Talk:Thermodynamic temperature

A scale is not defined by end points alone

This article states a number of times that the Kelvin scale is determined by the null point and the triple point. But fixing two points on the scale is not enough to determine the scale itself. One feels that some form of linearity of the scale is assumed (without being stated explicitly) but it is not clear what exactly that linearity should be - how could one split a temperature interval in two without using any other instrument (such as a thermometer) or concept?

The only way I see out of this is to base the linearity on heat - to divide the scale between 0 and the triple point linearly in terms of the heat transferred to or from the body, i.e. the halfway point between two temperatures is reached when half of the heat has been transferred. This is consistent with the kinetic definition (in terms of 1/2 m v^2).

How do others feel about this? 88.159.82.92 (talk) 06:31, 29 January 2013 (UTC)

The linearity is created by the definition of Kelvin, it is not assumed. Temperature is defined as the amount of kinetic energy of the vibrational motions of matter's particle constituents. All matter with no average kinetic energy is 0 K by definition. All matter that has the same average kinetic energy as water at its triple point has a temperature of 273.15K by definition. The temperature of any object with a different average kinetic energy is directly and linearly proportional to this average kinetic energy BY DEFINITION. The definition does not require a specified measurement method. It is the burden of the person measuring to demonstrate that the measurement technique used is an accurate method of measuring the average kinetic energy relative to the definition points. In your proposed method, how would you be able to tell when "half of the heat has been transferred?" To know this you would have to be able to measure the amount of heat (kinetic energy) that has been transfered. If you were able to measure this, wouldn't it be simpler just to measure and average the kinetic energy at the triple point of water and divide by 273.15?192.104.67.121 (talk) 15:50, 2 April 2013 (UTC)

Well, not quite. That's a definition that works for classical monatomic ideal gases. It doesn't work very near absolute zero, because of zero-point kinetic energy, and it doesn't work for things more complicated than monatomic gases even at normal temperatures unless you do some fancy bookkeeping.

The real definition of temperature is the one given at the very end of the definition section of the article, namely

${\frac {1}{T}}={\frac {dS}{dE}},$

This definition, which is the most fundamental one, is not based on kinetic energy at all. --Trovatore (talk) 16:15, 2 April 2013 (UTC)

" ...not based on kinetic energy "?

How do you mean? Is it then based on potential energy? Or perhaps not on energy at all?--Damorbel (talk) 14:20, 3 April 2013 (UTC)

It's based on the way that the entropy (defined as the logarithm of the number of quantum states having a given energy) varies with energy. --Trovatore (talk) 21:52, 3 April 2013 (UTC)

I also find it surprising that this definition of temperature is given so late in the article. I would expect it to be given in the very beginning. 130.233.174.49 (talk) 18:07, 20 May 2013 (UTC)

Single Particle Temperature

79.119.58.54 made this revision [1], claiming that:-

[it]might mislead someone to think that a single particle - as in a molecule - has a temperature

Since, through the Boltzmann constant, temperature is based on the energy of individual particles, it is self evident that individual particles also have a temperature - it is not possible to separate temperature from individual particle energy. --Damorbel (talk) 16:22, 25 March 2013 (UTC)

The Boltmann constant equations do not speak of single particle energies, but averages in energy of many particles. A single particle would have to be in many energy-states, over a long period of time, to get such an average. But a single particle at a single time (and a single energy) does not have a temperature. If it did, in some other reference frame, it would have a temperature of zero. That would result in any entropy it had in another frame, disappearing entirely in its own rest frame. But the universe doesn't work that way-- it's not that easy to get rid of entropy completely. SBHarris 02:39, 26 March 2013 (UTC)

I can find nothing about the Boltzmann constant other than it is defined as:

" energy per particle per K(elvin) "

My understanding of your argument is that the Boltzmann constant - k - is about the energy of a number of particles (n) > 1. Even if this were true surely there would be no loss if it were to be defined as k / n ?

