title: First law of thermodynamics

text: For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate the tank thermally, and move the paddle wheel with a pulley and a weight, we can relate the increase in temperature with the distance descended by the mass. Next, the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring...). In every case, the amount of work can be measured independently. The return to the initial state is not conducted by doing adiabatic work on the system. The evidence shows that the final state of the water (in particular, its temperature and volume) is the same in every case. It is irrelevant if the work is electrical, mechanical, chemical... or if done suddenly or slowly, as long as it is performed in an adiabatic way, that is to say, without heat transfer into or out of the system.


title: First law of thermodynamics

text: Evidence of this kind shows that to increase the temperature of the water in the tank, the qualitative kind of adiabatically performed work does not matter. No qualitative kind of adiabatic work has ever been observed to decrease the temperature of the water in the tank.


title: First law of thermodynamics

text: A change from one state to another, for example an increase of both temperature and volume, may be conducted in several stages, for example by externally supplied electrical work on a resistor in the body, and adiabatic expansion allowing the body to do work on the surroundings. It needs to be shown that the time order of the stages, and their relative magnitudes, does not affect the amount of adiabatic work that needs to be done for the change of state. According to one respected scholar: "Unfortunately, it does not seem that experiments of this kind have ever been carried out carefully. ... We must therefore admit that the statement which we have enunciated here, and which is equivalent to the first law of thermodynamics, is not well founded on direct experimental evidence." Another expression of this view is "... no systematic precise experiments to verify this generalization directly have ever been attempted."


title: First law of thermodynamics

text: This kind of evidence, of independence of sequence of stages, combined with the above-mentioned evidence, of independence of qualitative kind of work, would show the existence of an important state variable that corresponds with adiabatic work, but not that such a state variable represented a conserved quantity. For the latter, another step of evidence is needed, which may be related to the concept of reversibility, as mentioned below.


title: First law of thermodynamics

text: That important state variable was first recognized and denoted formula_6 by Clausius in 1850, but he did not then name it, and he defined it in terms not only of work but also of heat transfer in the same process. It was also independently recognized in 1850 by Rankine, who also denoted it formula_6 ; and in 1851 by Kelvin who then called it "mechanical energy", and later "intrinsic energy". In 1865, after some hestitation, Clausius began calling his state function formula_6 "energy". In 1882 it was named as the "internal energy" by Helmholtz. If only adiabatic processes were of interest, and heat could be ignored, the concept of internal energy would hardly arise or be needed. The relevant physics would be largely covered by the concept of potential energy, as was intended in the 1847 paper of Helmholtz on the principle of conservation of energy, though that did not deal with forces that cannot be described by a potential, and thus did not fully justify the principle. Moreover, that paper was critical of the early work of Joule that had by then been performed. A great merit of the internal energy concept is that it frees thermodynamics from a restriction to cyclic processes, and allows a treatment in terms of thermodynamic states.


title: First law of thermodynamics

text: In an adiabatic process, adiabatic work takes the system either from a reference state formula_31 with internal energy formula_32 to an arbitrary one formula_33 with internal energy formula_34, or from the state formula_33 to the state formula_31:


title: First law of thermodynamics

text: Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes  formula_38   or  formula_39  is empirically feasible by a simple application of externally supplied work. The reason for this is given as the second law of thermodynamics and is not considered in the present article.


title: First law of thermodynamics

text: The fact of such irreversibility may be dealt with in two main ways, according to different points of view:


title: First law of thermodynamics

text: The formula () above allows that to go by processes of quasi-static adiabatic work from the state formula_33 to the state formula_41 we can take a path that goes through the reference state formula_31, since the quasi-static adiabatic work is independent of the path


title: First law of thermodynamics

text: This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:


title: First law of thermodynamics

text: A complementary observable aspect of the first law is about heat transfer. Adynamic transfer of energy as heat can be measured empirically by changes in the surroundings of the system of interest by calorimetry. This again requires the existence of adiabatic enclosure of the entire process, system and surroundings, though the separating wall between the surroundings and the system is thermally conductive or radiatively permeable, not adiabatic. A calorimeter can rely on measurement of sensible heat, which requires the existence of thermometers and measurement of temperature change in bodies of known sensible heat capacity under specified conditions; or it can rely on the measurement of latent heat, through measurement of masses of material that change phase, at temperatures fixed by the occurrence of phase changes under specified conditions in bodies of known latent heat of phase change. The calorimeter can be calibrated by transferring an externally determined amount of heat into it, for instance from a resistive electrical heater inside the calorimeter through which a precisely known electric current is passed at a precisely known voltage for a precisely measured period of time. The calibration allows comparison of calorimetric measurement of quantity of heat transferred with quantity of energy transferred as (surroundings-based) work. According to one textbook, "The most common device for measuring formula_2 is an adiabatic bomb calorimeter." According to another textbook, "Calorimetry is widely used in present day laboratories." According to one opinion, "Most thermodynamic data come from calorimetry...".


title: First law of thermodynamics

text: When the system evolves with transfer of energy as heat, without energy being transferred as work, in an adynamic process, the heat transferred to the system is equal to the increase in its internal energy:


title: First law of thermodynamics

text: Heat transfer is practically reversible when it is driven by practically negligibly small temperature gradients. Work transfer is practically reversible when it occurs so slowly that there are no frictional effects within the system; frictional effects outside the system should also be zero if the process is to be reversible in the strict thermodynamic sense. For a particular reversible process in general, the work done reversibly on the system, formula_47, and the heat transferred reversibly to the system, formula_48 are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path, formula_49, through the space of thermodynamic states. Then the work and heat transfers can occur and be calculated simultaneously.


title: First law of thermodynamics

text: Putting the two complementary aspects together, the first law for a particular reversible process can be written


title: First law of thermodynamics

text: This combined statement is the expression the first law of thermodynamics for reversible processes for closed systems.


title: First law of thermodynamics

text: If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient, practically frictionless, and with nearly balanced forces, then the process is irreversible. Then the heat and work transfers may be difficult to calculate with high accuracy, although the simple equations for reversible processes still hold to a good approximation in the absence of composition changes. Importantly, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system, formula_52, and the heat transferred irreversibly to the system, formula_53, which belong to the same particular process defined by its particular irreversible path, formula_54, through the space of thermodynamic states.


title: First law of thermodynamics

text: This means that the internal energy formula_6 is a function of state and that the internal energy change formula_2 between two states is a function only of the two states.


title: First law of thermodynamics

text: The first law of thermodynamics is so general that its predictions cannot all be directly tested. In many properly conducted experiments it has been precisely supported, and never violated. Indeed, within its scope of applicability, the law is so reliably established, that, nowadays, rather than experiment being considered as testing the accuracy of the law, it is more practical and realistic to think of the law as testing the accuracy of experiment. An experimental result that seems to violate the law may be assumed to be inaccurate or wrongly conceived, for example due to failure to account for an important physical factor. Thus, some may regard it as a principle more abstract than a law.


