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The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically. Even a system at absolute zero, if it could somehow be achieved, would still possess quantum mechanical zero-point energy, the energy of its ground state at absolute zero; the kinetic energy of the ground state cannot be removed. |
Scientists and technologists routinely achieve temperatures close to absolute zero, where matter exhibits quantum effects such as Bose–Einstein condensate, superconductivity and superfluidity. |
Thermodynamics near absolute zero. |
At temperatures near , nearly all molecular motion ceases and Δ"S" = 0 for any adiabatic process, where "S" is the entropy. In such a circumstance, pure substances can (ideally) form perfect crystals with no structural imperfections as "T" → 0. Max Planck's strong form of the third law of thermodynamics states the entropy of a perfect crystal vanishes at absolute zero. The original Nernst "heat theorem" makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as "T" → 0: |
The implication is that the entropy of a perfect crystal approaches a constant value. An adiabat is a state with constant entropy, typically represented on a graph as a curve in a manner similar to isotherms and isobars. |
The Nernst postulate identifies the isotherm T = 0 as coincident with the adiabat S = 0, although other isotherms and adiabats are distinct. As no two adiabats intersect, no other adiabat can intersect the T = 0 isotherm. Consequently no adiabatic process initiated at nonzero temperature can lead to zero temperature. (≈ Callen, pp. 189–190) |
A perfect crystal is one in which the internal lattice structure extends uninterrupted in all directions. The perfect order can be represented by translational symmetry along three (not usually orthogonal) axes. Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping. For substances that exist in two (or more) stable crystalline forms, such as diamond and graphite for carbon, there is a kind of "chemical degeneracy". The question remains whether both can have zero entropy at "T" = 0 even though each is perfectly ordered. |
Perfect crystals never occur in practice; imperfections, and even entire amorphous material inclusions, can and do get "frozen in" at low temperatures, so transitions to more stable states do not occur. |
Using the Debye model, the specific heat and entropy of a pure crystal are proportional to "T" 3, while the enthalpy and chemical potential are proportional to "T" 4. (Guggenheim, p. 111) These quantities drop toward their "T" = 0 limiting values and approach with "zero" slopes. For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10 K. Even the less detailed Einstein model shows this curious drop in specific heats. In fact, all specific heats vanish at absolute zero, not just those of crystals. Likewise for the coefficient of thermal expansion. Maxwell's relations show that various other quantities also vanish. These phenomena were unanticipated. |
Since the relation between changes in Gibbs free energy ("G"), the enthalpy ("H") and the entropy is |
thus, as "T" decreases, Δ"G" and Δ"H" approach each other (so long as Δ"S" is bounded). Experimentally, it is found that all spontaneous processes (including chemical reactions) result in a decrease in "G" as they proceed toward equilibrium. If Δ"S" and/or "T" are small, the condition Δ"G" < 0 may imply that Δ"H" < 0, which would indicate an exothermic reaction. However, this is not required; endothermic reactions can proceed spontaneously if the "T"Δ"S" term is large enough. |
Moreover, the slopes of the derivatives of Δ"G" and Δ"H" converge and are equal to zero at "T" = 0. This ensures that Δ"G" and Δ"H" are nearly the same over a considerable range of temperatures and justifies the approximate empirical Principle of Thomsen and Berthelot, which states that "the equilibrium state to which a system proceeds is the one that evolves the greatest amount of heat", i.e., an actual process is the "most exothermic one". (Callen, pp. 186–187) |
One model that estimates the properties of an electron gas at absolute zero in metals is the Fermi gas. The electrons, being fermions, must be in different quantum states, which leads the electrons to get very high typical velocities, even at absolute zero. The maximum energy that electrons can have at absolute zero is called the Fermi energy. The Fermi temperature is defined as this maximum energy divided by the Boltzmann constant, and is on the order of 80,000 K for typical electron densities found in metals. For temperatures significantly below the Fermi temperature, the electrons behave in almost the same way as at absolute zero. This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century. |
Relation with Bose–Einstein condensate. |
A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero. Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. |
This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper from English to German and submitted it for Bose to the "Zeitschrift für Physik", which published it. Einstein then extended Bose's ideas to material particles (or matter) in two other papers. |
Seventy years later, in 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST-JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) (). |
A record cold temperature of 450 ± 80 picokelvin (pK) () in a BEC of sodium atoms was achieved in 2003 by researchers at the Massachusetts Institute of Technology (MIT). The associated black-body (peak emittance) wavelength of 6,400 kilometers is roughly the radius of Earth. |
Absolute temperature scales. |