I see no posssible argument aganst a particle having an instantaneous energy, and an average energy during a given time period and, since particle energy and temperature are directly related by the Boltzmann constant, individual particles must have a temperature. --Damorbel (talk) 14:07, 26 March 2013 (UTC)

Nota Bene. The importance of the word particle in the definition is very relevant because the energy of a particle cannot be subdivided without splitting the particle, i.e. creating a greater number of particles. --Damorbel (talk) 14:07, 26 March 2013 (UTC)

It is average energy per particle. If you find anything else in WP , it is incorrect and needs editing. You find me an equation in a reliable source. SBHarris 16:10, 26 March 2013 (UTC)

My previous response answers your request for a reliable source (the Boltzmann constant) completely. --Damorbel (talk) 07:28, 3 April 2013 (UTC)

No, that's wrong. See my remarks above in the "scale not defined by its endpoints" section. Energy per particle is an accident; the essential is the relationship between entropy and energy.

But in any case it makes no sense whatsoever to speak of the "temperature" of an individual particle. Temperature is an essentially statistical notion; only random components of energy are to be taken into account. Systematic energy is ignored. How an individual particle can have random energy is beyond me. --Trovatore (talk) 06:26, 4 April 2013 (UTC)

Trovatore, you write:-

"Temperature is an essentially statistical notion "

In this case what is the minimum statistical sample size that makes this true; so how many particles are needed to establish a temperature? --Damorbel (talk) 08:44, 4 April 2013 (UTC)

Not precisely defined. Nothing about temperature is precisely defined. That's why it makes sense only as a statistical notion. --Trovatore (talk) 15:02, 4 April 2013 (UTC)

What do you mean:-

" Nothing about temperature is precisely defined."?

The melting point of gallium is known to 7(6) figure accuracy - 302.9146 K & 29.7646 °C; many other elements to 5 figures.

What do you mean by 'not precisely defined'? --Damorbel (talk) 17:54, 4 April 2013 (UTC)

It's a statistical quantity of large aggregates. The aggregates are large enough that five-figure accuracy is certainly very plausible. Ask yourself at what temperature a single atom of gallium melts, and you'll maybe start to see what I'm talking about. Also see sorites paradox.

At this point we both need to remember (I'm not denying my responsibility) that this is not what article talk pages are for and stop arguing about it, unless it's connected with improving the article in some way. --Trovatore (talk) 17:59, 4 April 2013 (UTC)

Ask yourself at what temperature a single atom of gallium melts

Ho, Ho, Ho! What a joke! A single atom melts? You cannot be serious!

No, the melting point of crystalline materials is the temperature where the cystalline particles (atoms or molecules) have enough energy to break the bonds forming the crystal. A single, isolated atom has, by definition, no bonds to "melt".

Of course this is what 'temperature' is all about. If you care to glance at the link I gave, you will see that the atoms of Gallium break their crystalline bonds at the same temperature.

Again you wrote:-

How an individual particle can have random energy is beyond me.

I see. Mainly in a gas where, due to random collisions, the particles acquire energy with a random, (Maxwell Boltzmann) distribution; such a distribution can be in time or space; in time and in equilibrium conditions the individual particles have the same average energy (why should they be different?) because the energy is shared by the principle of equipartition, that is why the particles, individually, have a temperature. --Damorbel (talk) 20:49, 4 April 2013 (UTC)

This is not the place to discuss it. --Trovatore (talk) 22:01, 4 April 2013 (UTC)

I disagree. To establish your position on the article rather more than simple assertions are required. I have made a number of highly relevant points to which you have made no response, e.g. the Boltzmann constant, the time average of particle energy and the irrelevance of a single particle 'melting'. These lie at the heart of the concept of temperature and should be in the article. --Damorbel (talk) 05:39, 5 April 2013 (UTC)

Make specific proposals about what you think the article can say, and they can be discussed here. The general nature of temperature, separate from any such proposal, is off-topic on this page. --Trovatore (talk) 07:18, 5 April 2013 (UTC)

How can "The general nature of temperature..." be "...off-topic on this page"? The title of the article is Thermodynamic temperature; the points I have made are 100% relevant to any discussion on temperature, you appear to have nothing to say about them. --Damorbel (talk) 07:34, 5 April 2013 (UTC)

Please familiarize yourself with WP:TALK. --Trovatore (talk) 14:22, 5 April 2013 (UTC)

. . . where it says:-

The purpose of a Wikipedia talk page (accessible via the talk or discussion tab) is to provide space for editors to discuss changes to its associated article or project page.