title: First law of thermodynamics

text: When the heat and work transfers in the equations above are infinitesimal in magnitude, they are often denoted by , rather than exact differentials denoted by , as a reminder that heat and work do not describe the "state" of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential is always zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process.


title: First law of thermodynamics

text: The first law for a closed homogeneous system may be stated in terms that include concepts that are established in the second law. The internal energy may then be expressed as a function of the system's defining state variables , entropy, and , volume: . In these terms, , the system's temperature, and , its pressure, are partial derivatives of with respect to and . These variables are important throughout thermodynamics, though not necessary for the statement of the first law. Rigorously, they are defined only when the system is in its own state of internal thermodynamic equilibrium. For some purposes, the concepts provide good approximations for scenarios sufficiently near to the system's internal thermodynamic equilibrium.


title: First law of thermodynamics

text: Then, for the fictive case of a reversible process, can be written in terms of exact differentials. One may imagine reversible changes, such that there is at each instant negligible departure from thermodynamic equilibrium within the system and between system and surroundings. Then, mechanical work is given by and the quantity of heat added can be expressed as . For these conditions


title: First law of thermodynamics

text: While this has been shown here for reversible changes, it is valid more generally in the absence of chemical reactions or phase transitions, as can be considered as a thermodynamic state function of the defining state variables and :


title: First law of thermodynamics

text: Equation () is known as the fundamental thermodynamic relation for a closed system in the energy representation, for which the defining state variables are and , with respect to which and are partial derivatives of . It is only in the reversible case or for a quasistatic process without composition change that the work done and heat transferred are given by and .


title: First law of thermodynamics

text: In the case of a closed system in which the particles of the system are of different types and, because chemical reactions may occur, their respective numbers are not necessarily constant, the fundamental thermodynamic relation for d"U" becomes:


title: First law of thermodynamics

text: where d"N" is the (small) increase in number of type-i particles in the reaction, and "μ" is known as the chemical potential of the type-i particles in the system. If d"N" is expressed in mol then "μ" is expressed in J/mol. If the system has more external mechanical variables than just the volume that can change, the fundamental thermodynamic relation further generalizes to:


title: First law of thermodynamics

text: Here the "X" are the generalized forces corresponding to the external variables "x". The parameters "X" are independent of the size of the system and are called intensive parameters and the "x" are proportional to the size and called extensive parameters.


title: First law of thermodynamics

text: For an open system, there can be transfers of particles as well as energy into or out of the system during a process. For this case, the first law of thermodynamics still holds, in the form that the internal energy is a function of state and the change of internal energy in a process is a function only of its initial and final states, as noted in the section below headed First law of thermodynamics for open systems.


title: First law of thermodynamics

text: A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: d"U" = −"P" d"V". The pressure "P" can be viewed as a force (and in fact has units of force per unit area) while d"V"is the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred out of the system as a result of the process. If one were to make this term negative then this would be the work done on the system.


title: First law of thermodynamics

text: It is useful to view the "T" d"S" term in the same light: here the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement.


title: First law of thermodynamics

text: Similarly, a difference in chemical potential between groups of particles in the system drives a chemical reaction that changes the numbers of particles, and the corresponding product is the amount of chemical potential energy transformed in process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation that drives water molecules out of the liquid. There is a generalized "force" of condensation that drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal is there equilibrium, and the net rate of transfer zero.


title: First law of thermodynamics

text: The two thermodynamic parameters that form a generalized force-displacement pair are called "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and temperature-entropy.


title: First law of thermodynamics

text: Classical thermodynamics is initially focused on closed homogeneous systems (e.g. Planck 1897/1903), which might be regarded as 'zero-dimensional' in the sense that they have no spatial variation. But it is desired to study also systems with distinct internal motion and spatial inhomogeneity. For such systems, the principle of conservation of energy is expressed in terms not only of internal energy as defined for homogeneous systems, but also in terms of kinetic energy and potential energies of parts of the inhomogeneous system with respect to each other and with respect to long-range external forces. How the total energy of a system is allocated between these three more specific kinds of energy varies according to the purposes of different writers; this is because these components of energy are to some extent mathematical artefacts rather than actually measured physical quantities. For any closed homogeneous component of an inhomogeneous closed system, if formula_63 denotes the total energy of that component system, one may write


title: First law of thermodynamics

text: where formula_65 and formula_66 denote respectively the total kinetic energy and the total potential energy of the component closed homogeneous system, and formula_6 denotes its internal energy.


title: First law of thermodynamics

text: Potential energy can be exchanged with the surroundings of the system when the surroundings impose a force field, such as gravitational or electromagnetic, on the system.


title: First law of thermodynamics

text: A compound system consisting of two interacting closed homogeneous component subsystems has a potential energy of interaction formula_68 between the subsystems. Thus, in an obvious notation, one may write


title: First law of thermodynamics

text: The quantity formula_68 in general lacks an assignment to either subsystem in a way that is not arbitrary, and this stands in the way of a general non-arbitrary definition of transfer of energy as work. On occasions, authors make their various respective arbitrary assignments.


title: First law of thermodynamics

text: The distinction between internal and kinetic energy is hard to make in the presence of turbulent motion within the system, as friction gradually dissipates macroscopic kinetic energy of localised bulk flow into molecular random motion of molecules that is classified as internal energy. The rate of dissipation by friction of kinetic energy of localised bulk flow into internal energy, whether in turbulent or in streamlined flow, is an important quantity in non-equilibrium thermodynamics. This is a serious difficulty for attempts to define entropy for time-varying spatially inhomogeneous systems.


title: First law of thermodynamics

text: For the first law of thermodynamics, there is no trivial passage of physical conception from the closed system view to an open system view. For closed systems, the concepts of an adiabatic enclosure and of an adiabatic wall are fundamental. Matter and internal energy cannot permeate or penetrate such a wall. For an open system, there is a wall that allows penetration by matter. In general, matter in diffusive motion carries with it some internal energy, and some microscopic potential energy changes accompany the motion. An open system is not adiabatically enclosed.


title: First law of thermodynamics

text: There are some cases in which a process for an open system can, for particular purposes, be considered as if it were for a closed system. In an open system, by definition hypothetically or potentially, matter can pass between the system and its surroundings. But when, in a particular case, the process of interest involves only hypothetical or potential but no actual passage of matter, the process can be considered as if it were for a closed system.


title: First law of thermodynamics

text: Since the revised and more rigorous definition of the internal energy of a closed system rests upon the possibility of processes by which adiabatic work takes the system from one state to another, this leaves a problem for the definition of internal energy for an open system, for which adiabatic work is not in general possible. According to Max Born, the transfer of matter and energy across an open connection "cannot be reduced to mechanics". In contrast to the case of closed systems, for open systems, in the presence of diffusion, there is no unconstrained and unconditional physical distinction between convective transfer of internal energy by bulk flow of matter, the transfer of internal energy without transfer of matter (usually called heat conduction and work transfer), and change of various potential energies. The older traditional way and the conceptually revised (Carathéodory) way agree that there is no physically unique definition of heat and work transfer processes between open systems.


