Thermodynamics  

The classical Carnot heat engine  


Book:Thermodynamics  
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics.
Thermodynamic temperature is defined by the third law of thermodynamics in which the theoretically lowest temperature is the null or zero point. At this point, absolute zero, the particle constituents of matter have minimal motion and can become no colder.^{[1]}^{[2]} In the quantummechanical description, matter at absolute zero is in its ground state, which is its state of lowest energy. Thermodynamic temperature is often also called absolute temperature, for two reasons: one, proposed by Kelvin, that it does not depend on the properties of a particular material; two that it refers to an absolute zero according to the properties of the ideal gas.
The International System of Units specifies a particular scale for thermodynamic temperature. It uses the kelvin scale for measurement and selects the triple point of water at 273.16 K as the fundamental fixing point. Other scales have been in use historically. The Rankine scale, using the degree Fahrenheit as its unit interval, is still in use as part of the English Engineering Units in the United States in some engineering fields. ITS90 gives a practical means of estimating the thermodynamic temperature to a very high degree of accuracy.
Roughly, the temperature of a body at rest is a measure of the mean of the energy of the translational, vibrational and rotational motions of matter's particle constituents, such as molecules, atoms, and subatomic particles. The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, make up the total internal energy of a substance. Internal energy is loosely called the heat energy or thermal energy in conditions when no work is done upon the substance by its surroundings, or by the substance upon the surroundings. Internal energy may be stored in a number of ways within a substance, each way constituting a "degree of freedom". At equilibrium, each degree of freedom will have on average the same energy: where is the Boltzmann constant, unless that degree of freedom is in the quantum regime. The internal degrees of freedom (rotation, vibration, etc.) may be in the quantum regime at room temperature, but the translational degrees of freedom will be in the classical regime except at extremely low temperatures (fractions of kelvins) and it may be said that, for most situations, the thermodynamic temperature is specified by the average translational kinetic energy of the particles.
Temperature is a measure of the random submicroscopic motions and vibrations of the particle constituents of matter. These motions comprise the internal energy of a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy per classical (i.e., nonquantum) degree of freedom of its constituent particles. "Translational motions" are almost always in the classical regime. Translational motions are ordinary, wholebody movements in threedimensional space in which particles move about and exchange energy in collisions. Figure 1 below shows translational motion in gases; Figure 4 below shows translational motion in solids. Thermodynamic temperature's null point, absolute zero, is the temperature at which the particle constituents of matter are as close as possible to complete rest; that is, they have minimal motion, retaining only quantum mechanical motion.^{[3]} Zero kinetic energy remains in a substance at absolute zero (see Thermal energy at absolute zero, below).
Throughout the scientific world where measurements are made in SI units, thermodynamic temperature is measured in kelvins (symbol: K). Many engineering fields in the U.S. however, measure thermodynamic temperature using the Rankine scale.
By international agreement, the unit kelvin and its scale are defined by two points: absolute zero, and the triple point of Vienna Standard Mean Ocean Water (water with a specified blend of hydrogen and oxygen isotopes). Absolute zero, the lowest possible temperature, is defined as being precisely 0 K and −273.15 °C. The triple point of water is defined as being precisely 273.16 K and 0.01 °C. This definition does three things:
Temperatures expressed in kelvins (T_{K}) are converted to degrees Rankine (T_{°R}) simply by multiplying by 1.8 (T_{°R} = 1.8 × T_{K}). Temperatures expressed in degrees Rankine are converted to kelvins by dividing by 1.8 (T_{K} = T_{°R} ÷ 1.8).
Although the kelvin and Celsius scales are defined using absolute zero (0 K) and the triple point of water (273.16 K and 0.01 °C), it is impractical to use this definition at temperatures that are very different from the triple point of water. ITS90 is then designed to represent the thermodynamic temperature as closely as possible throughout its range. Many different thermometer designs are required to cover the entire range. These include helium vapor pressure thermometers, helium gas thermometers, standard platinum resistance thermometers (known as SPRTs, PRTs or Platinum RTDs) and monochromatic radiation thermometers.
For some types of thermometer the relationship between the property observed (e.g., length of a mercury column) and temperature, is close to linear, so for most purposes a linear scale is sufficient, without pointbypoint calibration. For others a calibration curve or equation is required. The mercury thermometer, invented before the thermodynamic temperature was understood, originally defined the temperature scale; its linearity made readings correlate well with true temperature, i.e. the "mercury" temperature scale was a close fit to the true scale.
The thermodynamic temperature is a measure of the average energy of the translational, vibrational, and rotational motions of matter's particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, contribute the total internal energy (loosely, the thermal energy) of a substance. Thus, internal energy may be stored in a number of ways (degrees of freedom) within a substance. When the degrees of freedom are in the classical regime ("unfrozen") the temperature is very simply related to the average energy of those degrees of freedom at equilibrium. The three translational degrees of freedom are unfrozen except for the very lowest temperatures, and their kinetic energy is simply related to the thermodynamic temperature over the widest range. The heat capacity, which relates heat input and temperature change, is discussed below.