Indeed. Now the Boltzmann constant relates energy to the individual particle temeperature thus relevant to the article and mean particle energy is equally related to temperature

Further according to WP:TALK:-

Article talk pages should not be used by editors as platforms for their personal views on a subject.

Neither the Boltzmann constant nor equipartition are my personal views, nor do you discuss them, so I do not understand what you wish to contribute to the article. --Damorbel (talk) 15:51, 5 April 2013 (UTC)

Footnote 8,

... where it is argued that modes of motion other than translational do not contribute to temperature is quite interesting but I saw no reference. Is this a well-known aspect of the microscopic/statistical treatment of temperature? — Preceding unsigned comment added by 94.54.71.115 (talk) 19:44, 3 April 2013 (UTC)

Temperature is the parameter that governs the transfer of heat energy. Accordng to the 2nd Law heat tranfers from a high temperature to a lower temperature; the critical matter is 'transfer' i.e. it changes place - 'moves' - has directional properties - thus is a vector quantity. Energy - even kinetic energy - is not a vector (v²); however momentum 'mv' is a vector and heat transfer is actually the transfer of momentum - see Einstein's 1917 paper 'On the Quantum Theory of Radiation', (The German for Momentum is 'Impuls'). Momentum corresponds to the translational aspects of particle energy. --Damorbel (talk) 12:49, 4 April 2013 (UTC)

Kinetic theory is *already* overemphasized here -- stop trying to make it worse

Damorbel keeps pushing the view that temperature is just kinetic energy. But that's flat wrong. See Kittel&Kromer. --Trovatore (talk) 17:32, 15 April 2013 (UTC)

But that's flat wrong

Then please explain why it is wrong!

I have a copy of Kittel & Kroemer and it has a whole chapter (Ch.14) devoted to kinetic theory. Here they explain about momentum exchange (eq.2).

In Ch.6 (p.164) the energy of an ideal gas is given as U = 3/2Nτ, which since an ideal gas has neither spin (rotational) energy, vibrational energy nor rotational energy, all that remains is translational energy. --Damorbel (talk) 18:20, 15 April 2013 (UTC)

Key words: Ideal gas. Your formulation does work for a monatomic ideal gas. It does not work in general. --Trovatore (talk)

Oh, not even quite that. Even a monatomic ideal gas does not have zero translational kinetic energy per particle at absolute zero. The zero-point kinetic energy remains. --Trovatore (talk) 18:24, 15 April 2013 (UTC)

I asked you to show how my contribution about kinetic energy was shown to be 'flat wrong' by Kittel & Kroemer - please give quotes! --Damorbel (talk) 20:50, 15 April 2013 (UTC)

Only kinetic energy contributes to temperature?

There are few very confusing statements in the article. For instance:

Internal energy may be stored in a number of ways within a substance, but only the kinetic energy of particles contributes to the substance's temperature.

Temperature arises from the random submicroscopic vibrations of the particle constituents of matter. These motions comprise the kinetic energy in a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy of a certain kind of vibrational motion of its constituent particles called translational motions.

What does it meant that "only the kinetic energy contributes to the temperature"?

To me this claim seems false. As an example, consider a nuclear spin system in a strong magnetic field. In such a system, the kinetic energy degree of freedom of the constituent particles is absolutely insignificant to the magnetic energy of the spins. Also the rotational degrees of freedom in many ordinary systems contribute to the temperature, not just the translational motion. — Preceding unsigned comment added by 130.233.174.49 (talk) 11:22, 18 May 2013 (UTC)

I think you're absolutely right. This is just a small piece of a larger issue in this article. The article is written mostly from the point of view of kinetic theory (ideal monatomic gases, billiard-ball-style interactions). The way it should be written is taking statistical mechanics as fundamental, and then present the kinetic theory as a special case. That's a serious rewrite and I'm not the best person to do it; I hope someone will step up. --Trovatore (talk) 22:08, 18 May 2013 (UTC)

I agree, except thermodynamic temperature is a thermodynamic concept, explained, but not defined by statistical mechanics. So thermodynamics is fundamental, but a statistical mechanical explanation should not make statements like "only the kinetic energy contributes to the temperature", etc. I have tried to edit out the above mentioned errors. Its not perfect, but certainly a step in the right direction. PAR (talk) 03:13, 19 May 2013 (UTC)

I think it isdefined by statistical mechanics. How else should it be defined? By the ideal gas law, when there aren't actually any ideal gases? I'm not sure what you mean by saying "thermodynamics is fundamental" as opposed to statistical mechanics; the way I think of it, thermodynamics is statistical mechanics. --Trovatore (talk) 17:33, 19 May 2013 (UTC)

Check out the introduction to Munster "Classical Thermodynamics", it gives a good explanation of this.