title: First law of thermodynamics

text: In particular, between two otherwise isolated open systems an adiabatic wall is by definition impossible. This problem is solved by recourse to the principle of conservation of energy. This principle allows a composite isolated system to be derived from two other component non-interacting isolated systems, in such a way that the total energy of the composite isolated system is equal to the sum of the total energies of the two component isolated systems. Two previously isolated systems can be subjected to the thermodynamic operation of placement between them of a wall permeable to matter and energy, followed by a time for establishment of a new thermodynamic state of internal equilibrium in the new single unpartitioned system. The internal energies of the initial two systems and of the final new system, considered respectively as closed systems as above, can be measured. Then the law of conservation of energy requires that


title: First law of thermodynamics

text: where and denote the changes in internal energy of the system and of its surroundings respectively. This is a statement of the first law of thermodynamics for a transfer between two otherwise isolated open systems, that fits well with the conceptually revised and rigorous statement of the law stated above.


title: First law of thermodynamics

text: For the thermodynamic operation of adding two systems with internal energies and , to produce a new system with internal energy , one may write ; the reference states for , and should be specified accordingly, maintaining also that the internal energy of a system be proportional to its mass, so that the internal energies are extensive variables.


title: First law of thermodynamics

text: There is a sense in which this kind of additivity expresses a fundamental postulate that goes beyond the simplest ideas of classical closed system thermodynamics; the extensivity of some variables is not obvious, and needs explicit expression; indeed one author goes so far as to say that it could be recognized as a fourth law of thermodynamics, though this is not repeated by other authors.


title: First law of thermodynamics

text: where and denote the changes in mole number of a component substance of the system and of its surroundings respectively. This is a statement of the law of conservation of mass.


title: First law of thermodynamics

text: A system connected to its surroundings only through contact by a single permeable wall, but otherwise isolated, is an open system. If it is initially in a state of contact equilibrium with a surrounding subsystem, a thermodynamic process of transfer of matter can be made to occur between them if the surrounding subsystem is subjected to some thermodynamic operation, for example, removal of a partition between it and some further surrounding subsystem. The removal of the partition in the surroundings initiates a process of exchange between the system and its contiguous surrounding subsystem.


title: First law of thermodynamics

text: An example is evaporation. One may consider an open system consisting of a collection of liquid, enclosed except where it is allowed to evaporate into or to receive condensate from its vapor above it, which may be considered as its contiguous surrounding subsystem, and subject to control of its volume and temperature.


title: First law of thermodynamics

text: A thermodynamic process might be initiated by a thermodynamic operation in the surroundings, that mechanically increases in the controlled volume of the vapor. Some mechanical work will be done within the surroundings by the vapor, but also some of the parent liquid will evaporate and enter the vapor collection which is the contiguous surrounding subsystem. Some internal energy will accompany the vapor that leaves the system, but it will not make sense to try to uniquely identify part of that internal energy as heat and part of it as work. Consequently, the energy transfer that accompanies the transfer of matter between the system and its surrounding subsystem cannot be uniquely split into heat and work transfers to or from the open system. The component of total energy transfer that accompanies the transfer of vapor into the surrounding subsystem is customarily called 'latent heat of evaporation', but this use of the word heat is a quirk of customary historical language, not in strict compliance with the thermodynamic definition of transfer of energy as heat. In this example, kinetic energy of bulk flow and potential energy with respect to long-range external forces such as gravity are both considered to be zero. The first law of thermodynamics refers to the change of internal energy of the open system, between its initial and final states of internal equilibrium.


title: First law of thermodynamics

text: This includes cases in which there is contact equilibrium between the system, and several subsystems in its surroundings, including separate connections with subsystems through walls that are permeable to the transfer of matter and internal energy as heat and allowing friction of passage of the transferred matter, but immovable, and separate connections through adiabatic walls with others, and separate connections through diathermic walls impermeable to matter with yet others. Because there are physically separate connections that are permeable to energy but impermeable to matter, between the system and its surroundings, energy transfers between them can occur with definite heat and work characters. Conceptually essential here is that the internal energy transferred with the transfer of matter is measured by a variable that is mathematically independent of the variables that measure heat and work.


title: First law of thermodynamics

text: With such independence of variables, the total increase of internal energy in the process is then determined as the sum of the internal energy transferred from the surroundings with the transfer of matter through the walls that are permeable to it, and of the internal energy transferred to the system as heat through the diathermic walls, and of the energy transferred to the system as work through the adiabatic walls, including the energy transferred to the system by long-range forces. These simultaneously transferred quantities of energy are defined by events in the surroundings of the system. Because the internal energy transferred with matter is not in general uniquely resolvable into heat and work components, the total energy transfer cannot in general be uniquely resolved into heat and work components. Under these conditions, the following formula can describe the process in terms of externally defined thermodynamic variables, as a statement of the first law of thermodynamics:


title: First law of thermodynamics

text: where Δ"U" denotes the change of internal energy of the system, and denotes the change of internal energy of the of the surrounding subsystems that are in open contact with the system, due to transfer between the system and that surrounding subsystem, and denotes the internal energy transferred as heat from the heat reservoir of the surroundings to the system, and denotes the energy transferred from the system to the surrounding subsystems that are in adiabatic connection with it. The case of a wall that is permeable to matter and can move so as to allow transfer of energy as work is not considered here.


title: First law of thermodynamics

text: If the system is described by the energetic fundamental equation, "U" = "U"("S", "V", "N"), and if the process can be described in the quasi-static formalism, in terms of the internal state variables of the system, then the process can also be described by a combination of the first and second laws of thermodynamics, by the formula


title: First law of thermodynamics

text: where there are "n" chemical constituents of the system and permeably connected surrounding subsystems, and where "T", "S", "P", "V", "N", and "μ", are defined as above.


title: First law of thermodynamics

text: For a general natural process, there is no immediate term-wise correspondence between equations () and (), because they describe the process in different conceptual frames.


title: First law of thermodynamics

text: Nevertheless, a conditional correspondence exists. There are three relevant kinds of wall here: purely diathermal, adiabatic, and permeable to matter. If two of those kinds of wall are sealed off, leaving only one that permits transfers of energy, as work, as heat, or with matter, then the remaining permitted terms correspond precisely. If two of the kinds of wall are left unsealed, then energy transfer can be shared between them, so that the two remaining permitted terms do not correspond precisely.


title: First law of thermodynamics

text: For the special fictive case of quasi-static transfers, there is a simple correspondence. For this, it is supposed that the system has multiple areas of contact with its surroundings. There are pistons that allow adiabatic work, purely diathermal walls, and open connections with surrounding subsystems of completely controllable chemical potential (or equivalent controls for charged species). Then, for a suitable fictive quasi-static transfer, one can write


title: First law of thermodynamics

text: where formula_74 is the added amount of species formula_75 and formula_76 is the corresponding molar entropy.


