On the beauty of a mysterious thermodynamic parameter –Temperature.

In this article, I discuss an overview of temperature by briefly touching upon both the microscopic and the macroscopic viewpoint. I also discuss some special aspects of temperature towards the latter half. I must, however, say that the article is by no means exhaustive. I hope that reading this article would motivate the readers to appreciate the fact that temperature is a lot more than what it seems at the face of it. And inside the intricate details of temperature lies some immense beauty.

Introduction

The concept of temperature is something that is often viewed as trivial. It is something that a small kid doing a backyard science experiment is aware of and on the other so is a scientist doing ‘serious’ science experiments. But if I may ask you what temperature is, what would you say? Something that tells you how hot or cold an object is, right?

A professor during one of my undergraduate courses asked this question in his thermodynamics lecture and I had replied that it is the ‘degree of hotness and coldness of an object’. After hearing this he was fairly disgusted and gave me a serious look. This made it clear to me that temperature is far away from being something trivial, the way we are introduced to it as children.

Brief digression to Kinetic Theory

Let me walk the readers through a brief discussion of the way a collection of gas molecules behave which I will use to introduce more about temperature.

Almost every undergraduate science student takes a course where Kinetic Theory of Gases (KTG) is discussed in a quantitative form. It may a physics lecture for some or a physical chemistry lecture for some, nevertheless, the ideas are more or less similar. In KTG we look at the ‘average’ behavior of a collection of gas molecules because it is cumbersome to individually keep track of the motion of each gas molecules as the number of variables becomes too large. This simple argument shows how –

Consider a collection of N monoatomic gas molecules you would have 3 momentum (p_x, p_y, p_z) and 3 velocity (v_x, v_y, v_z) coordinates for a single molecule. Thus a total of 6N coordinates for the entire collection. Typically N is of the order of 10²³, so that gives us an overwhelming number of variables to keep track of!

To circumvent the above problem of having to deal with such a large number of variables simultaneously, we choose to study the ‘average’ behavior of the collection of these molecules and relate it quantitatively to the macroscopic thermodynamic variables like temperature (something I will extensively talk about here), pressure, volume etc. However I should mention that there are perils of this averaging. By doing so we lose certain information on the dynamic variables which might reveal interesting information about the system.

KTG talks about the beautiful Maxwell-Boltzmann (M-B) distribution of velocities and from there we note a very important fact that there exists some inherent internal heterogeneity in a collection of gas molecules. Say we have a collection of N gas molecules in a cubical (you can choose any other geometry of the container too but I have just chosen a cube because it’s symmetric, nice and neat to imagine) container of volume V. We can measure the pressure (P) of the gas molecules inside this V volume and we can measure the temperature (T) of the gas too. But say one has a (hypothetical) pressure measuring device that has very quick detection time and we measure the pressure at different points (locations) inside this cubical container. There is no guarantee that the device will record the same pressure at all of these points. There will be some ‘fluctuations’ in the recorded value from point to point. Under equilibrium situation these fluctuations will be very small and in order to detect them, our hypothetical device should be very sensitive. Have you ever wondered what might cause these fluctuations at all? What is the origin of these fluctuations?

A closer look tells us that these fluctuations are a result of the intrinsic internal heterogeneity that is present in the collection of gas molecules. This heterogeneity is beautifully engraved in the M-B distribution of velocities itself.

According to the M-B distribution –

To begin with we have f(v) d³v as the probability that a molecule has a velocity in the infinitesimal range v to v+dv.

Some very elegant theoretical arguments give the following explicit form of the function as –

where m = mass of the gas molecule, k = Boltzmann’s constant, v = velocity of the molecule

and lastly we have T = thermodynamic temperature.

By analyzing the symmetry of f(v) we can integrate over the solid angle (Ω) by using the relation d³v = v² dv dΩ to get the speed distribution function as –

Note that both the distributions have a parametric temperature dependency which can be emphasized by writing f(v) as f(v;T)

Let’s use now the above speed distribution function to qualitatively microscopic origins of the pressure fluctuations that we discussed earlier.

The M-B distribution tells us that not all molecules move with the same speed. But we have several patches of molecules moving in several patches of speeds accessible which theoretically ranges from 0 to ∞

Pressure of a gas arises due to collisions between the gas molecules with the walls of the container. During these wall collisions there is exchange in momentum between the gas molecule and the wall, which gives rise to a pressure. So if all molecules were to be moving with the same speed then our device would have recorded the same pressure at every point. But since the gas molecules don’t have all the same speeds hence this pressure heterogeneity is somehow linked to the intrinsic heterogeneity present in the speeds with which the gas molecules move.