Absolute, or thermodynamic, temperature is conventionally measured in kelvin (Celsius-scaled increments) and in the Rankine scale (Fahrenheit-scaled increments) with increasing rarity. Absolute temperature measurement is uniquely determined by a multiplicative constant which specifies the size of the "degree", so the "ratios" of two absolute temperatures, "T"2/"T"1, are the same in all scales. The most transparent definition of this standard comes from the Maxwell–Boltzmann distribution. It can also be found in Fermi–Dirac statistics (for particles of half-integer spin) and Bose–Einstein statistics (for particles of integer spin). All of these define the relative numbers of particles in a system as decreasing exponential functions of energy (at the particle level) over "kT", with "k" representing the Boltzmann constant and "T" representing the temperature observed at the macroscopic level. |
Negative temperatures. |
Temperatures that are expressed as negative numbers on the familiar Celsius or Fahrenheit scales are simply colder than the zero points of those scales. Certain systems can achieve truly negative temperatures; that is, their thermodynamic temperature (expressed in kelvins) can be of a negative quantity. A system with a truly negative temperature is not colder than absolute zero. Rather, a system with a negative temperature is hotter than "any" system with a positive temperature, in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat flows from the negative to the positive-temperature system. |
Most familiar systems cannot achieve negative temperatures because adding energy always increases their entropy. However, some systems have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease. Because temperature is defined by the relationship between energy and entropy, such a system's temperature becomes negative, even though energy is being added. As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy. Therefore, no complete system, i.e. including the electromagnetic modes, can have negative temperatures, since there is no highest energy state, so that the sum of the probabilities of the states would diverge for negative temperatures. However, for quasi-equilibrium systems (e.g. spins out of equilibrium with the electromagnetic field) this argument does not apply, and negative effective temperatures are attainable. |
On 3 January 2013, physicists announced that for the first time they had created a quantum gas made up of potassium atoms with a negative temperature in motional degrees of freedom. |
One of the first to discuss the possibility of an absolute minimal temperature was Robert Boyle. His 1665 "New Experiments and Observations touching Cold", articulated the dispute known as the "primum frigidum". The concept was well known among naturalists of the time. Some contended an absolute minimum temperature occurred within earth (as one of the four classical elements), others within water, others air, and some more recently within nitre. But all of them seemed to agree that, "There is some body or other that is of its own nature supremely cold and by participation of which all other bodies obtain that quality." |
Limit to the "degree of cold". |
The question whether there is a limit to the degree of coldness possible, and, if so, where the zero must be placed, was first addressed by the French physicist Guillaume Amontons in 1703, in connection with his improvements in the air thermometer. His instrument indicated temperatures by the height at which a certain mass of air sustained a column of mercury—the pressure, or "spring" of the air varying with temperature. Amontons therefore argued that the zero of his thermometer would be that temperature at which the spring of the air was reduced to nothing. He used a scale that marked the boiling point of water at +73 and the melting point of ice at +, so that the zero was equivalent to about −240 on the Celsius scale. Amontons held that the absolute zero cannot be reached, so never attempted to compute it explicitly. |
The value of −240 °C, or "431 divisions [in Fahrenheit's thermometer] below the cold of freezing water" was published by George Martine in 1740. |
This close approximation to the modern value of −273.15 °C for the zero of the air thermometer was further improved upon in 1779 by Johann Heinrich Lambert, who observed that might be regarded as absolute cold. |
Values of this order for the absolute zero were not, however, universally accepted about this period. Pierre-Simon Laplace and Antoine Lavoisier, in their 1780 treatise on heat, arrived at values ranging from 1,500 to 3,000 below the freezing point of water, and thought that in any case it must be at least 600 below. John Dalton in his "Chemical Philosophy" gave ten calculations of this value, and finally adopted −3,000 °C as the natural zero of temperature. |
From 1787 to 1802, it was determined by Jacques Charles (unpublished), John Dalton, and Joseph Louis Gay-Lussac that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. This suggested that the volume of a gas cooled at about −273 °C would reach zero. |
After James Prescott Joule had determined the mechanical equivalent of heat, Lord Kelvin approached the question from an entirely different point of view, and in 1848 devised a scale of absolute temperature that was independent of the properties of any particular substance and was based on Carnot's theory of the Motive Power of Heat and data published by Henri Victor Regnault. It followed from the principles on which this scale was constructed that its zero was placed at −273 °C, at almost precisely the same point as the zero of the air thermometer, where the air volume would reach "nothing". This value was not immediately accepted; values ranging from to , derived from laboratory measurements and observations of astronomical refraction, remained in use in the early 20th century. |
The race to absolute zero. |
With a better theoretical understanding of absolute zero, scientists were eager to reach this temperature in the lab. By 1845, Michael Faraday had managed to liquefy most gases then known to exist, and reached a new record for lowest temperatures by reaching . Faraday believed that certain gases, such as oxygen, nitrogen, and hydrogen, were permanent gases and could not be liquefied. Decades later, in 1873 Dutch theoretical scientist Johannes Diderik van der Waals demonstrated that these gases could be liquefied, but only under conditions of very high pressure and very low temperatures. In 1877, Louis Paul Cailletet in France and Raoul Pictet in Switzerland succeeded in producing the first droplets of liquid air at . This was followed in 1883 by the production of liquid oxygen by the Polish professors Zygmunt Wróblewski and Karol Olszewski. |
