Bose gas

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https://en.wikipedia.org/wiki/Bose_gas 

An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.

Introduction and examples

Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin. These particles can be classified as elementary: these are the Higgs boson, the photon, the gluon, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of ¹⁶O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in more complex systems can also be considered bosons like the plasmons (quanta of charge density waves).

The first model that treated a gas with several bosons, was the photon gas, a gas of photons, developed by Bose. This model leads to a better understanding of Planck's law and the black-body radiation. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The phonon gas, also known as Debye model, is an example where the normal modes of vibration of the crystal lattice of a metal, can be treated as effective massless bosons. Peter Debye used the phonon gas model to explain the behaviour of heat capacity of metals at low temperature.

An interesting example of a Bose gas is an ensemble of helium-4 atoms. When a system of ⁴He atoms is cooled down to temperature near absolute zero, many quantum mechanical effects are present. Below 2.17 kelvins, the ensemble starts to behave as a superfluid, a fluid with almost zero viscosity. The Bose gas is the most simple quantitative model that explains this phase transition. Mainly when a gas of bosons is cooled down, it forms a Bose–Einstein condensate, a state where a large number of bosons occupy the lowest energy, the ground state, and quantum effects are macroscopically visible like wave interference.

The theory of Bose-Einstein condensates and Bose gases can also explain some features of superconductivity where charge carriers couple in pairs (Cooper pairs) and behave like bosons. As a result, superconductors behave like having no electrical resistivity at low temperatures.

The equivalent model for half-integer particles (like electrons or helium-3 atoms), that follow Fermi–Dirac statistics, is called the Fermi gas (an ensemble of non-interacting fermions). At low enough particle number density and high temperature, both the Fermi gas and the Bose gas behave like a classical ideal gas.

Macroscopic limit

The thermodynamics of an ideal Bose gas is best calculated using the grand canonical ensemble. The grand potential for a Bose gas is given by:

${\displaystyle \mathrm{\Omega }=-\ln (\mathcal{Z})=\sum _i{g}_i\ln \left(1-z{e}^{-\beta {ϵ}_i}\right).}$

where each term in the sum corresponds to a particular single-particle energy level ε_i ; g_i  is the number of states with energy ε_i ; z  is the absolute activity (or "fugacity"), which may also be expressed in terms of the chemical potential μ by defining:

${\displaystyle z(\beta ,\mu )=e^{\beta \mu }}$

and β defined as:

${\displaystyle \beta =\frac{1}{k_{\mathrm{B}}T}}$

where k_B  is Boltzmann's constant and T  is the temperature. All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variables z , β (or T ), and V . All partial derivatives are taken with respect to one of these three variables while the other two are held constant.

The permissible range of z is from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0).

Macroscopic limit, result for uncondensed fraction

Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas, Bose gas) in three dimensions. The Bose gas pressure is lower than an equivalent classical gas, especially below the critical temperature (marked with ★) where particles begin moving en masse into the zero-pressure condensed phase.

Following the procedure described in the gas in a box article, we can apply the Thomas–Fermi approximation which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral. This replacement gives the macroscopic grand potential function ${\displaystyle {\mathrm{\Omega }}_m}$, which is close to ${\displaystyle \mathrm{\Omega }}$:

${\displaystyle {\mathrm{\Omega }}_{\mathrm{m}}={\int }_0^{\mathrm{\infty }}\ln \left(1-z{e}^{-\beta E}\right)dg\approx \mathrm{\Omega }.}$

The degeneracy dg  may be expressed for many different situations by the general formula:

${\displaystyle dg=\frac{1}{\mathrm{\Gamma }(\alpha )}\frac{E^{\alpha -1}}{E_{\mathrm{c}}^{\alpha }}dE}$

where α is a constant, E_c is a critical energy, and Γ is the Gamma function. For example, for a massive Bose gas in a box, α=3/2 and the critical energy is given by:

${\displaystyle \frac{1}{(\beta E_{\mathrm{c}}{)}^{\alpha }}=\frac{Vf}{{\mathrm{\Lambda }}^3}}$

where Λ is the thermal wavelength, and f is a degeneracy factor (f=1 for simple spinless bosons). For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by:

${\displaystyle \frac{1}{(\beta E_c{)}^{\alpha }}=\frac{f}{(\hslash \omega \beta {)}^3}}$

where V(r)=mω²r²/2  is the harmonic potential. It is seen that E_c  is a function of volume only.