At this point I ask the readers a question, how are we able to assign the pressure of the cubical gas box definitely then?

We must appreciate that this pressure we are talking about is a collective property of all the gas molecules confined in the box and not of one single gas molecule. So this macroscopically measured pressure is simply the average pressure and it is during this ‘averaging’ process, the fluctuations in the pressure mutually cancel out (internal compensation) to zero. Please note that this can only happen if these fluctuations are completely random. Such things are said to be stochastic in nature.

I would make a short comment on the fact that the science of measurement is fairly complex. If you were observant enough then you should’ve noted that I made two comments regarding the pressure device –

• It should be sensitive enough to detect these small fluctuations in systems which are equilibrated
• It should have a really fast detection time scale, one which is at least 10³ times faster than the time scale of these fluctuations for the measurement to be reliable.

Relating KTG with temperature from a microscopic point of view

Now that we have talked briefly about the M-B distribution let’s come back to our main topic, temperature.

In the speed distribution function stated above, we called T as the thermodynamic temperature. But what is it? Thermodynamic temperature is defined by the third law of thermodynamics. Which helps in defining the ‘zero’ of the temperature scale. This mysterious ‘zero’ of the temperature scale is something we cannot ever attain in reality. The absolute scale of temperature is called the Kelvin scale and the lowest possible temperature one can theoretically reach is 0K (absolute zero). There’s nothing beyond 0 K, it is the rock bottom of the temperature scale.

Let’s revisit the M-B speed distribution again. We can calculate the rms (root mean square) speed (v_rms), the average speed (v_avg) and the most probable speed (v_mp) using this distribution using some elementary calculus. The results are simply stated below –

The common denominator in the above equation is the fact that all kinds of speed depend on the square root of thermodynamic temperature. Which means that temperature has something to do with motion of particles at the microscopic level.

The above M-B speed distribution function can be simply converted to the energy frame from the speed frame by making use of the relation ∈ = 0.5mv² , that gives the energy distribution function. But this is subject to the caveat that the energy possessed by the gas molecules is wholly kinetic. In fact the latter is an important assumption on which KTG is built on. The gas molecules are point masses and they are quasi-independent in the sense that they don’t have any potential (attractive or repulsive) between them. I use the word quasi-independent instead of simply using independent because when the molecules collide amongst each other, they undergo redistribution of energy by exchanging momentum. So strictly speaking they do ‘interact’ but not strongly enough. These assumptions are also familiar to us from our study of the behavior of ideal gases that follow PV = NkT equation of state.

And in similar way used for calculating the various speeds above, we can calculate the rms (root mean square) energy (∈_rms), the average energy (∈_avg) and the most probable energy (∈_mp). For a monoatomic ideal gas ∈_avg = 1.5kT

The common theme here is that all of these different energies are directly proportional to the thermodynamic temperature T. So by now we have some idea of what thermodynamic temperature relates with. It relates with the energy possessed by the particles which in turn relates to the speed with which they move. So we define thermodynamic temperature as the measure of mean translational kinetic energy.

But in reality, not all molecules are monoatomic. Moreover, they do interact with each other (attractively or repulsively) and they are not structureless blobs. They have a complicated (but beautiful) internal structure. These molecules have other internal motions that we have ignored so far. They can rotate, they can vibrate (except for monoatomic molecules which can only translate) and thus their energies cannot be wholly kinetic. The energy content of the molecule is channelized into different ‘degrees of freedom’. So we need to refine our definition of temperature.

The concept of non-kinetic temperatures

We can tacitly extend our definition of thermodynamic temperature and say that it is the measure of the average energy of the translational, vibrational and rotational modes of motion (in principle one should also consider the electronic degrees of freedom and the spin/nuclear degrees of freedom too to be exact, but I am intentionally skipping that discussion to keep things simple). This new definition is somewhat better as it accounts for the so-called internal structure. In the context of degrees of freedom, temperature solely derived from kinetic energy is called the kinetic temperature and likewise, we have the rotational temperature and the vibrational temperature.

A molecule moving faster would have greater translational temperature than a molecule moving slower. Similarly, a molecule vibrating with higher frequency would have a higher vibrational temperature compared to a molecule vibrating with lower frequency. One should also note that molecular vibrations and molecular rotations are quantized. I shall talk about this later.