Scottish chemist and physicist James Dewar and Dutch physicist Heike Kamerlingh Onnes took on the challenge to liquefy the remaining gases, hydrogen and helium. In 1898, after 20 years of effort, Dewar was the first to liquefy hydrogen, reaching a new low-temperature record of . However, Kamerlingh Onnes, his rival, was the first to liquefy helium, in 1908, using several precooling stages and the Hampson–Linde cycle. He lowered the temperature to the boiling point of helium . By reducing the pressure of the liquid helium, he achieved an even lower temperature, near 1.5 K. These were the coldest temperatures achieved on Earth at the time and his achievement earned him the Nobel Prize in 1913. Kamerlingh Onnes would continue to study the properties of materials at temperatures near absolute zero, describing superconductivity and superfluids for the first time. |
Very low temperatures. |
The average temperature of the universe today is approximately , or about −270.42 °C, based on measurements of cosmic microwave background radiation. Standard models of the future expansion of the universe predict that the average temperature of the universe is decreasing over time. This temperature is calculated as the mean density of energy in space; it should not be confused with the mean electron temperature (total energy divided by particle count) which has increased over time. |
Absolute zero cannot be achieved, although it is possible to reach temperatures close to it through the use of evaporative cooling, cryocoolers, dilution refrigerators, and nuclear adiabatic demagnetization. The use of laser cooling has produced temperatures of less than a billionth of a kelvin. At very low temperatures in the vicinity of absolute zero, matter exhibits many unusual properties, including superconductivity, superfluidity, and Bose–Einstein condensation. To study such phenomena, scientists have worked to obtain even lower temperatures. |
An adiabatic process (Greek: "adiábatos", "impassable") is a type of thermodynamic process |
that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, an adiabatic process transfers energy to the surroundings only as work. As a key concept in thermodynamics, the adiabatic process supports the theory that explains the first law of thermodynamics. |
Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation". For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings. |
In meteorology, adiabatic expansion and cooling of moist air, which can be triggered by winds flowing up and over a mountain for example, can cause the water vapor pressure to exceed the saturation vapor pressure. Expansion and cooling beyond the saturation vapor pressure is often idealized as a "pseudo-adiabatic process" whereby excess vapor instantaneously precipitates into water droplets. The change in temperature of an air undergoing pseudo-adiabatic expansion differs from air undergoing adiabatic expansion because latent heat is released by precipitation. |
A process without transfer of heat to or from a system, so that , is called adiabatic, and such a system is said to be adiabatically isolated. The simplifying assumption frequently made is that a process is adiabatic. For example, the compression of a gas within a cylinder of an engine is assumed to occur so rapidly that on the time scale of the compression process, little of the system's energy can be transferred out as heat to the surroundings. Even though the cylinders are not insulated and are quite conductive, that process is idealized to be adiabatic. The same can be said to be true for the expansion process of such a system. |
The assumption of adiabatic isolation is useful and often combined with other such idealizations to calculate a good first approximation of a system's behaviour. For example, according to Laplace, when sound travels in a gas, there is no time for heat conduction in the medium, and so the propagation of sound is adiabatic. For such an adiabatic process, the modulus of elasticity (Young's modulus) can be expressed as , where is the ratio of specific heats at constant pressure and at constant volume () and is the pressure of the gas. |
Various applications of the adiabatic assumption. |
For a closed system, one may write the first law of thermodynamics as , where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings. |
Naturally occurring adiabatic processes are irreversible (entropy is produced). |
The transfer of energy as work into an adiabatically isolated system can be imagined as being of two idealized extreme kinds. In one such kind, no entropy is produced within the system (no friction, viscous dissipation, etc.), and the work is only pressure-volume work (denoted by ). In nature, this ideal kind occurs only approximately because it demands an infinitely slow process and no sources of dissipation. |
The other extreme kind of work is isochoric work (), for which energy is added as work solely through friction or viscous dissipation within the system. A stirrer that transfers energy to a viscous fluid of an adiabatically isolated system with rigid walls, without phase change, will cause a rise in temperature of the fluid, but that work is not recoverable. Isochoric work is irreversible. The second law of thermodynamics observes that a natural process, of transfer of energy as work, always consists at least of isochoric work and often both of these extreme kinds of work. Every natural process, adiabatic or not, is irreversible, with , as friction or viscosity are always present to some extent. |
Adiabatic compression and expansion. |
The adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. In contrast, free expansion is an isothermal process for an ideal gas. |
Adiabatic compression occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. This finds practical application in diesel engines which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it. |