The relationship of kinetic energy, mass, and velocity is given by the formula E_{k} = ^{1}⁄_{2}mv^{2}.^{[4]} Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity.
Except in the quantum regime at extremely low temperatures, the thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three x, y, and z–axis dimensions of space means the particles move in the three spatial degrees of freedom. The temperature derived from this translational kinetic energy is sometimes referred to as kinetic temperature and is equal to the thermodynamic temperature over a very wide range of temperatures. Since there are three translational degrees of freedom (e.g., motion along the x, y, and z axes), the translational kinetic energy is related to the kinetic temperature by:
where:
While the Boltzmann constant is useful for finding the mean kinetic energy of a particle, it's important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occur across a wide range of speeds (see animation in Figure 1 above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in Fig. 2 shows the speed distribution of 5500 K helium atoms. They have a most probable speed of 4.780 km/s. However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the x–axis to the right). This graph uses inverse speed for its x–axis so the shape of the curve can easily be compared to the curves in Figure 5 below. In both graphs, zero on the x–axis represents infinite temperature. Additionally, the x and y–axis on both graphs are scaled proportionally.
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast^{[5]} and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a recordsetting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool caesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second in order to calculate their temperature.^{[6]} Formulas for calculating the velocity and speed of translational motion are given in the following footnote.^{[7]}
There are other forms of internal energy besides the kinetic energy of translational motion. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements. These are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom. Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an internal temperature. Even though these motions are called internal, the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the twoway exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as energy is removed from molecules, both their kinetic temperature (the temperature derived from the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active (i.e. unfrozen) degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of nonlocal thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.
The kinetic energy stored internally in molecules causes substances to contain more internal energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions is not at that same instant contributing to the molecules' translational motions.^{[8]} This extra thermal energy simply increases the amount of energy a substance absorbs for a given temperature rise. This property is known as a substance's specific heat capacity.
Different molecules absorb different amounts of thermal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, nitrogen, which is a diatomic molecule, has five active degrees of freedom at room temperature: the three comprising translational motion plus two rotational degrees of freedom internally. Since the two internal degrees of freedom are essentially unfrozen, in accordance with the equipartition theorem, nitrogen has fivethirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases.^{[9]} Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of thermal energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.
Heat conduction is the diffusion of thermal energy from hot parts of a system to cold. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term bulk in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever thermal energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).
One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more.^{[10]}
Translational motion in solids, however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets that travel at a given substance's speed of sound. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phononbased heat conduction is usually inefficient^{[11]} and such solids are considered thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.
Metals however, are not restricted to only phononbased heat conduction. Thermal energy conducts through metals extraordinarily quickly because instead of direct moleculetomolecule collisions, the vast majority of thermal energy is mediated via very light, mobile conduction electrons. This is why there is a nearperfect correlation between metals' thermal conductivity and their electrical conductivity.^{[12]} Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of zeropoint energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only ^{1}⁄_{1836}th that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his third law of motion,
Law #3: All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.
However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they're delocalized and very fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.
Thermal radiation is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal photons (known as blackbody radiation). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even individual molecules with internal temperatures greater than absolute zero also emit blackbody radiation from their atoms. In any bulk quantity of a substance at equilibrium, blackbody photons are emitted across a range of wavelengths in a spectrum that has a bell curvelike shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve (the peak emittance wavelength) is located in a particular part of the electromagnetic spectrum depending on the temperature of the blackbody. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see Table of common temperatures).
Blackbody radiation diffuses thermal energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Blackbody photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.
As established by the Stefan–Boltzmann law, the intensity of blackbody radiation increases as the fourth power of absolute temperature. Thus, a blackbody at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, blackbody radiation can be the principal mechanism by which thermal energy escapes a system.
The full range of the thermodynamic temperature scale, from absolute zero to absolute hot, and some notable points between them are shown in the table below.