It was mentioned earlier that there is a lower limit to temperature called the absolute zero. But is there any upper limit to it? It is believed that the maximum attainable temperature is approximately a whopping 1.417 × 10³² K! This is called the Planck Temperature (T_p). The latter can be expressed in the system of natural units and on the Planck scale of temperature it is assigned a value of 1 and absolute zero is assigned the value 0. As opposed to absolute zero, the Planck temperature is called the absolute hot. Cosmologists believe that the universe passed through this temperature for a very short time of, 10^-42 seconds (one Planck time in the system of natural units.)

Temperature from a macroscopic point of view and thermometry

Now that we have looked at the microscopic view of temperature (where we talk about the internal structure at the molecular/atomic level and the quantum mechanical states) let’s have a deeper look into the macroscopic view temperature and how we ‘measure’ it. As mentioned earlier, the degree hotness or coldness is the anthropomorphic description of temperature. It is neither correct nor wrong, but it is not quite scientific. A proper macroscopic understanding of temperature comes from the zeroth law of thermodynamics which introduced the concept of thermal equilibrium. The zeroth law states that two distinct systems A and B are in thermal equilibrium with another system C then they are also in thermal equilibrium with each other. Thermal equilibrium is a situation when two systems on being connected by a diathermal (thermally conducting) interface will show no net transfer of heat.

In view of the zeroth law one can say that temperature of a system is simply that property of the system which dictates whether or not it is in thermal equilibrium with other systems.

This gives the necessary and sufficient condition of thermal equilibrium between two systems which is equivalency of their respective temperature.

Macroscopically we measure the temperature of an object by using a thermometer. Temperature is a scalar quantity so it can be measured by assigning some value to it. Thermometers measure temperature with the help of an empirical temperature scale. Typically one chooses two suitable thermodynamic variables (macroscopic) α and β. This way we have a α-β plane where we can draw isotherms which is variation of β with α at a fixed value of T. These isotherms can be thought of sections through the α-β-T surface [f(α,β,T) = 0] by making cuts parallel to the T axis. This would give a two dimensional projection of the α-β plane with their mutual variation studied a function of temperature. Now by choosing a convenient path in this α-β plane say for example one parallel to α axis. This path will cut the α-β isotherm at a point and give a certain value of α and T. Now by making cuts parallel to this path on the α-β-T surface we can study the variation of α and T. This enables us to write α=α(T) thermometric function (at fixed β) and the inverse function gives us temperature as a function of α. Since α is something we can measure physically, that is how we indirectly end up ‘measuring’ temperature.

The above description is fairly abstract, this can be better understood with an example. Consider a mercury thermometer that we commonly use in our day to day life. The appropriate thermodynamic variables here would be pressure and volume. The atmospheric pressure is under a good assumption constant. Then one measures the temperature by the volume change (expansion or contraction) of the mercury enclosed in the thermometer (bulb + limb). The higher the temperature the more is the volume expansion and the reading is noted using the scale used to calibrate the thermometer. It should be noted that this scale is an empirical temperature scale and we arbitrarily (but with some logic) define two reference points on it, one being the freezing point of pure water at 1 atm which is 0^oC and the other being the boiling point of pure water at 1 atm, which is 100⁰C (this scale is called the Celsius scale). Since all our measurements for this thermometer is at constant pressure so this is an isobaric thermometer.

Similarly one can think of other experimental ways of measuring thermodynamic temperature and thus devise different thermometers. So of the examples are the gas thermometers, bi-metallic stem thermometers, vapor pressure thermometer, resistance thermometer etc.

Revisiting the very exotic absolute zero!

Previously we have discussed absolute zero and it was said that it is the lowest possible temperature that can be attained (theoretically). While discussing the microscopic point of view we have also alluded the connection between temperature and molecular motion. Something which nicely comes out of KTG too. Now let us revisit absolute zero and take a finer look.

If one puts T=0K in any of the speeds from M-B distribution (rms or avg or mp) then they come out to be zero. This is also case for the energies derived in the same manner. Does this mean that all kinds of motion cease at absolute zero? Is energy of a gas (which is directly proportional to T) zero at absolute zero?

First of all there will be no gas at 0K. From our day to day experiences we know that ice melts into water on heating and water vaporizes into water vapor on heating. This is something physical chemists called phase transitions. Again as we cool water vapor it condenses to form water, which on further cooling gives ice. So this is a case of a reversible phase transition.

