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From equations () and (), taking squared magnitudes, we find that the "reflectivity" (ratio of reflected power to incident power) is |
for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cos"θ", the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power "transmission" (below), these factors must be taken into account. |
The simplest way to obtain the power transmission coefficient ("transmissivity", the ratio of transmitted power to incident power "in the direction normal to the interface", i.e. the direction) is to use (conservation of energy). In this way we find |
In the case of an interface between two lossless media (for which ϵ and μ are "real" and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations () and (). But, for given amplitude (as noted above), the component of the Poynting vector in the direction is proportional to the geometric factor and inversely proportional to the wave impedance . Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient: |
for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, ). |
From equations () and (), we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have (that is, the transmitted ray is undeviated), so that the cosines in equations (), (), (), (), and () to () cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence. When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering. |
Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing () by ()) yields |
For non-magnetic media we can substitute the vacuum permeability "μ"0 for "μ", so that |
that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations () to () and equations () to (), the factor "cμ"0 cancels out. For the amplitude coefficients we obtain: |
For the case of normal incidence these reduce to: |
The power transmissions can then be found from . |
For equal permeabilities (e.g., non-magnetic media), if and are "complementary", we can substitute for and for so that the numerator in equation () becomes which is zero (by Snell's law). Hence and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting for in Snell's law, we readily obtain |
This switch of polarizations has an analog in the old mechanical theory of light waves (see "§History", above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different "densities" and that the vibrations were "normal" to what was then called the plane of polarization, or by supposing (like MacCullagh and Neumann) that different refractive indices were due to different "elasticities" and that the vibrations were "parallel" to that plane. Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest. |
The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. Entropy predicts the direction of spontaneous processes, and determines whether they are irreversible or impossible, despite obeying the requirement of conservation of energy, which is established in the first law of thermodynamics. The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest. If all processes in the system are reversible, the entropy is constant. |
An increase in entropy accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time. |
Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics provides a microscopic explanation of the law in terms of probability distributions of the states of large assemblies of atoms or molecules. |
The second law has been expressed in many ways. Its first formulation, which preceded the proper definition of entropy and was based on caloric theory, is Carnot's theorem, credited to the French scientist Sadi Carnot, who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit. The first rigorous definition of the second law based on the concept of entropy came from German scientist Rudolph Clausius in the 1850s including his statement that heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. |
The second law of thermodynamics can also be used to define the concept of thermodynamic temperature, but this is usually delegated to the zeroth law of thermodynamics. |
The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases. |
In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat: |
Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality |
This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings ("surr"). Note that the equality still applies for pure heat flow, |
which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. Introducing a set of internal variables formula_4 to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure "P" and temperature "T") from the chemical equilibrium state, one can record the equality |
The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of "the second principle of thermodynamics" – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy. |
The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body. |
The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent. |
In modern terms, Carnot's principle may be stated more precisely: |
The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the "Clausius statement": |
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. |
The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other. |
Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system. |
Lord Kelvin expressed the second law in several wordings. |
Equivalence of the Clausius and the Kelvin statements. |
Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law. |
Relation between Kelvin's statement and Planck's proposition. |
It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland. |
The Kelvin–Planck statement (or the "heat engine statement") of the second law of thermodynamics states that |
Planck stated the second law as follows. |
Rather like Planck's statement is that of Uhlenbeck and Ford for "irreversible phenomena". |
Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows: |
In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S. |
With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, formula_9. |
Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. |
In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle |
This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible." |
Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy. |
A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law: |
Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy. |
Statement for a system that has a known expression of its internal energy as a function of its extensive state variables. |
The second law has been shown to be equivalent to the internal energy "U" being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). |
Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines. |
Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states: |
In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realize the Carnot cycle's reversibility and was condemned to be less efficient. |
Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law. |
The Clausius theorem (1854) states that in a cyclic process |
The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality. |
For an arbitrary heat engine, the efficiency is: |
where "W"n is for the net 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 temperatures only: |
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 and"T"3. This can only be the case if |
Now consider the case where formula_12 is a fixed reference temperature: the temperature of the triple point of water. Then for any "T"2 and "T"3, |
Therefore, if thermodynamic temperature is defined by |
then the function "f", viewed as a function of thermodynamic temperature, is simply |
and the reference temperature "T"1 will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.) |
According to the Clausius equality, for a reversible process |
That means the line integral formula_17 is path independent for reversible processes. |
So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies |
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that "S" = 0 at absolute zero for perfect crystals. |
For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy. |
Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop, |
where the equality holds if the transformation is reversible. |
Notice that if the process is an adiabatic process, then formula_21, so formula_22. |
An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an "unlimited" heat reservoir at temperature "TR" and pressure "PR" so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain "TR"; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain "PR". |
Whatever changes to "dS" and "dSR" occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy "S"tot of the isolated total system must not decrease: |
According to the first law of thermodynamics, the change "dU" in the internal energy of the sub-system is the sum of the heat "δq" added to the sub-system, "less" any work "δw" done "by" the sub-system, "plus" any net chemical energy entering the sub-system "d" ∑"μiRNi", so that: |
where "μ""iR" are the chemical potentials of chemical species in the external surroundings. |
Now the heat leaving the reservoir and entering the sub-system is |
where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above. |
It therefore follows that any net work "δw" done by the sub-system must obey |
It is useful to separate the work "δw" done by the subsystem into the "useful" work "δwu" that can be done "by" the sub-system, over and beyond the work "pR dV" done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done: |
It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the "availability" or "exergy" "E" of the subsystem, |
The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact, |
i.e. the change in the subsystem's exergy plus the useful work done "by" the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done "on" the system) must be less than or equal to zero. |
In sum, if a proper "infinite-reservoir-like" reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in "E" for an irreversible process and no change for a reversible process. |
This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. ("Also, see process engineer"). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found ("See second law efficiency".) |
This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines. |
For a spontaneous chemical process in a closed system at constant temperature and pressure without non-"PV" work, the Clausius inequality Δ"S" > "Q/T"surr transforms into a condition for the change in Gibbs free energy |
or d"G" < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, formula_33. Thus, a negative value of the change in free energy ("G" or "A") is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant "T" and "p" without electrical work is d"G" = 0. |
The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment. |
Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow "spontaneously" from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865). |
Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius. |
The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same. |
There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent. |
In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form: |
where "Q" is heat, "T" is temperature and "N" is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes: |
The entropy of the universe tends to a maximum. |
This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description. |
In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is: |
The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is: |
with formula_38 the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature formula_39 it gives the so-called dissipated energy formula_40. |
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