Thermodynamics is a theory of measurements performed on macroscopic "systems" while statistical mechanics explains, but does not replace, thermodynamics by introducing the particulate theory of matter and viewing thermodynamic variables in statistical terms. Thermodynamics deals with measurements, statistical mechanics deals with thermodynamics.

Thermodynamics makes predictions and places constraints on the results of macroscopic thermodynamic measurements. Those constraints are mostly contained in the various laws of thermodynamics. There are fundamental measurements which are "outside" of thermodynamics - measurement of mass, volume, pressure, and work. The laws then form concepts like internal energy, temperature, entropy and chemical potential, tell you how to measure them macroscopically using the fundamental measurements, and puts constraints on what your results will be. This is all done without reference to particles or statistics. For example the first law says that there is a state variable called the internal energy and when you perform measurable work on a system, that energy is added to the internal energy, and any mismatch is the result of heating the system. Not a single reference to particles or statistics there. The second law likewise defines entropy and temperature as state variables, ultimately measurable without reference to particles or statistics or Boltzmann's constant. The DEFINITIONS of internal energy, entropy, temperature, etc. are found in thermodynamics. Materials are characterized by their measurable thermodynamic parameters (viscosity, compressibility, etc.) all of which are measured without reference to particles or statistics. It's a fairly complete theory that does not require statistical mechanics in the least for its formulation.

Statistical mechanics is an explanation of the results of thermodynamics using the particulate theory of matter and viewing most of the thermodynamic parameters in statistical terms. Statistical mechanics defines NONE of the thermodynamic variables and parameters. It only explains them and that explanation allows you theoretically describe why the measured parameters behave as they do. Thermodynamics cannot and does not attempt to describe why materials exhibit their measured properties, only that if they do, then in a particular scenario you will get such and such results. It also states that if you measure this or that parameter, then some other parameter is constrained to be such and such. Statistical mechanics allows you to "see behind the curtain" and understand the microscopic reasons for the thermodynamic laws and constraints. It allows you to go beyond thermodynamics, since it can make predictions for small (i.e. non-thermodynamic) systems, and can deal with many situations that are too rapid for thermodynamics to deal with. However, if your statistical mechanical theory gives a prediction that is in conflict with thermodynamics, it is the statistical mechanical theory that is at fault, not thermodynamics, because thermodynamics is measured reality. The most famous link between thermodynamics and statistical mechanics is Boltzmann's equation $S=k\ln W$. On the left is a thermodynamic parameter, the entropy S, and on the right is a statistical mechanical parameter W, the number of microstates which yield the thermodynamically defined macrostate.

I was taught the same way, thermodynamics and statistical mechanics all mushed together, and it doesn't really matter if you are not interested in the theoretical understanding of the two theories. Also, the idea that statistical mechanics must be superior in some sense because it encompasses thermodynamics and more, but when you realize that thermodynamics is the "real thing" - a logical condensation of an immense amount of data from actual experiments, while statistical mechanics never touches reality when explaining thermodynamics except through a thermodynamic measurement, thermodynamics is seen to be more primary, but less informative than statistical mechanics. PAR (talk) 21:16, 19 May 2013 (UTC)

Well, let's take temperature, then. How do you "thermodynamically measure" a negative temperature? Obviously you're not using the ideal gas law. You certainly can measure negative temperatures; we have articles that cite papers from workers that do it. But aren't they using a definition taken from statistical mechanics? Granted, the microscopic properties are inferred rather than directly measured, but it seems to me that the definition is in terms of the statistical and microscopic, not in terms of the way that those are inferred. --Trovatore (talk) 01:05, 21 May 2013 (UTC)