Similarly as we cool our gas, we are reducing its temperature or ‘stealing away energy’ from it and dumping it to some sink. This of course reduces the thermal (random) motion of the gas molecules and thus the probability that they come closer increases. This means the molecules start interacting with each other (remember that ideal gas exists only on paper) and eventually a temperature comes when the gas liquefies. Subsequent cooling will make the liquid freeze into a solid. So the existence of gas at such low temperature regimes is not possible. Liquid and solids are called condensed phases of matter. The theories of condensed phases are significantly more difficult than the theories of gas phase. Quite obviously the translational degrees of freedom reduce as we move from gas to liquid to solid. But that doesn’t imply a total cessation of motion. The constituent of a solid vibrate about their equilibrium positions in a lattice.

The truth is that matter starts behaving unusually when we reach such low temperatures. This unusual behavior gives rise to the so called ‘exotic state’ of matter. One such largely celebrated example is the Bose-Einstein condensate (BEC) which was first predicted (theoretically) by S.N. Bose and Albert Einstein in 1924. It took about 70 years for the first experimental realization of a true BEC which was done by Carl Wieman and Eric Cornell. During this same time another physicist named Wolfgang Ketterle made important independent contributions to the experimental study of BEC. This led to the 2001 Nobel Prize in Physics which was shared by Wieman, Cornell and Ketterle. BEC is simply a group of atoms cooled to very very close to absolute zero (note that second law of thermodynamics tells us that it is impossible to reach absolute zero in finite number of steps). Under such circumstances the atoms tend to clump to each other and enter the same energy state. Another exotic activity seen at such low temperature regimes are superconductivity, superfluidity and quantum vortices. I just mention all of this very briefly here just because they are really exciting, but it is beyond the scope of this article to discuss these in details.

What I want to emphasize on is that the connection of atomic/molecular motion with thermodynamic temperature and its apparent cessation at absolute zero is somewhat erroneous. A very powerful branch of science called Statistical Mechanics links microscopic properties to macroscopic thermodynamic variables. This is why it is sometimes also called Statistical Thermodynamics. When we correlate atomic/molecular motion with thermodynamic temperature it is necessary to append classical Statistical Mechanics with concepts of Quantum Mechanics to void the erroneous conclusion that atomic motion ceases at absolute zero. Quantum mechanics introduces a vital concept of Zero Point Energy (ZPE) that one can say is the finite amount of residual vibrational energy that a system must necessarily have even at absolute zero. ZPE is strictly non-tradable which means we can’t take it away at any cost. This in accord to the Heisenberg Uncertainity Principle.

Negative temperatures!!!

All throughout the article so far I have repeatedly mentioned that we can’t go beyond absolute zero (T ≥ 0) then have I gone crazy talking about T < 0 ?!?!?

The answer is NO. Let’s try and stay calm and have a look at what do we mean by negative temperature.

The concept of negative temperature was first predicted by Onsager in 1949. A system with a negative temperature is said to be hotter than other system. We know that, naturally heat flows from hotter objects to colder objects that is from high temperature to low temperature such that direction of heat flow in always in the direction where the temperature gradient is always negative. For the reverse to happen we must supply work from outside, a day to day example being refrigeration.

But from thermodynamics and statistical mechanical considerations it is seen that heat flows form a negative temperature system to a positive temperature system when they are in thermal contact! What is happening? Why is our basic thermodynamic equation being strongly challenged now?

Negative temperature corresponds to a higher energy Boltzmann state.

To understand this let us consider the Boltzmann distribution which has the form –

Where –

N_i = Population of the i^th quantum state with energy E_i and statistical weight (degeneracy) g_i

An alternative form of expressing this is –

is called the sum over all states (Zustandsumme) or the partition function. It tells us how the molecules/atoms are partitioned over the available quantum states.

It is imperative to define which is one of the most important equations in Statistical Mechanics and it is known as the thermodynamic beta or coldness. It is worthwhile to mention that 1/T is more ‘fundamental’ that T itself and it is expressed as

in thermodynamics.

is simply the probability of finding the particle in the i^th quantum state with energy E_i and we see that at a given T, as E_i increases decreases exponentially.

This means under thermal equilibrium, lower energy states are populated more than higher energy states. We can provide a statistical interpretation of work and heat using the Boltzmann distribution. Work is defined as changing the values by keeping the population over the states fixed and heat is defined as changing the population of the states by keeping the values fixed.