kelvin  Peak emittance wavelength^{[13]} of blackbody photons  
Absolute zero (precisely by definition) 
0 K  ∞ ^{[3]} 
Coldest measured temperature^{ }^{[14]} 
450 pK  6,400 kilometers 
One millikelvin (precisely by definition) 
0.001 K  2.897 77 meters (Radio, FM band)^{[15]} 
Cosmic Microwave Background Radiation  2.725 48(57) K  1.063 mm (peak wavelength) 
Water's triple point (precisely by definition) 
273.16 K  10,608.3 nm (Long wavelength I.R.) 
Incandescent lamp^{B}  2500 K  1160 nm (Near infrared)^{C} 
Sun’s visible surface^{C}^{[16]}  5778 K  501.5 nm (Green light) 
Lightning bolt’s channel 
28,000 K  100 nm (Far Ultraviolet light) 
Sun’s core  16 MK  0.18 nm (Xrays) 
Thermonuclear explosion (peak temperature)^{[17]} 
350 MK  8.3 × 10^{−3} nm (Gamma rays) 
Sandia National Labs’ Z machine^{ D}^{[18]} 
2 GK  1.4 × 10^{−3} nm (Gamma rays) 
Core of a high–mass star on its last day^{[19]} 
3 GK  1 × 10^{−3} nm (Gamma rays) 
Merging binary neutron star system ^{[20]} 
350 GK  8 × 10^{−6} nm (Gamma rays) 
Gammaray burst progenitors^{[21]} 
1 TK  3 × 10^{−6} nm (Gamma rays) 
Relativistic Heavy Ion Collider^{[22]} 
1 TK  3 × 10^{−6} nm (Gamma rays) 
CERN’s proton vs. nucleus collisions^{[23]} 
10 TK  3 × 10^{−7} nm (Gamma rays) 
Universe 5.391 × 10^{−44} s after the Big Bang 
1.417 × 10^{32} K  1.616 × 10^{−26} nm (Planck frequency)^{[24]} 
^{A} The 2500 K value is approximate.
^{B} For a true blackbody (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.
^{C} Effective photosphere temperature.
^{D} For a true blackbody (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.
The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is phase transitions, which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The thermal energy required for a phase transition is called latent heat. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to break chemical bonds (such as during evaporation and melting). Almost everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.
Even though thermal energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right). Consider one particular type of phase transition: melting. When a solid is melting, crystal lattice chemical bonds are being broken apart; the substance is transitioning from what is known as a more ordered state to a less ordered state. In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green.
At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are allornothing forces: they either hold fast, or break; there is no inbetween state. Consequently, when a substance is at its melting point, every joule of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules,^{[25]} converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional thermal energy can't make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance.
As stated above, the thermal energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it's called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements.^{[26]} If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole.^{[27]} Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the thermal energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times.^{[28]} And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a onedegree increase.^{[29]}
Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above). In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a subambient wetbulb temperature that is dependent on relative humidity). Water's highly energetic enthalpy of vaporization is also an important factor underlying why solar pool covers (floating, insulated blankets that cover swimming pools when not in use) are so effective at reducing heating costs: they prevent evaporation. For instance, the evaporation of just 20 mm of water from a 1.29meterdeep pool chills its water 8.4 degrees Celsius (15.1 °F).
The total energy of all particle motion translational and internal, including that of conduction electrons, plus the potential energy of phase changes, plus zeropoint energy^{[3]} comprise the internal energy of a substance.
As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic temperature decreases); the internal motions of molecules diminish (their internal temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower;^{[30]} and blackbody radiation's peak emittance wavelength increases (the photons' energy decreases). When the particles of a substance are as close as possible to complete rest and retain only ZPEinduced quantum mechanical motion, the substance is at the temperature of absolute zero (T=0).
Note that whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero thermal energy; one must be very precise with what one means by internal energy. Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, T=0 helium remains liquid at room pressure and must be under a pressure of at least 25 bar (2.5 MPa) to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motioninducing effect of zeropoint energy is sufficient to prevent it from freezing at lower pressures. Only if under at least 25 bar (2.5 MPa) of pressure will this latent thermal energy be liberated as helium freezes while approaching absolute zero. A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as solidsolid phase transitions wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.
The above complexities make for rather cumbersome blanket statements regarding the internal energy in T=0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowestenergy crystal lattice such those with a closestpacked arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the everpresent background of zeropoint energy.^{[3]}^{ }^{[31]} One can also say that for a given substance at constant pressure, absolute zero is the point of lowest enthalpy (a measure of work potential that takes internal energy, pressure, and volume into consideration).^{[32]} Lastly, it is always true to say that all T=0 substances contain zero kinetic thermal energy.^{[3]}^{ }^{[7]}
Thermodynamic temperature is useful not only for scientists, it can also be useful for laypeople in many disciplines involving gases. By expressing variables in absolute terms and applying Gay–Lussac's law of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a relatively cold pressure of 200 kPagage , then in absolute terms (relative to a vacuum), its pressure is 300 kPaabsolute.^{[33]}^{[34]}^{[35]} Room temperature ("cold" in tire terms) is 296 K. If the tire pressure is 20 °C hotter (20 kelvins), the solution is calculated as ^{316 K}⁄_{296 K} = 6.8% greater thermodynamic temperature and absolute pressure; that is, a pressure of 320 kPaabsolute, which is 220 kPagage.
The thermodynamic temperature is defined by the ideal gas law and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratio T_{2}/T_{1} of two temperaturesT_{1} andT_{2} is the same in all absolute scales.