To be honest, I have not studied negative temperatures. However, looking at the Wikipedia page, it seems to me that it is measured in terms of thermodynamic quantities $(T=\delta Q/S)$. I don't think the definition is taken from statistical mechanics, but I think that statistical mechanics is vital to microscopically understand negative temperatures. I'm guessing that the definition is in fact phenomenological - i.e. thermodynamic, but I can't say for sure. If you could show that a negative temperature cannot be measured without invoking statistical mechanics, I would certainly be interested. PAR (talk) 04:52, 21 May 2013 (UTC)

The article is much better now, but I still do not understand why the translational d.o.f. is emphasized so strongly throughout the text.130.233.174.49 (talk) 18:03, 20 May 2013 (UTC)

Because they are the last degrees of freedom to "freeze" as you lower the temperature. As you lower the temperature, various dof's start becoming quantum mechanical, not classical - their average energy is of the order of the spacing between energy levels and the nice formula $E=kT/2$ for the average energy goes wrong. When they totally freeze, practically all of the particles are in the ground state for that dof, and when you add energy, it's like they're not even there. Vibrational, rotational, etc, they mostly all freeze at temperatures well above zero, some are even frozen at room temperature, but the translational dof's don't freeze until its really cold - fractions of a Kelvin. So unless you are concentrating on that very cold regime, using the simple formula $E=kT/2$ and only dealing with the translational dof's works really well. Sometimes people act like the translational dof's never freeze and $E=kT/2$ is universally true for them, which is wrong. PAR (talk) 19:22, 20 May 2013 (UTC)

Sorry for reopening the can of worms, but aren't the rotational and vibrational kinetic energies primary contributors to changes in the statistical temperature under certain conditions? Suppose I radiate a liquid with IR then stop, the non-radiative relaxation from the excited state would increase the temperature no? Or does the absorbed photon energy convert to translational energy in that case? It's easy to demonstrate temperature = translational energy for monoatomic ideal gases but as a learner struggling with these concepts my confusion is that temperature in general is often explained as translational energy e.g.: http://www.chem1.com/acad/webtext/energetics/CE-3.html "We also pointed out that temperature is a measure of the average kinetic energy due to translational motions of molecules. If vibrational or rotational motions are also active, these will also accept thermal energy and reduce the amount that goes into translational motions. Because the temperature depends only on the latter,..." In its current form, the Wiki entry remains somewhat ambiguous on the matter. It would be great if this could be clarified - not necessarily by 'picking a side'. 144.122.162.45 (talk) 11:40, 27 September 2013 (UTC)

Definition of thermodynamic temperature

I deleted this section because it contains no refs., it has of course been replaced with the observation :-

Put cite needed if you like, but don't delete large sections of good material without TALK consensus

I have not the slightest objection to, in this case, material being replaced. But why no refs.?

For example 1/ the first paragraph opens with this:-

The thermodynamic temperature is defined by the second law of thermodynamics and its consequences

But 2nd Law is about a difference in temperature, it applies to a difference anywhere on the temperature scale between 100K and 200K as well as 1100K and 1200K. There is no definition of temperature here.

The argument is justifed further by saying:-

...and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines

In no way is the efficiency of a heat engine defined by the temperature scale used, this has got to be absurd!

I propose it be deleted.

2/ the 2^nd para. opens with:-

Strictly speaking, the temperature of a system is well-defined only if it is in thermal equilibrium

Nothing to do with Strictly speaking. If the system is not in equilibrium, either more than one temperature is needed to describe it and/or the temperature(s) is/are changing with time. This not a suitable defintion for a Wiki article.

3/ The 3^rd para. has:-

This suggests that there should be a relationship between temperature and entropy

How can this be justified? To assign a temperature to any system it must be in a state of maximum entropy i.e. in equilibrium. If the system is not in equilibrium its entropy is completely undefined since different systems may well have the same sub-maximum entropy.

4/The final paragraph begins:-

Loosely stated, temperature differences control the flow of heat between two systems

The section is about the defintion of Thermodynamic temperature, it is not about some process that depends on the difference of temperatures, but this is the whole content of the last section.