It is also important to note that Boltzmann distribution is an equilibrium distribution and care should be taken while applying it to situations as we must first examine if we are dealing with equilibrium or non-equilibrium cases.

Careful observation of the argument of the exponential function in the above equations is always negative since both ≥ 0 and β ≥ 0. That means in order to populate a higher energy state more than a lower energy state (anti Boltzmann distribution) one must have to make β < 0 which would require T< 0 !

Using some elementary calculus it can be shown that for a two level system (two energy states 1 and 2 with say E₂ > E₁) it is impossible to attain a population inversion (i.e. a situation when population of state 2- which is higher in energy is greater than the population of state 1 – which is lower in energy). At best we can reach N₂ = N₁ and this is call the saturation condition. Population inversion is in no way an equilibrium situation it is a highly non-equilibrium situation.

For LASERs (Light Amplification through Stimulated Emission of Radiation) it is necessary to have population inversion between the lasing sates and the better the population inversion, the better will be the lasing. By this is mean to say 5:4 is also a population inversion but so is 8:1 and quite obviously the latter is a better population inversion.

We have already discussed how there are other degrees of freedom (except translational or kinetic degrees of freedom) among which energy can be distributed. When we associate temperature with kinetic energy of atoms/molecules, since there is no upper bound to the momentum so there’s no upper bound to the number of state available and hence the concept of negative temperature cannot arise. Negative temperatures can only exist in those systems which have a limited number of states available to them. LASER is a textbook example of such a system and from the above discussion it follows that the most primitive laser has to be at least a 3 level laser as we need to necessarily (but not sufficiently because some other criteria must also need to be satisfied simultaneously for lasing to occur but it is beyond the scope of this article to discuss) have a population inversion.

References:

• For further reading on temperature:
• • Heat & Thermodynamics – Mark W. Zeemanksy and Richard H. Dittman
  • An Introduction To Thermal Physics – Daniel V. Schroeder
  • Fundamentals Of Statistical & Thermal Physics – Frederick Reif

• For further reading on negative temperature:
• • Heat & Thermodynamics – Mark W. Zeemanksy and Richard H. Dittman
  • Fundamentals Of Statistical & Thermal Physics – Frederick Reif
  • Negative Kelvin temperatures : Some anomalies and a speculation – R.J. Tykodi, American Journal of Physics 43, 271 (1975); https://doi.org/10.1119/1.10069

• • Comment on “Negative Kelvin temperatures : Some anomalies and a speculation” – André Marie Tremblay, American Journal of Physics 44, 994 (1976); https://doi.org/10.1119/1.10248

• • Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures – Norman F. Ramsey, Phys. Rev. 103, 20 (1956); https://doi.org/10.1101) 3/PhysRev.103.20

• For further reading on Kinetic Theory of Gases
  • Physical Chemistry (volume 1) – Ashish Kumar Nag
  • Physical Chemistry – Silbey,Alberty,Bawendi
  • Chemical Kinetics and Reaction Dynamics – Paul Houston
  • Physical Chemistry – Gilbert W. Castellan
  • The Feynman Lectures On Physics (volume 1) – Richard Feynman

• For further reading on different internal degrees of freedom, non-kinetic temperatures and their application to molecular systems
• • Statistical Mechanics – Donald McQuairre
  • Heat & Thermodynamics – Mark W. Zeemanksy and Richard H. Dittman
  • Fundamentals Of Statistical & Thermal Physics – Frederick Reif
  • Physical Chemistry – Silbey, Alberty, Bawendi

• For further reading on LASERs
• • How The Laser Happened: Adventures Of A Scientist – Charles H. Townes
  • Physical Chemistry – Silbey, Alberty, Bawendi
  • Principles Of Lasers – Orazio Svelto

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By Chirag Arora, Department of Chemistry, IIT Kharagpur.

Also by the author: Origin of biological chirality & natural left preference

  About the author: Chirag is 1^st year MSc (Chemistry) student at Indian Institute of Technology (Kharagpur). His concentration lies in Physical Chemistry and Chemical Physics. He is captivated by the science of light-matter interaction and quantum effects in chemical systems. He likes to study about ultrashort laser pulses and time-resolved spectroscopy. Apart from studying he likes eating, procrastinating and getting loaded on caffeine. He likes music and has a strong admiration for Taylor Swift! He writes guest articles for The Qrius Rhino. He also has his own blog : https://chemchirag13.wixsite.com/lumierelmi