This integral expression for the grand potential evaluates to:

${\displaystyle {\mathrm{\Omega }}_{\mathrm{m}}=-\frac{{\text{Li}}_{\alpha +1}(z)}{{\left(\beta E_c\right)}^{\alpha }},}$

where Li_s(x) is the polylogarithm function.

The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the Bose–Einstein condensate and will be dealt with in the next sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the un-condensed portion of the gas.

Limit on number of particles in uncondensed phase, critical temperature

The total number of particles is found from the grand potential by

${\displaystyle N_{\mathrm{m}}=-z\frac{\mathrm{\partial }{\mathrm{\Omega }}_m}{\mathrm{\partial }z}=\frac{{\text{Li}}_{\alpha }(z)}{(\beta E_c{)}^{\alpha }}.}$

This increases monotonically with z (up to the maximum z = +1). The behaviour when approaching z = 1 is however crucially dependent on the value of α (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).

For α > 1, the number of particles only increases up to a finite maximum value, i.e., ${\displaystyle N_{\mathrm{m}}}$ is finite at z = 1:

${\displaystyle N_{\mathrm{m},\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{\zeta (\alpha )}{(\beta E_{\mathrm{c}}{)}^{\alpha }},}$

where ζ(α) is the Riemann zeta function (using Li_α(1) = ζ(α)). Thus, for a fixed number of particles ${\displaystyle N_{\mathrm{m}}}$, the largest possible value that β can have is a critical value β_c. This corresponds to a critical temperature T_c=1/k_Bβ_c, below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature:

${\displaystyle T_{\mathrm{c}}={\left(\frac{N}{\zeta (\alpha )}\right)}^{1/\alpha }\frac{E_{\mathrm{c}}}{k_{\mathrm{B}}}}$

For example, for the three-dimensional Bose gas in a box (${\displaystyle \alpha =3/2}$ and using the above noted value of $E_{\mathrm{c}}$) we get:

${\displaystyle T_{\mathrm{c}}={\left(\frac{N}{Vf\zeta (3/2)}\right)}^{2/3}\frac{h^2}{2\pi m{k}_{\mathrm{B}}}}$

For α ≤ 1, there is no upper limit on the number of particles (${\displaystyle N_{\mathrm{m}}}$ diverges as z approaches 1), and thus for example for a gas in a one- or two-dimensional box (${\displaystyle \alpha =1/2}$ and ${\displaystyle \alpha =1}$ respectively) there is no critical temperature.

Inclusion of the ground state

The above problem raises the question for α > 1: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? The problem here is that the Thomas–Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so particles simply 'disappear' from the continuum of states. It turns out, however, that the macroscopic equation gives an accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term to accept the particles that fall out of the continuum:

${\displaystyle N=N_0+N_{\mathrm{m}}=N_0+\frac{{\text{Li}}_{\alpha }(z)}{(\beta E_{\mathrm{c}}{)}^{\alpha }}}$

where N₀ is the number of particles in the ground state condensate.

Thus in the macroscopic limit, when T < T_c, the value of z is pinned to 1 and N₀ takes up the remainder of particles. For T > T_c there is the normal behaviour, with N₀ = 0. This approach gives the fraction of condensed particles in the macroscopic limit:

${\displaystyle \frac{N_0}{N}=\{\begin{array}{ll}1-{\left(\frac{T}{T_{\mathrm{c}}}\right)}^{\alpha }& \text{if }\alpha >1\text{ and }T<T_{\mathrm{c}},\\ 0& \text{otherwise}.\end{array}}$

Limitations of the macroscopic Bose gas model

The above standard treatment of a macroscopic Bose gas is straight-forward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow a geometric distribution, meaning that when condensation happens at T < T_c and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the compressibility becomes unbounded for T < T_c. Calculations can instead be performed in the canonical ensemble, which fixes the total particle number, however the calculations are not as easy.

Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite). See the article Bose–Einstein condensate.

Approximate behaviour in small gases

Figure 1: Various Bose gas parameters as a function of normalized temperature τ. The value of α is 3/2. Solid lines are for N=10,000, dotted lines are for N=1000. Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential μ is shown in red, and green lines are the values of z. It has been assumed that k =ε_c=1.