Adiabatic compression occurs in the Earth's atmosphere when an air mass descends, for example, in a Katabatic wind, Foehn wind, or Chinook wind flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Because of this increase in pressure, the parcel's volume decreases and its temperature increases as work is done on the parcel of air, thus increasing its internal energy, which manifests itself by a rise in the temperature of that mass of air. The parcel of air can only slowly dissipate the energy by conduction or radiation (heat), and to a first approximation it can be considered adiabatically isolated and the process an adiabatic process. |
Adiabatic expansion occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand in size, thus causing it to do work on its surroundings. When the pressure applied on a parcel of gas is reduced, the gas in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic expansion occurs in the Earth's atmosphere with orographic lifting and lee waves, and this can form pilei or lenticular clouds. |
Due in part to adiabatic expansion in mountainous areas, snowfall infrequently occurs in some parts of the Sahara desert. |
Adiabatic expansion does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is via adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic expansion. Also, the contents of an expanding universe can be described (to first order) as an adiabatically expanding fluid. (See heat death of the universe.) |
Rising magma also undergoes adiabatic expansion before eruption, particularly significant in the case of magmas that rise quickly from great depths such as kimberlites. |
In the Earth's convecting mantle (the asthenosphere) beneath the lithosphere, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth. |
Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes. |
In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist. |
Ideal gas (reversible process). |
The mathematical equation for an ideal gas undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the polytropic process equation |
where is pressure, is volume, and is the adiabatic index or heat capacity ratio defined as |
Here is the specific heat for constant pressure, is the specific heat for constant volume, and is the number of degrees of freedom (3 for a monatomic gas, 5 for a diatomic gas or a gas of linear molecules such as carbon dioxide). |
For a monatomic ideal gas, , and for a diatomic gas (such as nitrogen and oxygen, the main components of air), . Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not Bose–Einstein or Fermi gases. |
One can also use the ideal gas law to rewrite the above relationship between and as |
where "T" is the absolute or thermodynamic temperature. |
Example of adiabatic compression. |
The compression stroke in a gasoline engine can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure). |
so the adiabatic constant for this example is about 6.31 Pa m4.2. |
The gas is now compressed to a 0.1 L (0.0001 m3) volume, which we assume happens quickly enough that no heat enters or leaves the gas through the walls. The adiabatic constant remains the same, but with the resulting pressure unknown |
We can now solve for the final pressure |
or 25.1 bar. This pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas also increases its internal energy, which manifests itself by a rise in the gas temperature and an additional rise in pressure above what would result from a simplistic calculation of 10 times the original pressure. |
We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, "PV" = "nRT" ("n" is amount of gas in moles and "R" the gas constant for that gas). Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant ("nR") is: |
We know the compressed gas has = 0.1 L and = , so we can solve for temperature: |
That is a final temperature of 753 K, or 479 °C, or 896 °F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite (which would cause engine knocking when operated under these conditions of temperature and pressure), or that a supercharger with an intercooler to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas pressure, which ensures immediate ignition of the injected fuel. |
Adiabatic free expansion of a gas. |
For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. |
Derivation of "P"–"V" relation for adiabatic compression and expansion. |
The definition of an adiabatic process is that heat transfer to the system is zero, . Then, according to the first law of thermodynamics, |
where is the change in the internal energy of the system and is work done "by" the system. Any work () done must be done at the expense of internal energy , since no heat is being supplied from the surroundings. Pressure–volume work done "by" the system is defined as |
However, does not remain constant during an adiabatic process but instead changes along with . |
It is desired to know how the values of and relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law ) the internal energy is given by |
where is the number of degrees of freedom divided by 2, is the universal gas constant and is the number of moles in the system (a constant). |
Differentiating equation (a3) yields |
Equation (a4) is often expressed as because . |
Now substitute equations (a2) and (a4) into equation (a1) to obtain |
and divide both sides by : |
After integrating the left and right sides from to and from to and changing the sides respectively, |
Exponentiate both sides, substitute with , the heat capacity ratio |
and eliminate the negative sign to obtain |
At the same time, the work done by the pressure–volume changes as a result from this process, is equal to |
Since we require the process to be adiabatic, the following equation needs to be true |
By the previous derivation, |
Rearranging (b4) gives |
Substituting this into (b2) gives |
Integrating we obtain the expression for work, |
Substituting in second term, |
Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), |
By the continuous formula, |