title: First law of thermodynamics

text: For fictive quasi-static transfers for which the chemical potentials in the connected surrounding subsystems are suitably controlled, these can be put into equation (4) to yield


title: First law of thermodynamics

text: The transfer of energy between an open system and a single contiguous subsystem of its surroundings is considered also in non-equilibrium thermodynamics. The problem of definition arises also in this case. It may be allowed that the wall between the system and the subsystem is not only permeable to matter and to internal energy, but also may be movable so as to allow work to be done when the two systems have different pressures. In this case, the transfer of energy as heat is not defined.


title: First law of thermodynamics

text: The first law of thermodynamics for any process on the specification of equation (3) can be defined as


title: First law of thermodynamics

text: where Δ"U" denotes the change of internal energy of the system, denotes the internal energy transferred as heat from the heat reservoir of the surroundings to the system, denotes the work of the system and formula_77 is the molar enthalpy of species formula_75, coming into the system from the surrounding that is in contact with the system.


title: First law of thermodynamics

text: Formula (6) is valid in general case, both for quasi-static and for irreversible processes. The situation of the quasi-static process is considered in the previous Section, which in our terms defines


title: First law of thermodynamics

text: To describe deviation of the thermodynamic system from equilibrium, in addition to fundamental variables that are used to fix the equilibrium state, as was described above, a set of variables formula_81 that are called "internal variables" have been introduced, which allows to formulate for the general case


title: First law of thermodynamics

text: Methods for study of non-equilibrium processes mostly deal with spatially continuous flow systems. In this case, the open connection between system and surroundings is usually taken to fully surround the system, so that there are no separate connections impermeable to matter but permeable to heat. Except for the special case mentioned above when there is no actual transfer of matter, which can be treated as if for a closed system, in strictly defined thermodynamic terms, it follows that transfer of energy as heat is not defined. In this sense, there is no such thing as 'heat flow' for a continuous-flow open system. Properly, for closed systems, one speaks of transfer of internal energy as heat, but in general, for open systems, one can speak safely only of transfer of internal energy. A factor here is that there are often cross-effects between distinct transfers, for example that transfer of one substance may cause transfer of another even when the latter has zero chemical potential gradient.


title: First law of thermodynamics

text: Usually transfer between a system and its surroundings applies to transfer of a state variable, and obeys a balance law, that the amount lost by the donor system is equal to the amount gained by the receptor system. Heat is not a state variable. For his 1947 definition of "heat transfer" for discrete open systems, the author Prigogine carefully explains at some length that his definition of it does not obey a balance law. He describes this as paradoxical.


title: First law of thermodynamics

text: The situation is clarified by Gyarmati, who shows that his definition of "heat transfer", for continuous-flow systems, really refers not specifically to heat, but rather to transfer of internal energy, as follows. He considers a conceptual small cell in a situation of continuous-flow as a system defined in the so-called Lagrangian way, moving with the local center of mass. The flow of matter across the boundary is zero when considered as a flow of total mass. Nevertheless, if the material constitution is of several chemically distinct components that can diffuse with respect to one another, the system is considered to be open, the diffusive flows of the components being defined with respect to the center of mass of the system, and balancing one another as to mass transfer. Still there can be a distinction between bulk flow of internal energy and diffusive flow of internal energy in this case, because the internal energy density does not have to be constant per unit mass of material, and allowing for non-conservation of internal energy because of local conversion of kinetic energy of bulk flow to internal energy by viscosity.


title: First law of thermodynamics

text: Gyarmati shows that his definition of "the heat flow vector" is strictly speaking a definition of flow of internal energy, not specifically of heat, and so it turns out that his use here of the word heat is contrary to the strict thermodynamic definition of heat, though it is more or less compatible with historical custom, that often enough did not clearly distinguish between heat and internal energy; he writes "that this relation must be considered to be the exact definition of the concept of heat flow, fairly loosely used in experimental physics and heat technics." Apparently in a different frame of thinking from that of the above-mentioned paradoxical usage in the earlier sections of the historic 1947 work by Prigogine, about discrete systems, this usage of Gyarmati is consistent with the later sections of the same 1947 work by Prigogine, about continuous-flow systems, which use the term "heat flux" in just this way. This usage is also followed by Glansdorff and Prigogine in their 1971 text about continuous-flow systems. They write: "Again the flow of internal energy may be split into a convection flow and a conduction flow. This conduction flow is by definition the heat flow . Therefore: where denotes the [internal] energy per unit mass. [These authors actually use the symbols and to denote internal energy but their notation has been changed here to accord with the notation of the present article. These authors actually use the symbol to refer to total energy, including kinetic energy of bulk flow.]" This usage is followed also by other writers on non-equilibrium thermodynamics such as Lebon, Jou, and Casas-Vásquez, and de Groot and Mazur. This usage is described by Bailyn as stating the non-convective flow of internal energy, and is listed as his definition number 1, according to the first law of thermodynamics. This usage is also followed by workers in the kinetic theory of gases. This is not the "ad hoc" definition of "reduced heat flux" of Haase.


title: First law of thermodynamics

text: In the case of a flowing system of only one chemical constituent, in the Lagrangian representation, there is no distinction between bulk flow and diffusion of matter. Moreover, the flow of matter is zero into or out of the cell that moves with the local center of mass. In effect, in this description, one is dealing with a system effectively closed to the transfer of matter. But still one can validly talk of a distinction between bulk flow and diffusive flow of internal energy, the latter driven by a temperature gradient within the flowing material, and being defined with respect to the local center of mass of the bulk flow. In this case of a virtually closed system, because of the zero matter transfer, as noted above, one can safely distinguish between transfer of energy as work, and transfer of internal energy as heat.


title: P versus NP problem

text: The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.


title: P versus NP problem

text: The informal term "quickly", used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be "verified" in polynomial time is NP, which stands for "nondeterministic polynomial time".


title: P versus NP problem

text: An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turns out that P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time.


title: P versus NP problem

text: The problem has been called the most important open problem in computer science. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields.


title: P versus NP problem

text: It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution.


title: P versus NP problem

text: Consider Sudoku, a game where the player is given a partially filled-in grid of numbers and attempts to complete the grid following certain rules. Given an incomplete Sudoku grid, of any size, is there at least one legal solution? Any proposed solution is easily verified, and the time to check a solution grows slowly (polynomially) as the grid gets bigger. However, all known algorithms for finding solutions take, for difficult examples, time that grows exponentially as the grid gets bigger. So, Sudoku is in NP (quickly checkable) but does not seem to be in P (quickly solvable). Thousands of other problems seem similar, in that they are fast to check but slow to solve. Researchers have shown that many of the problems in NP have the extra property that a fast solution to any one of them could be used to build a quick solution to any other problem in NP, a property called NP-completeness. Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly. This, however, has never been proven.


title: P versus NP problem

text: The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" (and independently by Leonid Levin in 1973).


