the Plank-Kelvin formulation<\/a> of the 2nd<\/sup> Law is stated as follows:<\/p>\n

\n

“No process is possible whose\u00a0sole<\/span>\u00a0result is the absorption of heat from a reservoir and the conversion of this heat into work.”<\/em><\/p>\n<\/blockquote>\n

For the purposes of this section, however, the 2nd<\/sup> Law can be thought of as stating that the total disorder of the universe (entropy<\/a>) tends to increase over time. Energy is still conserved, but over time, less of it is available to do work. So, if the entropy of a system decreases (becomes more ordered), then that will be compensated for by an increase in the entropy (or disorder) of the surroundings, and vice versa.<\/p>\n

Note that although the loose definition of entropy as a measure of disorder suffices for the purposes of this discussion, it is not strictly correct. In classical thermodynamics, the details of a system\u2019s constituents are not considered, but are instead described by their macroscopically averaged behavior. Definitions of the properties of thermodynamic systems were also presumed to be in equilibrium, and were only much later extended to non-equilibrium situations. From this view, entropy represents the amount of energy per unit temperature of a system that is unavailable to do useful work.<\/p>\n

In statistical mechanics, the entropy of a system is defined in terms of the statistics of the motions of its constituent particles. Namely, for a given macrostate of a system, its entropy is given by kB<\/sub>ln(\u03a9), where kB<\/sub> is called the Boltzmann constant, which is equal to 1.38*10-23<\/sup> Joules per Kelvin in SI units, and \u03a9 represents the number of possible microstates<\/a> of the particles that together would comprise the same macrostate<\/a> \u2013 a concept which relies on the indistinguishability of certain configurations of identical particles. This has since even been extended to incorporate quantum statistics.<\/p>\n

Another description of entropy I like, one which is particularly applicable to information theory, was one I stumbled across in an article by Johannes Koelman (the Hammock Physicist<\/a>). Johannes stated it thusly<\/a>:<\/p>\n

\n

“The entropy of a physical system is the minimum number of bits you need to fully describe the detailed state of the system”<\/em><\/p>\n<\/blockquote>\n

If all of that seems too confusing, don\u2019t worry about it. Trust me. You\u2019re not alone. Instead, just think of entropy as a measurement of randomness or disorder for now. For more on entropy and the 2nd<\/sup> Law, see here<\/a>.<\/p>\n

\"\"<\/a>

Lord Entropy: an anthropomorphized comic personification of the Entropy concept (just because this awesome song<\/a><\/em> about him by my friends Tombstone da Deadman and Greydon Square wouldn\u2019t stop replaying in my head while I was writing that last section).\ud83d\ude0e<\/span>\u00a0<\/em><\/p><\/div>\n

I intend to refer back to these ideas in a separate article in which I plan to unpack the concepts of heat, work, and internal energy. The following section on conservation of energy in GR is not directly pertinent to that goal, but is just a tangential issue I thought was cool enough to mention for the benefit of any of my readers as fascinated as I am with the science of cosmology.<\/p>\n

*More on energy conservation in relativistic cosmology<\/em>: For example, the red shift of the Cosmic Microwave Background Radiation (CMB) due to universal expansion entails that the photons must lose energy over time, because longer wavelengths correspond to lower energies, and it\u2019s not clear whether (and\/or the sense in which) that energy might be compensated for as gravitational potential energy.<\/p>\n

Needless to say, adding dark energy into the mix doesn\u2019t exactly make this any clearer. This has led some physicists and cosmologists, including popular science communicator extraordinaire, Sean Carroll<\/a>, to say that the total energy of the universe isn\u2019t<\/em> conserved; \u201cit changes because spacetime does.\u201d<\/p>\n

The Law is sometimes reformulated in General Relativity as the statement that the covariant derivative of the stress-energy tensor is equal to zero <\/em>in all reference frames. But what this means insofar as how we ought to conceptualize physical reality is not clear, and is further confounded by differences in the transformation rules governing mathematical objects called tensors<\/em>, whose equations are valid in all possible coordinate systems (if they\u2019re valid in any), vs so-called pseudo-tensors<\/em>, which arise in certain models, but whose usefulness is contingent upon the scientist\u2019s choice of coordinate systems.<\/p>\n