Either the section should be deleted or heavily revised. --Damorbel (talk) 12:54, 15 June 2013 (UTC)

Thermodynamics temperature is defined by second law or third law of thermodynamics?? BJKShah (talk) 07:21, 8 October 2017 (UTC)

Thermodynamic temperature vs Temperature articles

Why are there two separate articles, Thermodynamic temperature and Temperature? Couldn't they be mereged in some way? DavRosen (talk) 06:03, 21 July 2013 (UTC)

I've had the same question for years. As far as I'm concerned, the Temperature page has become so comprehensive as to essentially contain everything covered in the Thermodynamic temperature page. JCMPC (talk) 21:35, 17 July 2014 (UTC)

Moreover the Temperature article is more accurate than this one in some respects. For example it correctly uses the rms velocity in the computation of temperature where this article incorrectly uses the mean velocity (changing it to mean speed does not fix this, and median velocity is even worse). The incorrect statement was introduced over eight years ago, in a footnote on 24 Oct. 2006. And where is "vector-isolated velocity" defined? Does the term exist anywhere besides material derived from this article? Vaughan Pratt (talk) 18:10, 5 November 2014 (UTC)

I also think this is a problem. The title of this article makes it even worse since it does not show up in a search for an article titled "temperature". That could be fixed by changing the title of this article to "Temperature (thermodynamic)". It does seem that there is a lot of overlap between the two articles, but I have not examined either enough to comment on relative accuracy. Retired Pchem Prof (talk) 17:39, 17 January 2016 (UTC)

@Retired Pchem Prof: I think you may have a point that people looking for this page may not always start with the word "Thermodynamic". And since there already are the redirects Absolute temperature and Kelvin temperature to this page, I don't see why there can not be one more. The redirect Temperature (thermodynamic) now exist to direct readers here. w.carter-Talk 18:16, 17 January 2016 (UTC)

Both this article and the Temperature article contain major, fundamental errors and need to be re-written. Why there are two articles is hard to understand. Temperature is the same whether it is defined in terms of thermodynamics or kinetic theory. And since when is kinetic theory not a part of thermodynamics, anyway?

Temperature is NOT "a measure of the mean of the energy of the translational, vibrational and rotational motions of matter's particle constituents". This shows a fundamental misunderstanding the relationship between internal energy and temperature - or the relationship between heat (heat flow) and temperature. For example, the constituent particles of a monatomic gas (such as Helium) and of a diatomic gas (such as Nitrogen) at the same temperature do not have the same average internal kinetic energies i.e. the same average energies of their "translational, vibrational and rotational motions". They have the same average translational kinetic energies. The average vibrational and rotational kinetic energies do not contribute to temperature. That is why these gases have different heat capacities. Hyperphysics ( http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper.html )contains a much more accurate and complete explanation of temperature. These articles will simply confuse and mislead readers.

The two articles should be combined into one and completely rewritten. AMSask (talk) 07:55, 17 December 2020 (UTC)

Footnote 20

The text given there (``Based on a computer model ... 5% greater than its rest mass (m0).) is not in the paper referred to, and googling for it only finds thoughtless copies of this wikipedia article. I propose deletion. 130.233.138.9 (talk) 12:49, 26 March 2014 (UTC)

The footnote is discussing Model S1216 of the paper, which is singled out for special attention in Figures 5 and 6. Is there any specific number in the footnote that you feel is not supported by the paper? Vaughan Pratt (talk) 16:08, 6 November 2014 (UTC)

{{vague}} template

I have removed the following template from the lead section, in context:

At this point, absolute zero, the particle constituents of matter have minimal motion and can become no colder{{vague|What does {{''}}colder{{''}} mean? If it means a lower temperature, negative thermodynamic temperatures are possible. Would {{''}}lower energy{{''}} perhaps be more suitable?|date=August 2014}}.

I believe "colder" is well-understood here. A body would be "colder" than absolute zero if, when you put them in thermal contact, heat energy would flow from the one at absolute zero to the colder one. By definition of absolute zero, this cannot happen.
As to the point about negative temperatures, they are lower numerically, but they are not colder; they are in fact hotter, as the linked article explains. If you put a system at negative temperature in thermal contact with one at positive temperature, heat energy flows from the former to the latter.
So there still might be a question about the best wording. If it confused the person who put the {{vague}} template there, it might confuse someone else. I don't think the {{vague}} template is the right way of addressing this, but perhaps it should be addressed. Suggestions? Maybe just an explanatory footnote, with an explanation something like the one I give above? --Trovatore (talk) 21:48, 28 June 2015 (UTC)

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