title: P versus NP problem

text: Although the P versus NP problem was formally defined in 1971, there were previous inklings of the problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA, in which he speculated that cracking a sufficiently complex code would require time exponential in the length of the key. If proved (and Nash was suitably skeptical), this would imply what is now called P ≠ NP, since a proposed key can easily be verified in polynomial time. Another mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether theorem-proving (now known to be co-NP-complete) could be solved in quadratic or linear time, and pointed out one of the most important consequences – that if so, then the discovery of mathematical proofs could be automated.


title: P versus NP problem

text: The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem).


title: P versus NP problem

text: In such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that the computer is "deterministic" (given the computer's present state and any inputs, there is only one possible action that the computer might take) and "sequential" (it performs actions one after the other).


title: P versus NP problem

text: In this theory, the class P consists of all those "decision problems" (defined below) that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine. Clearly, P ⊆ NP. Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes:


title: P versus NP problem

text: Since 2002, William Gasarch has conducted three polls of researchers concerning this and related questions. Confidence that P ≠ NP has been increasing – in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002. When restricted to experts, the 2019 answers became 99% believed P ≠ NP. These polls do not imply anything about whether P = NP is true, as stated by Gasarch himself: "This does not bring us any closer to solving P=?NP or to knowing when it will be solved, but it attempts to be an objective report on the subjective opinion of this era."


title: P versus NP problem

text: To attack the P = NP question, the concept of NP-completeness is very useful. NP-complete problems are a set of problems to each of which any other NP problem can be reduced in polynomial time and whose solution may still be verified in polynomial time. That is, any NP problem can be transformed into any of the NP-complete problems. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP.


title: P versus NP problem

text: NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time.


title: P versus NP problem

text: For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, so "any" instance of "any" problem in NP can be transformed mechanically into an instance of the Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many such NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. However, many important problems have been shown to be NP-complete, and no fast algorithm for any of them is known.


title: P versus NP problem

text: Based on the definition alone it is not obvious that NP-complete problems exist; however, a trivial and contrived NP-complete problem can be formulated as follows: given a description of a Turing machine "M" guaranteed to halt in polynomial time, does there exist a polynomial-size input that "M" will accept? It is in NP because (given an input) it is simple to check whether "M" accepts the input by simulating "M"; it is NP-complete because the verifier for any particular instance of a problem in NP can be encoded as a polynomial-time machine "M" that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists.


title: P versus NP problem

text: The first natural problem proven to be NP-complete was the Boolean satisfiability problem, also known as SAT. As noted above, this is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete, including the game Sudoku discussed earlier. In this case, the proof shows that a solution of Sudoku in polynomial time could also be used to complete Latin squares in polynomial time. This in turn gives a solution to the problem of partitioning tri-partite graphs into triangles, which could then be used to find solutions for the special case of SAT known as 3-SAT, which then provides a solution for general Boolean satisfiability. So a polynomial-time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-problem in polynomial time. Using transformations like this, a vast class of seemingly unrelated problems are all reducible to one another, and are in a sense "the same problem".


title: P versus NP problem

text: Although it is unknown whether P = NP, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have "exponential" running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O(2) time, where "p"("n") is a polynomial function of "n". A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it. A number of problems are known to be EXPTIME-complete. Because it can be shown that P ≠ EXPTIME, these problems are outside P, and so require more than polynomial time. In fact, by the time hierarchy theorem, they cannot be solved in significantly less than exponential time. Examples include finding a perfect strategy for chess positions on an "N" × "N" board and similar problems for other board games.


title: P versus NP problem

text: The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer and Rabin proved in 1974 that every algorithm that decides the truth of Presburger statements of length "n" has a runtime of at least formula_1 for some constant "c". Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all.


title: P versus NP problem

text: It is also possible to consider questions other than decision problems. One such class, consisting of counting problems, is called #P: whereas an NP problem asks "Are there any solutions?", the corresponding #P problem asks "How many solutions are there?". Clearly, a #P problem must be at least as hard as the corresponding NP problem, since a count of solutions immediately tells if at least one solution exists, if the count is greater than zero. Surprisingly, some #P problems that are believed to be difficult correspond to easy (for example linear-time) P problems. For these problems, it is very easy to tell whether solutions exist, but thought to be very hard to tell how many. Many of these problems are #P-complete, and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems.


title: P versus NP problem

text: In 1975, Richard E. Ladner showed that if P ≠ NP, then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem, and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.


title: P versus NP problem

text: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai, runs in quasi-polynomial time.


title: P versus NP problem

text: The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than "k". No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP = co-NP). The most efficient known algorithm for integer factorization is the general number field sieve, which takes expected time


title: P versus NP problem

text: to factor an "n"-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time, although this does not indicate where the problem lies with respect to non-quantum complexity classes.


title: P versus NP problem

text: All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as "Cobham's thesis". It is a common and reasonably accurate assumption in complexity theory; however, it has some caveats.


title: P versus NP problem

text: First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, thus rendering it impractical. For example, the problem of deciding whether a graph "G" contains "H" as a minor, where "H" is fixed, can be solved in a running time of "O"("n"), where "n" is the number of vertices in "G". However, the big O notation hides a constant that depends superexponentially on "H". The constant is greater than formula_3 (using Knuth's up-arrow notation), and where "h" is the number of vertices in "H".


title: P versus NP problem

text: On the other hand, even if a problem is shown to be NP-complete, and even if P ≠ NP, there may still be effective approaches to tackling the problem in practice. There are algorithms for many NP-complete problems, such as the knapsack problem, the traveling salesman problem, and the Boolean satisfiability problem, that can solve to optimality many real-world instances in reasonable time. The empirical average-case complexity (time vs. problem size) of such algorithms can be surprisingly low. An example is the simplex algorithm in linear programming, which works surprisingly well in practice; despite having exponential worst-case time complexity, it runs on par with the best known polynomial-time algorithms.


title: P versus NP problem

text: Finally, there are types of computations which do not conform to the Turing machine model on which P and NP are defined, such as quantum computation and randomized algorithms.


title: P versus NP problem

text: Cook provides a restatement of the problem in "The P Versus NP Problem" as "Does P = NP?" According to polls, most computer scientists believe that P ≠ NP. A key reason for this belief is that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than 3000 important known NP-complete problems (see List of NP-complete problems). These algorithms were sought long before the concept of NP-completeness was even defined (Karp's 21 NP-complete problems, among the first found, were all well-known existing problems at the time they were shown to be NP-complete). Furthermore, the result P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NP and P = PH.


title: P versus NP problem

text: It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience.


title: P versus NP problem

text: On the other hand, some researchers believe that there is overconfidence in believing P ≠ NP and that researchers should explore proofs of P = NP as well. For example, in 2002 these statements were made:


title: P versus NP problem

text: One of the reasons the problem attracts so much attention is the consequences of the possible answers. Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well.