For smaller, mesoscopic, systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy ε=0 in the grand potential:

${\displaystyle \mathrm{\Omega }=g_0\ln (1-z)+{\mathrm{\Omega }}_{\mathrm{m}}}$

which gives instead ${\displaystyle N_0=\frac{g_{0}z}{1-z}}$. Now, the behaviour is smooth when crossing the critical temperature, and z approaches 1 very closely but does not reach it.

This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for α=3/2, with k=ε_c=1 which corresponds to a gas of bosons in a box. The solid black line is the fraction of excited states 1-N₀/N  for N =10,000 and the dotted black line is the solution for N =1000. The blue lines are the fraction of condensed particles N₀/N  The red lines plot values of the negative of the chemical potential μ and the green lines plot the corresponding values of z . The horizontal axis is the normalized temperature τ defined by

${\displaystyle \tau =\frac{T}{T_{\mathrm{c}}}}$

It can be seen that each of these parameters become linear in τ^α in the limit of low temperature and, except for the chemical potential, linear in 1/τ^α in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.

The equation for the number of particles can be written in terms of the normalized temperature as:

${\displaystyle N=\frac{g_{0}z}{1-z}+N\frac{{\text{Li}}_{\alpha }(z)}{\zeta (\alpha )}{\tau }^{\alpha }}$

For a given N  and τ, this equation can be solved for τ^α and then a series solution for z  can be found by the method of inversion of series, either in powers of τ^α or as an asymptotic expansion in inverse powers of τ^α. From these expansions, we can find the behavior of the gas near T =0 and in the Maxwell–Boltzmann as T  approaches infinity. In particular, we are interested in the limit as N  approaches infinity, which can be easily determined from these expansions.

This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number, which is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.

Thermodynamics

Expanded out, the grand potential is:

${\displaystyle \mathrm{\Omega }=g_0\ln (1-z)-\frac{{\text{Li}}_{\alpha +1}(z)}{{\left(\beta E_{\mathrm{c}}\right)}^{\alpha }}}$

All thermodynamic properties can be computed from this potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in ${\displaystyle {\tau }^{\alpha }}$ is shown.

Quantity	General	${\displaystyle T\ll T_c}$	${\displaystyle T\gg T_c}$
${\displaystyle z}$		${\displaystyle \approx \frac{\zeta (\alpha )}{{\tau }^{\alpha }}-\frac{{\zeta }^2(\alpha )}{2^{\alpha }{\tau }^{2\alpha }}}$	${\displaystyle =1}$
Vapor fraction ${\displaystyle 1-\frac{N_0}{N}}$	${\displaystyle =\frac{{\text{Li}}_{\alpha }(z)}{\zeta (\alpha )}{\tau }^{\alpha }}$	${\displaystyle ={\tau }^{\alpha }}$	${\displaystyle =1}$
Equation of state ${\displaystyle \frac{PV\beta }{N}=-\frac{\mathrm{\Omega }}{N}}$	${\displaystyle =\frac{{\text{Li}}_{\alpha +1}(z)}{\zeta (\alpha )}{\tau }^{\alpha }}$	${\displaystyle =\frac{\zeta (\alpha +1)}{\zeta (\alpha )}{\tau }^{\alpha }}$	${\displaystyle \approx 1-\frac{\zeta (\alpha )}{2^{\alpha +1}{\tau }^{\alpha }}}$
Gibbs Free Energy ${\displaystyle G=\ln (z)}$	${\displaystyle =\ln (z)}$	${\displaystyle =0}$	${\displaystyle \approx \ln \left(\frac{\zeta (\alpha )}{{\tau }^{\alpha }}\right)-\frac{\zeta (\alpha )}{2^{\alpha }{\tau }^{\alpha }}}$

It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures:

${\displaystyle U=\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }\beta }=\alpha PV}$

A similar situation holds for the specific heat at constant volume

${\displaystyle C_V=\frac{\mathrm{\partial }U}{\mathrm{\partial }T}=k_{\mathrm{B}}(\alpha +1)U\beta }$

The entropy is given by:

${\displaystyle TS=U+PV-G}$

Note that in the limit of high temperature, we have

${\displaystyle TS=(\alpha +1)+\ln \left(\frac{{\tau }^{\alpha }}{\zeta (\alpha )}\right)}$

which, for α=3/2 is simply a restatement of the Sackur–Tetrode equation. In one dimension bosons with delta interaction behave as fermions, they obey Pauli exclusion principle. In one dimension Bose gas with delta interaction can be solved exactly by Bethe ansatz. The bulk free energy and thermodynamic potentials were calculated by Chen-Ning Yang. In one dimensional case correlation functions also were evaluated. In one dimension Bose gas is equivalent to quantum non-linear Schrödinger equation.