Strictly speaking, the temperature of a system is welldefined only if it is at thermal equilibrium. From a microscopic viewpoint, a material is at thermal equilibrium if the quantity of heat between its individual particles cancel out. There are many possible scales of temperature, derived from a variety of observations of physical phenomena.
Loosely stated, temperature differences dictate the direction of heat between two systems such that their combined energy is maximally distributed among their lowest possible states. We call this distribution "entropy". To better understand the relationship between temperature and entropy, consider the relationship between heat, work and temperature illustrated in the Carnot heat engine. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, T_{H}, and a lower temperature heat sync, T_{C}, through a gas filled piston. The work done per cycle is equal to the difference between the heat supplied to the engine by T_{H}, q_{H}, and the heat supplied to T_{C} by the engine, q_{C}. The efficiency of the engine is the work divided by the heat put into the system or
where w_{cy} is the work done per cycle. Thus the efficiency depends only on q_{C}/q_{H}.
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T_{1} and T_{2} must have the same efficiency, that is to say, the efficiency is the function of only temperatures
In addition, a reversible heat engine operating between temperatures T_{1} and T_{3} must have the same efficiency as one consisting of two cycles, one between T_{1} and another (intermediate) temperature T_{2}, and the second between T_{2} andT_{3}. If this were not the case, then energy (in the form of Q) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles.
With this understanding of Q_{1}, Q_{2} and Q_{3}, we note also that mathematically,
But the first function is NOT a function of T_{2}, therefore the product of the final two functions MUST result in the removal of T_{2} as a variable. The only way is therefore to define the function f as follows:
and
so that
i.e. The ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our ; it is a matter of convenience and convention that we choose . Choosing then one fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale.
It is to be noted that such a definition coincides with that of the ideal gas derivation; also it is this definition of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of T_{H} and T_{C}, and hence derive that the (complete) Carnot cycle is isentropic:
Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:
Notice that for T_{C}=0 the efficiency is 100% and that efficiency becomes greater than 100% for T_{C}<0, which cases are unrealistic. Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives
where the negative sign indicates heat ejected from the system. The generalization of this equation is Clausius theorem, which suggests the existence of a state function S (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by
where the subscript indicates heat transfer in a reversible process. The function S corresponds to the entropy of the system, mentioned previously, and the change of S around any cycle is zero (as is necessary for any state function). Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid logic loop, we should first define entropy through statistical mechanics):
For a system in which the entropy S is a function S(E) of its energy E, the thermodynamic temperature T is therefore given by
so that the reciprocal of the thermodynamic temperature is the rate of increase of entropy with energy.
Although absolute zero (T=0) is not a state of zero molecular motion, it is the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two T=0 helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620 pm between them (a gap of ten atomic diameters). It's an "average" separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvins (1 yK = ×10^{−24} K). This is done in a way that directs this atom's velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620 pm gap through their common 1barycenter would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the Fig. 1 animation above. Before being given the kinetic kick, both T=0 atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At T=0, no kinetic energy is available for transfer to other systems. The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of chaos, i.e., unpredictability, to rebound kinetics; it is as likely that there will be less ZPEinduced particle motion after a given collision as more. This random nature of ZPE is why it has no net effect upon either the pressure or volume of any bulk quantity (a statistically significant quantity of particles) of T>0 K gases. However, in T=0 condensed matter; e.g., solids and liquids, ZPE causes interatomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the realworld effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won't freeze unless under a pressure of at least 25 bar or 2.5 MPa), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy.
Note too that absolute zero serves as the baseline atop which thermodynamics and its equations are founded because they deal with the exchange of thermal energy between "systems" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPEinduced particle motion within a system that is at absolute zero but there can never be a net outflow of thermal energy from such a system. Also, the peak emittance wavelength of blackbody radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and blackbody photons can no longer escape. Because of ZPE, however, virtual photons are still emitted at T=0. Such photons are called "virtual" because they can't be intercepted and observed. Furthermore, this zeropoint radiation has a unique zeropoint spectrum. However, even though a T=0 system emits zeropoint radiation, no net heat flow Q out of such a system can occur because if the surrounding environment is at a temperature greater than T=0, heat will flow inward, and if the surrounding environment is at T=0, there will be an equal flux of ZP radiation both inward and outward. A similar Q equilibrium exists at T=0 with the ZPEinduced spontaneous emission of photons (which is more properly called a stimulated emission in this context). The graph at upper right illustrates the relationship of absolute zero to zeropoint energy. The graph also helps in the understanding of how zeropoint energy got its name: it is the vibrational energy matter retains at the zerokelvin point. Derivation of the classical electromagnetic zeropoint radiation spectrum via a classical thermodynamic operation involving van der Waals forces, Daniel C. Cole, Physical Review A, 42 (1990) 1847.