title: P versus NP problem

text: A proof that P = NP could have stunning practical consequences if the proof leads to efficient methods for solving some of the important problems in NP. The potential consequences, both positive and negative, arise since various NP-complete problems are fundamental in many fields.


title: P versus NP problem

text: It is also very possible that a proof would "not" lead to practical algorithms for NP-complete problems. The formulation of the problem does not require that the bounding polynomial be small or even specifically known. A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice. In this case the initial proof would be mainly of interest to theoreticians, but the knowledge that polynomial time solutions are possible would surely spur research into better (and possibly practical) methods to achieve them.


title: P versus NP problem

text: An example of a field that could be upended by a solution showing P = NP is cryptography, which relies on certain problems being difficult. A constructive and efficient solution to an NP-complete problem such as 3-SAT would break most existing cryptosystems including:


title: P versus NP problem

text: These would need to be modified or replaced by information-theoretically secure solutions not inherently based on P–NP inequivalence.


title: P versus NP problem

text: On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as some types of integer programming and the travelling salesman problem. Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in protein structure prediction, are also NP-complete; if these problems were efficiently solvable, it could spur considerable advances in life sciences and biotechnology.


title: P versus NP problem

text: But such changes may pale in significance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. Gödel, in his early thoughts on computational complexity, noted that a mechanical method that could solve any problem would revolutionize mathematics:


title: P versus NP problem

text: Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove. A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle.


title: P versus NP problem

text: Donald Knuth has stated that he has come to believe that P = NP, but is reserved about the impact of a possible proof:


title: P versus NP problem

text: A proof showing that P ≠ NP would lack the practical computational benefits of a proof that P = NP, but would nevertheless represent a very significant advance in computational complexity theory and provide guidance for future research. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place.


title: P versus NP problem

text: Also, P ≠ NP still leaves open the average-case complexity of hard problems in NP. For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable. Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question. These range from "Algorithmica", where P = NP and problems like SAT can be solved efficiently in all instances, to "Cryptomania", where P ≠ NP and generating hard instances of problems outside P is easy, with three intermediate possibilities reflecting different possible distributions of difficulty over instances of NP-hard problems. The "world" where P ≠ NP but all problems in NP are tractable in the average case is called "Heuristica" in the paper. A Princeton University workshop in 2009 studied the status of the five worlds.


title: P versus NP problem

text: Although the P = NP problem itself remains open despite a million-dollar prize and a huge amount of dedicated research, efforts to solve the problem have led to several new techniques. In particular, some of the most fruitful research related to the P = NP problem has been in showing that existing proof techniques are not powerful enough to answer the question, thus suggesting that novel technical approaches are required.


title: P versus NP problem

text: As additional evidence for the difficulty of the problem, essentially all known proof techniques in computational complexity theory fall into one of the following classifications, each of which is known to be insufficient to prove that P ≠ NP:


title: P versus NP problem

text: These barriers are another reason why NP-complete problems are useful: if a polynomial-time algorithm can be demonstrated for an NP-complete problem, this would solve the P = NP problem in a way not excluded by the above results.


title: P versus NP problem

text: These barriers have also led some computer scientists to suggest that the P versus NP problem may be independent of standard axiom systems like ZFC (cannot be proved or disproved within them). The interpretation of an independence result could be that either no polynomial-time algorithm exists for any NP-complete problem, and such a proof cannot be constructed in (e.g.) ZFC, or that polynomial-time algorithms for NP-complete problems may exist, but it is impossible to prove in ZFC that such algorithms are correct. However, if it can be shown, using techniques of the sort that are currently known to be applicable, that the problem cannot be decided even with much weaker assumptions extending the Peano axioms (PA) for integer arithmetic, then there would necessarily exist nearly polynomial-time algorithms for every problem in NP. Therefore, if one believes (as most complexity theorists do) that not all problems in NP have efficient algorithms, it would follow that proofs of independence using those techniques cannot be possible. Additionally, this result implies that proving independence from PA or ZFC using currently known techniques is no easier than proving the existence of efficient algorithms for all problems in NP.


title: P versus NP problem

text: While the P versus NP problem is generally considered unsolved, many amateur and some professional researchers have claimed solutions. Gerhard J. Woeginger compiled a list of 62 purported proofs of P = NP from 1986 to 2016, of which 50 were proofs of P ≠ NP, 2 were proofs the problem is unprovable, and one was a proof that it is undecidable. Some attempts at resolving P versus NP have received brief media attention, though these attempts have since been refuted.


title: P versus NP problem

text: The P = NP problem can be restated in terms of expressible certain classes of logical statements, as a result of work in descriptive complexity.


title: P versus NP problem

text: Consider all languages of finite structures with a fixed signature including a linear order relation. Then, all such languages in P can be expressed in first-order logic with the addition of a suitable least fixed-point combinator. Effectively, this, in combination with the order, allows the definition of recursive functions. As long as the signature contains at least one predicate or function in addition to the distinguished order relation, so that the amount of space taken to store such finite structures is actually polynomial in the number of elements in the structure, this precisely characterizes P.


title: P versus NP problem

text: Similarly, NP is the set of languages expressible in existential second-order logic—that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets. The languages in the polynomial hierarchy, PH, correspond to all of second-order logic. Thus, the question "is P a proper subset of NP" can be reformulated as "is existential second-order logic able to describe languages (of finite linearly ordered structures with nontrivial signature) that first-order logic with least fixed point cannot?". The word "existential" can even be dropped from the previous characterization, since P = NP if and only if P = PH (as the former would establish that NP = co-NP, which in turn implies that NP = PH).


title: P versus NP problem

text: No algorithm for any NP-complete problem is known to run in polynomial time. However, there are algorithms known for NP-complete problems with the property that if P = NP, then the algorithm runs in polynomial time on accepting instances (although with enormous constants, making the algorithm impractical). However, these algorithms do not qualify as polynomial time because their running time on rejecting instances are not polynomial. The following algorithm, due to Levin (without any citation), is such an example below. It correctly accepts the NP-complete language SUBSET-SUM. It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP:


title: P versus NP problem

text: If, and only if, P = NP, then this is a polynomial-time algorithm accepting an NP-complete language. "Accepting" means it gives "yes" answers in polynomial time, but is allowed to run forever when the answer is "no" (also known as a "semi-algorithm").


title: P versus NP problem

text: This algorithm is enormously impractical, even if P = NP. If the shortest program that can solve SUBSET-SUM in polynomial time is "b" bits long, the above algorithm will try at least other programs first.


title: P versus NP problem

text: Conceptually speaking, a "decision problem" is a problem that takes as input some string "w" over an alphabet Σ, and outputs "yes" or "no". If there is an algorithm (say a Turing machine, or a computer program with unbounded memory) that can produce the correct answer for any input string of length "n" in at most "cn" steps, where "k" and "c" are constants independent of the input string, then we say that the problem can be solved in "polynomial time" and we place it in the class P. Formally, P is defined as the set of all languages that can be decided by a deterministic polynomial-time Turing machine. That is,


title: P versus NP problem

text: and a deterministic polynomial-time Turing machine is a deterministic Turing machine "M" that satisfies the following two conditions:


title: P versus NP problem

text: NP can be defined similarly using nondeterministic Turing machines (the traditional way). However, a modern approach to define NP is to use the concept of "certificate" and "verifier". Formally, NP is defined as the set of languages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" is defined as follows.