Carnot cycle

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Carnot_cycle

A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodynamic engine during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference through the application of work to the system.

In a Carnot cycle, a system or engine transfers energy in the form of heat between two thermal reservoirs at temperatures ${\displaystyle T_H}$ and ${\displaystyle T_C}$ (referred to as the hot and cold reservoirs, respectively), and a part of this transferred energy is converted to the work done by the system. The cycle is reversible, and there is no generation of entropy. (In other words, entropy is conserved; entropy is only transferred between the thermal reservoirs and the system without gain or loss of it.) When work is applied to the system, heat moves from the cold to hot reservoir (heat pump or refrigeration). When heat moves from the hot to the cold reservoir, the system applies work to the environment. The work ${\displaystyle W}$ done by the system or engine to the environment per Carnot cycle depends on the temperatures of the thermal reservoirs and the entropy transferred from the hot reservoir to the system ${\displaystyle \mathrm{\Delta }S}$ per cycle such as ${\displaystyle W=(T_H-T_C)\mathrm{\Delta }S=(T_H-T_C)\frac{Q_H}{T_H}}$, where ${\displaystyle Q_H}$ is heat transferred from the hot reservoir to the system per cycle.

Stages

A Carnot cycle as an idealized thermodynamic cycle performed by a heat engine (Carnot heat engine) consists of the following steps.

1. 

   Isothermal expansion. Heat (as an energy) is transferred reversibly from hot temperature reservoir at constant temperature T_H to the gas at temperature infinitesimally less than T_H (to allow heat transfer to the gas without practically changing the gas temperature so isothermal heat addition or absorption). During this step (1 to 2 on Figure 1, A to B in Figure 2), the gas is thermally in contact with the hot temperature reservoir (while thermally isolated from the cold temperature reservoir) and the gas is allowed to expand, doing work on the surroundings by gas pushing up the piston (stage 1 figure, right). Although the pressure drops from points 1 to 2 (figure 1) the temperature of the gas does not change during the process because the heat transferred from the hot temperature reservoir to the gas is exactly used to do work on the surroundings by the gas, so no gas internal energy changes (no gas temperature change for an ideal gas). Heat Q_H > 0 is absorbed from the hot temperature reservoir, resulting in an increase in the entropy ${\displaystyle S}$ of the gas by the amount ${\displaystyle \mathrm{\Delta }{S}_H=Q_H/T_H}$.

2. 

   Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (2 to 3 on Figure 1, B to C in Figure 2) the gas in the engine is thermally insulated from both the hot and cold reservoirs, thus they neither gain nor lose heat, an 'adiabatic' process. The gas continues to expand with reduction of its pressure, doing work on the surroundings (raising the piston; stage 2 figure, right), and losing an amount of internal energy equal to the work done. The gas expansion without heat input causes the gas to cool to the "cold" temperature (by losing its internal energy), that is infinitesimally higher than the cold reservoir temperature T_C. The entropy remains unchanged as no heat Q transfers (Q = 0) between the system (the gas) and its surroundings, so an isentropic process, meaning no entropy change in the process).

3. 

   Isothermal compression. Heat transferred reversibly to low temperature reservoir at constant temperature T_C (isothermal heat rejection). In this step (3 to 4 on Figure 1, C to D on Figure 2), the gas in the engine is in thermal contact with the cold reservoir at temperature T_C (while thermally isolated from the hot temperature reservoir) and the gas temperature is infinitesimally higher than this temperature (to allow heat transfer from the gas to the cold reservoir without practically changing the gas temperature). The surroundings do work on the gas, pushing the piston down (stage 3 figure, right). An amount of energy earned by the gas from this work exactly transfers as a heat energy Q_C < 0 (negative as leaving from the system, according to the universal convention in thermodynamics) to the cold reservoir so the entropy of the system decreases by the amount ${\displaystyle \mathrm{\Delta }{S}_C=Q_C/T_C}$. ${\displaystyle \mathrm{\Delta }{S}_C<0}$ because the isothermal compression decreases the multiplicity of the gas.