To be clear, none of this has to do with any disagreement among physicists as to what is actually happening physically. Rather, it\u2019s a matter of how best to translate certain implications of GR into conceptually intelligible terms for people not intimately familiar with the mathematical details of the theory. To quote Sean Carroll once again,<\/p>\n

\n

\u201cAll of the experts agree on what\u2019s happening; this is an issue of translation, not of physics. And in my experience, saying \u201cthere\u2019s energy in the gravitational field, but it\u2019s negative, so it exactly cancels the energy you think is being gained in the matter fields\u201d does not actually increase anyone\u2019s understanding \u2014 it just quiets them down. Whereas if you say, \u201cin general relativity spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn\u2019t conserved,\u201d they might be surprised but I think most people do actually gain some understanding thereby.\u201d<\/em><\/p>\n<\/blockquote>\n

References <\/strong><\/p>\n

State vs. Path Functions<\/em>. (2018).\u00a0Chemistry LibreTexts<\/em>. Retrieved 2 July 2018, from https:\/\/goo.gl\/eoEQBd<\/a><\/p>\n

Feynman, R. P., Leighton, R. B., & Sands, M. (2011).\u00a0The Feynman lectures on physics, Vol. I: The new millennium edition: mainly mechanics, radiation, and heat<\/em>\u00a0(Vol. 1). Basic books.<\/p>\n

Weiss, M., & Baez, J. (2017). Is Energy Conserved in General Relativity?.<\/em> (2018). Math.ucr.edu. Retrieved 2 July 2018, from http:\/\/math.ucr.edu\/home\/baez\/physics\/Relativity\/GR\/energy_gr.html<\/a><\/p>\n

5.1 Concept and Statements of the Second Law.<\/em> (2018). Web.mit.edu. Retrieved 2 July 2018, from http:\/\/web.mit.edu\/16.unified\/www\/FALL\/thermodynamics\/notes\/node37.html<\/a><\/p>\n

An introduction to thermodynamics.<\/em> (2018). Google Books. Retrieved 2 July 2018, from https:\/\/books.google.com\/books?id=iYWiCXziWsEC&pg=PA213#v=onepage&q&f=false<\/a><\/p>\n

A Course in Classical Physics 2\u2014Fluids and Thermodynamics<\/em>. (2018). Google Books. Retrieved 2 July 2018, from https:\/\/goo.gl\/Y9A6Sv<\/a><\/p>\n

Equilibrium Thermodynamics. (2018).<\/em> Google Books. Retrieved 2 July 2018, from https:\/\/goo.gl\/qiq56n<\/a><\/p>\n

Entropy | Definition and Equation.<\/em> (2018). Encyclopedia Britannica. Retrieved 2 July 2018, from https:\/\/www.britannica.com\/science\/entropy-physics<\/a><\/p>\n

Microstates.<\/em> (2018). Chemistry LibreTexts. Retrieved 3 July 2018, from https:\/\/goo.gl\/VQNCkU<\/a><\/p>\n

Entropy?, W. (2018).\u00a0What Is Entropy?<\/em>.\u00a0Science 2.0<\/em>. Retrieved 3 July 2018, from http:\/\/www.science20.com\/hammock_physicist\/what_entropy-89730<\/a><\/p>\n

Entropy. (2013). Chemistry LibreTexts. Retrieved 3 July 2018, from https:\/\/goo.gl\/ZTJ6WR<\/a><\/p>\n

Energy Is Not Conserved<\/em>. (2010).\u00a0Sean Carroll<\/em>. Retrieved 3 July 2018, from http:\/\/www.preposterousuniverse.com\/blog\/2010\/02\/22\/energy-is-not-conserved\/<\/a><\/strong><\/p>\n

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