title: P versus NP problem

text: "L" ∈ NP if, and only if, there exists a binary relation formula_10 and a positive integer "k" such that the following two conditions are satisfied:


title: P versus NP problem

text: A Turing machine that decides "L" is called a "verifier" for "L" and a "y" such that ("x", "y") ∈ "R" is called a "certificate of membership" of "x" in "L".


title: P versus NP problem

text: In general, a verifier does not have to be polynomial-time. However, for "L" to be in NP, there must be a verifier that runs in polynomial time.


title: P versus NP problem

text: Clearly, the question of whether a given "x" is a composite is equivalent to the question of whether "x" is a member of COMPOSITE. It can be shown that COMPOSITE ∈ NP by verifying that it satisfies the above definition (if we identify natural numbers with their binary representations).


title: P versus NP problem

text: Alternatively, if "L" ∈ NP, and there is another NP-complete problem that can be polynomial-time reduced to "L", then "L" is NP-complete. This is a common way of proving some new problem is NP-complete.


title: P versus NP problem

text: The film "Travelling Salesman", by director Timothy Lanzone, is the story of four mathematicians hired by the US government to solve the P versus NP problem.


title: P versus NP problem

text: In the sixth episode of "The Simpsons" seventh season "Treehouse of Horror VI", the equation P = NP is seen shortly after Homer accidentally stumbles into the "third dimension".


title: P versus NP problem

text: In the second episode of season 2 of "Elementary", "Solve for X" revolves around Sherlock and Watson investigating the murders of mathematicians who were attempting to solve P versus NP.


title: Ed Skrein

text: Edward George Skrein (; born 29 March 1983) is an English actor, film director, screenwriter and rapper. He is known as an actor in independent films or television series. In the years that followed, he became known for portraying Daario Naharis in the third season of the television series "Game of Thrones". Skrein later rose to greater recognition portraying the villain Francis Freeman/Ajax in the superhero-comedy film "Deadpool" (2016). Skrein went on to replace Jason Statham in the action film "The Transporter: Refueled" (2015) and portrayed the cocky bounty-hunting cyborg Zapan in "" (2019). He later starred as Lt. Dick Best in the war film "Midway" (2019).


title: Ed Skrein

text: Skrein made his directorial debut with the short film "Little River Run" (2020), which he released on his Instagram.


title: Ed Skrein

text: Skrein was born in Camden, London, England, and was raised there and in other London boroughs, Haringey and Islington. He is of English and Austrian Jewish descent. Skrein attended Fortismere School. He graduated from Byam Shaw School of Art, a public tertiary art school, with a BA Hons degree in fine art painting.


title: Ed Skrein

text: In 2004, Skrein released a three track EP under the record label Dented Records. In 2007, he released his first album, "The Eat Up". He has collaborated with a variety of artists including Foreign Beggars, Asian Dub Foundation, Plan B, Dubbledge and Doc Brown. Later that year, Skrein released a collaborative EP, "Pre-Emptive Nostalgia", with the group A State of Mind. In 2009 Skrein collaborated with rapper Dr Syntax to bring the collaborative album "Scene Stealers" under the name Skreintax.


title: Ed Skrein

text: Skrein made his acting debut in Plan B's short film "Michelle". Skrein got his first lead role in Plan B's "Ill Manors". Skrein portrayed Daario Naharis in the third season of the television series "Game of Thrones". However, in the fourth season of the series, he was replaced by Dutch actor Michiel Huisman. Skrein states that it was not his choice to leave the show. In 2014, Skrein was cast in "", replacing Jason Statham as the lead. Though the film was critically panned, his performance was praised. In 2016, Skrein played the lead villain, Ajax, in the blockbuster action film "Deadpool". In 2017 Skrein also appeared in the music video for 'Eye Contact' by Ocean Wisdom playing the bad guy.


title: Ed Skrein

text: In 2018, Skrein starred in psychological revenge thriller, "In Darkness", alongside Natalie Dormer and Stacy Martin; principal photography began in early 2016. He was slated to star in the "Hellboy" reboot as Ben Daimio, a Japanese-American character; after controversy over his casting, Skrein announced on Twitter that he was stepping down from the role in the hopes an actor of the proper ethnicity would be cast.


title: Ed Skrein

text: On 8 November 2019, he starred in Roland Emmerich’s blockbuster movie "Midway", co-starring Mandy Moore, Patrick Wilson, Luke Evans, Aaron Eckhart, Nick Jonas, Dennis Quaid and Woody Harrelson.


title: Ed Skrein

text: In 2022, Skrein starred as Vince in "I Used to Be Famous". Leslie Felperin, writing for "The Guardian", gave the film 3/5 stars, calling it a "sweet-natured but predictable comedy drama".


title: Ed Skrein

text: Skrein made his directing debut by directing and writing the short drama film "Little River Run" which he released on 20 July 2020 at his Instagram.


title: Ed Skrein

text: Since he was 12 years old, Skrein has been a swimming coach for Greenwich Leisure Limited working at leisure centres such as Caledonian Road Pool and Gym and Archway Pool.


title: Bernardo Silva

text: Bernardo Mota Veiga de Carvalho e Silva (; born 10 August 1994), known as Bernardo Silva or simply Bernardo, is a Portuguese professional footballer who plays as an attacking midfielder or a winger for club Manchester City and the Portugal national team. Considered one of the best midfielders in the world and amongst the best Portuguese players, Silva is known for his stamina, agility and work-rate.


title: Bernardo Silva

text: Born in Lisbon, Silva came through Benfica's youth academy. He began playing for Benfica B in 2013 and was promoted to the first-team in 2014, playing a few minutes with the latter. He was on loan at the Ligue 1 side Monaco during the 2014–15 season, with "Les Monégasques" making the move permanent in winter 2015. After winning the league title with them in 2017, he was signed by English club Manchester City for a reported fee of £43.5 million. He subsequently won the Premier League and EFL Cup in his first season in England, following this up with a domestic treble the following season. He had a significant role in City becoming the first men's team in England to win a domestic treble, being named Manchester City's Player of the Year in 2019 and being part of the PFA Premier League Team of the Year. Afterwards, Silva was listed among the 30-man shortlist for the Ballon d'Or. In 2020–21, he had a significant role on Manchester City reaching their first UEFA Champions League final.


title: Bernardo Silva

text: Silva made his senior debut for Portugal in 2015 after previously being capped by the nations's youth teams at under-19 and under-21 levels. He was chosen in Portugal's squads for the 2017 FIFA Confederations Cup, 2018 FIFA World Cup, 2019 UEFA Nations League Finals, UEFA Euro 2020 and 2022 FIFA World Cup, winning the 2019 tournament on home soil while also being named as the Player of the Tournament.