4. 

   Isentropic compression. (4 to 1 on Figure 1, D to A on Figure 2) Once again the gas in the engine is thermally insulated from the hot and cold reservoirs, and the engine is assumed to be frictionless and the process is slow enough, hence reversible. During this step, the surroundings do work on the gas, pushing the piston down further (stage 4 figure, right), increasing its internal energy, compressing it, and causing its temperature to rise back to the temperature infinitesimally less than T_H due solely to the work added to the system, but the entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.

Figure 1: A Carnot cycle illustrated on a PV diagram to illustrate the work done. 1-to-2 (isothermal expansion), 2-to-3 (isentropic expansion), 3-to-4 (isothermal compression), 4-to-1 (isentropic compression).

In this case, since it is a reversible thermodynamic cycle (no net change in the system and its surroundings per cycle)

$${\displaystyle \mathrm{\Delta }{S}_H+\mathrm{\Delta }{S}_C=\mathrm{\Delta }{S}_{\text{cycle}}=0,}$$

or,

$${\displaystyle \frac{Q_H}{T_H}=-\frac{Q_C}{T_C}.}$$

This is true as ${\displaystyle Q_C}$ and ${\displaystyle T_C}$ are both smaller in magnitude and in fact are in the same ratio as ${\displaystyle Q_H/T_H}$.

The pressure–volume graph

When a Carnot cycle is plotted on a pressure–volume diagram (Figure 1), the isothermal stages follow the isotherm lines for the working fluid, the adiabatic stages move between isotherms, and the area bounded by the complete cycle path represents the total work that can be done during one cycle. From point 1 to 2 and point 3 to 4 the temperature is constant (isothermal process). Heat transfer from point 4 to 1 and point 2 to 3 are equal to zero (adiabatic process).

Properties and significance

The temperature–entropy diagram

Main article: Temperature–entropy diagram

Figure 2: A Carnot cycle as an idealized thermodynamic cycle performed by a heat engine (Carnot heat engine), illustrated on a TS (temperature T–entropy S) diagram. The cycle takes place between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. The vertical axis is the system temperature, the horizontal axis is the system entropy. A-to-B (isothermal expansion), B-to-C (isentropic expansion), C-to-D (isothermal compression), D-to-A (isentropic compression).

Figure 3: A generalized thermodynamic cycle taking place between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. By the second law of thermodynamics, the cycle cannot extend outside the temperature band from T_C to T_H. The area in red, |Q_C|, is the amount of energy exchanged between the system and the cold reservoir. The area in white, W, is the amount of work energy exchanged by the system with its surroundings. The amount of heat exchanged with the hot reservoir is the sum of the two. If the system is behaving as an engine, the process moves clockwise around the loop, and moves counter-clockwise if it is behaving as a refrigerator. The efficiency to the cycle is the ratio of the white area (work) divided by the sum of the white and red areas (heat absorbed from the hot reservoir).
Q _C (energy lost to the cold reservoir) can be seen as a direct subtraction, or expressed as the sum of a negative quantity, which can lead to different conventions.

The behavior of a Carnot engine or refrigerator is best understood by using a temperature–entropy diagram (T–S diagram), in which the thermodynamic state is specified by a point on a graph with entropy (S) as the horizontal axis and temperature (T) as the vertical axis (Figure 2). For a simple closed system (control mass analysis), any point on the graph represents a particular state of the system. A thermodynamic process is represented by a curve connecting an initial state (A) and a final state (B). The area under the curve is:

${\displaystyle Q={\int }_A^{B}dQ={\int }_A^{B}TdS}$

(1)

which is the amount heat transferred in the process. If the process moves the system to greater entropy, the area under the curve is the amount of heat absorbed by the system in that process; otherwise, it is the amount of heat removed from or leaving from the system. For any cyclic process, there is an upper portion of the cycle and a lower portion. In T-S diagrams for a clockwise cycle, the area under the upper portion will be the energy absorbed by the system during the cycle, while the area under the lower portion will be the energy removed from the system during the cycle. The area inside the cycle is then the difference between the two (the absorbed net heat energy), but since the internal energy of the system must have returned to its initial value, this difference must be the amount of work done by the system per cycle. Referring to Figure 1, mathematically, for a reversible process, we may write the amount of work done over a cyclic process as:

${\displaystyle W=\oint PdV=\oint (dQ-dU)=\oint (TdS-dU)=\oint TdS-\oint dU=\oint TdS}$

(2)

Since dU is an exact differential, its integral over any closed loop is zero and it follows that the area inside the loop on a T–S diagram is equal to the total work performed by the system on the surroundings if the loop is traversed in a clockwise direction, and is equal to the total work done on the system by the surroundings as the loop is traversed in a counterclockwise direction.

Figure 4: A Carnot cycle taking place between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C.

The Carnot cycle

Figure 5: A visualization of a Carnot cycle

Evaluation of the above integral is particularly simple for a Carnot cycle. The amount of energy transferred as work is

$${\displaystyle W=\oint PdV=\oint TdS=(T_H-T_C)(S_B-S_A)}$$

The total amount of heat transferred from the hot reservoir to the system (in the isothermal expansion) will be

$${\displaystyle Q_H=T_H(S_B-S_A)=T_H\mathrm{\Delta }{S}_H}$$

and the total amount of heat transferred from the system to the cold reservoir (in the isothermal compression) will be

$${\displaystyle Q_C=T_C(S_A-S_B)=T_C\mathrm{\Delta }{S}_C<0}$$

Due to energy conservation, the net heat transferred, ${\displaystyle Q}$, is equal to the work performed

$${\displaystyle W=Q=Q_H+Q_C}$$

The efficiency ${\displaystyle \eta }$ is defined to be:

${\displaystyle \eta =\frac{W}{Q_H}=\frac{Q_H+Q_C}{Q_H}=1-\frac{T_C}{T_H}}$

(3)

where

• W is the work done by the system (energy exiting the system as work),
• ${\displaystyle Q_C}$ < 0 is the heat taken from the system (heat energy leaving the system),
• ${\displaystyle Q_H}$ > 0 is the heat put into the system (heat energy entering the system),
• ${\displaystyle T_C}$ is the absolute temperature of the cold reservoir, and
• ${\displaystyle T_H}$ is the absolute temperature of the hot reservoir.
• ${\displaystyle S_B}$ is the maximum system entropy
• ${\displaystyle S_A}$ is the minimum system entropy

The expression with the temperature ${\displaystyle \eta =1-\frac{T_C}{T_H}}$ can be derived from the expressions above with the entropy: ${\displaystyle Q_H=T_H(S_B-S_A)=T_H\mathrm{\Delta }{S}_H}$ and ${\displaystyle Q_C=T_C(S_A-S_B)=T_C\mathrm{\Delta }{S}_C<0}$. Since ${\displaystyle \mathrm{\Delta }{S}_C=S_A-S_B=-\mathrm{\Delta }{S}_H}$, a minus sign appears in the final expression for ${\displaystyle \eta }$.

This is the Carnot heat engine working efficiency definition as the fraction of the work done by the system to the thermal energy received by the system from the hot reservoir per cycle. This thermal energy is the cycle initiator.

Reversed Carnot cycle

A Carnot heat-engine cycle described is a totally reversible cycle. That is, all the processes that compose it can be reversed, in which case it becomes the Carnot heat pump and refrigeration cycle. This time, the cycle remains exactly the same except that the directions of any heat and work interactions are reversed. Heat is absorbed from the low-temperature reservoir, heat is rejected to a high-temperature reservoir, and a work input is required to accomplish all this. The P–V diagram of the reversed Carnot cycle is the same as for the Carnot heat-engine cycle except that the directions of the processes are reversed.

Carnot's theorem

Main article: Carnot's theorem (thermodynamics)

It can be seen from the above diagram that for any cycle operating between temperatures ${\displaystyle T_H}$ and ${\displaystyle T_C}$, none can exceed the efficiency of a Carnot cycle.

Figure 6: A real engine (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a T–S diagram. For this figure, the curve indicates a vapor-liquid equilibrium (See Rankine cycle). Irreversible systems and losses of energy (for example, work due to friction and heat losses) prevent the ideal from taking place at every step.

Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir. Looking at this formula an interesting fact becomes apparent: Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature.

In other words, the maximum efficiency is achieved if and only if entropy does not change per cycle. An entropy change per cycle is made, for example, if there is friction leading to dissipation of work into heat. In that case, the cycle is not reversible and the Clausius theorem becomes an inequality rather than an equality. Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a (minimal) reduction in efficiency. So Equation 3 gives the efficiency of any reversible heat engine.

In mesoscopic heat engines, work per cycle of operation in general fluctuates due to thermal noise. If the cycle is performed quasi-statically, the fluctuations vanish even on the mesoscale. However, if the cycle is performed faster than the relaxation time of the working medium, the fluctuations of work are inevitable. Nevertheless, when work and heat fluctuations are counted, an exact equality relates the exponential average of work performed by any heat engine to the heat transfer from the hotter heat bath.

Efficiency of real heat engines

See also: Heat Engine § Efficiency

Carnot realized that, in reality, it is not possible to build a thermodynamically reversible engine. So, real heat engines are even less efficient than indicated by Equation 3. In addition, real engines that operate along the Carnot cycle style (isothermal expansion / isentropic expansion / isothermal compression / isentropic compression) are rare. Nevertheless, Equation 3 is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.

Although Carnot's cycle is an idealization, Equation 3 as the expression of the Carnot efficiency is still useful. Consider the average temperatures,

$${\displaystyle ⟨T_H⟩=\frac{1}{\mathrm{\Delta }S}{\int }_{Q_{\text{in}}}TdS}$$

$${\displaystyle ⟨T_C⟩=\frac{1}{\mathrm{\Delta }S}{\int }_{Q_{\text{out}}}TdS}$$

at which the first integral is over a part of a cycle where heat goes into the system and the second integral is over a cycle part where heat goes out from the system. Then, replace T_H and T_C in Equation 3 by ⟨T_H⟩ and ⟨T_C⟩, respectively, to estimate the efficiency a heat engine.

For the Carnot cycle, or its equivalent, the average value ⟨T_H⟩ will equal the highest temperature available, namely T_H, and ⟨T_C⟩ the lowest, namely T_C. For other less efficient thermodynamic cycles, ⟨T_H⟩ will be lower than T_H, and ⟨T_C⟩ will be higher than T_C. This can help illustrate, for example, why a reheater or a regenerator can improve the thermal efficiency of steam power plants and why the thermal efficiency of combined-cycle power plants (which incorporate gas turbines operating at even higher temperatures) exceeds that of conventional steam plants. The first prototype of the diesel engine was based on the Carnot cycle.

Carnot heat engine as an impractical macroscopic construct

A Carnot heat engine is a heat engine performing a Carnot cycle, and its realization on a macroscopic scale is impractical. For example, for the isothermal expansion part of the Carnot cycle, the following conditions must be satisfied simultaneously at every step in the expansion:

• The hot reservoir temperature T_H is infinitesimally higher than the system gas temperature T so heat flow (energy transfer) from the hot reservoir to the gas is made without increasing T (via infinitesimal work on the surroundings by the gas as another energy transfer); if T_H is significantly higher than T, then T may be not uniform through the gas so the system would deviate from thermal equilibrium as well as not being a reversible process (i.e. not a Carnot cycle) or T might increase noticeably so it would not be an isothermal process.
• The force externally applied on the piston (opposite to the internal force on the piston by the gas) needs to be infinitesimally reduced somehow. Without this external assistance, it would not be possible to follow a gas PV (Pressure-Volume) curve downward at a constant T since following this curve means that the gas-to-piston force decreases (P decreases) as the volume expands (the piston moves outward). If this assistance is so strong that the volume expansion is significant, the system may deviate from thermal equilibrium as well as not being a reversible process (i.e. not a Carnot cycle).

These (and other) "infinitesimal" requirements make the Carnot cycle take an infinite amount of time. Other practical requirements that make the Carnot cycle hard to realize (e.g., fine control of the gas, thermal contact with the surroundings including high and low temperature reservoirs), so the Carnot engine should be thought as the theoretical limit of macroscopic scale heat engines rather than a practical device that could ever be built.