title: Bernardo Silva

text: A product of S.L. Benfica's youth system, Silva played for its ranks, and in 2013 they won the 2012–13 Portuguese Juniors Championship. He made his debut for Benfica B in a Segunda Liga match against Trofense on 10 August 2013 (Matchday 1).


title: Bernardo Silva

text: On 19 October 2013, Silva made his debut for Benfica at the age of 19, in a 2013–14 Taça de Portugal third round 1–0 away victory against Campeonato Nacional de Seniores club C.D. Cinfães, coming out of the bench in the 80th minute. His good performance for Benfica B in the 2013–14 Segunda Liga earned him the league's Breakthrough Player of the Year award. He was a member of Benfica's domestic treble-winning team of the 2013–14 season.


title: Bernardo Silva

text: On 7 August 2014, Silva joined Monaco on one-year loan deal. He made his debut on 17 August in a Ligue 1 away match against Bordeaux, replacing Lucas Ocampos in the second half. On 21 September, he made his first start, in a 1–0 home win against Guingamp. On 14 December, he scored the only goal of a home win against Marseille.


title: Bernardo Silva

text: On 20 January 2015, Benfica announced that Silva's economic and sports rights had been sold to Monaco for €15.75 million; he signed a contract for Monaco that would expire on 30 June 2019. On 10 April, he scored twice in a 3–0 away win against Caen. On 10 August 2015, Silva extended his contract by one year, tying him to the club until June 2020.


title: Bernardo Silva

text: In the 2016–17 Champions League group stage away match against CSKA Moscow on 18 October 2016, Silva scored Monaco's equalizer in the 87th minute to ensure that the match finished in 1–1 draw. On 15 January 2017, he scored Monaco's last two goals in a 4–1 away win over Marseille to help Monaco move to the top of the Ligue 1 table for the first time since Week 5 of the current Ligue 1 season. The following 29 January, Silva scored an added-time equaliser in the 1–1 draw against reigning league champions Paris Saint-Germain at the Parc des Princes, putting his team on top of the league. He finished the 2016–17 season with 8 goals and 9 assists in the league and 11 goals and 12 assists in 58 matches across all competitions.


title: Bernardo Silva

text: The calibre of his performances against Manchester City for Monaco in the UEFA Champions League knockout legs in the 2016–17 season was noted by head coach Pep Guardiola and his backroom staff. On 26 May 2017, Manchester City confirmed the signing of Silva on a five-year contract after passing his medical tests. Although the transfer fee remains undisclosed, it has been reported to amount to €50 million (£43.5 million), which could reach €70 million with add-ons. Silva officially joined the club on 1 July 2017, ahead of the 2017–18 season.


title: Bernardo Silva

text: On 13 February 2018, Silva scored his first Champions League goal for the City in a 4–0 away victory against Basel in the first leg of the round of 16 stage. On 5 March, Silva scored the solitary winning goal against Chelsea at home – a key win which put them only three wins away from winning the Premier League title. He established himself as a consistent performer, playing the most matches of any City player in the season up to the end of February 2018. By the end of the season, he had played for Manchester City 53 times in all competitions – the most of any City player that season and helped them to a record 100 points in the Premier League and the EFL Cup.


title: Bernardo Silva

text: Silva made 51 appearances for Manchester City during the 2018–19 season, scoring 13 goals and adding 14 assists in all competitions. On 24 April 2019, Silva scored the opening goal for City against Manchester United in the Manchester Derby, and the 2–0 win at Old Trafford put his team on top of the league. Silva's consistent and influential performances helped his team to win Premier League title, filling in Kevin De Bruyne's position who missed the majority of the season with an injury. As a result, he was voted into the PFA Team of the Year alongside four other City players and won the Manchester City Player of the Season award, voted by the fans.


title: Bernardo Silva

text: In the 2019–20 season, Silva scored a hat-trick in an 8–0 thrashing of Watford in the Premier League on 21 September. Days later, he was accused of being racist towards his teammate Benjamin Mendy in a tweet which likened the player's appearance to that of a dark-skinned advertising mascot; Mendy said that he was not offended. In November, Silva was banned for one match and fined £50,000 for the tweet. The FA said that Silva did not intend the post to be insulting or in any way racist. In October 2019, Silva was nominated as one amongst 30 candidates for Ballon d´Or.


title: Bernardo Silva

text: On 10 January 2021, Silva scored a brace in a 3–0 home win over Championship side Birmingham City in the Third round of the FA Cup. On 20 January 2021, Silva scored his first league goal of the season in a 2–0 home win over Aston Villa. On 24 February 2021, Silva scored his first Champions League goal of the season in a 2–0 away win over Borussia Mönchengladbach in the first leg of the round of 16 tie.


title: Bernardo Silva

text: In the start of the 2021–22 season, Manchester City manager Pep Guardiola revealed that Silva was one of three players that wanted to leave the club in the summer transfer window. It was reported that, he was unhappy in Manchester, with Spanish La Liga side Atlético Madrid and Serie A side A.C. Milan being interested in signing him. After failling to leave the club in the transfer window, Silva began displaying impressive performances in the midfield for Manchester City, most notably in the 5–0 victory against Arsenal, scoring the only goal in a 1–0 win over Leicester City, and against Liverpool and Chelsea, being named man of the match in the latter, with manager Pep Guardiola describing Silva as "one of the best players in the world". For his performances, Silva was awarded Manchester City's Player of the Month by the club's supporters for two consecutive months in September and October.


title: Bernardo Silva

text: On 6 November, in the Manchester derby, Silva scored City's second goal in a 2–0 victory against Manchester United at Old Trafford. On 24 November, Silva became the first player in the Champions League to register a 100% passing rate, being named man of the match, as he provided an assist for Gabriel Jesus, in City's 2–1 home win against Paris Saint-Germain in a Champions League group stage match, to ensure his team qualification to the round of sixteen. For his performances, Silva was named Manchester City's Player of the Month for November, becoming the first player to win the award in three consecutive months.


title: Bernardo Silva

text: On 15 February 2022, Silva scored a brace and provided an assist in a 5–0 away win over Sporting CP in the first leg of the Champions League round of 16 tie, being named man of the match for the second match in a row.


title: Bernardo Silva

text: Having spent the majority of his first four seasons in Manchester providing width on the right, he was transformed into the ultimate utility player over the course of this season. Though functionally a central midfielder, he has spent time at both centre forward (he scored from there against Manchester United) and defensive midfield. He ended the season with 13 goals and 7 assists in all competitions.


title: Bernardo Silva

text: In 2013, Silva represented Portugal at the 2013 European Under-19 Championship, reaching the semi-finals. For his performances, he was named among the top 10 talents under the age of 19 in Europe by a selection of UEFA reporters.


title: Bernardo Silva

text: On 31 March 2015, Silva made his senior debut for Portugal, starting in a 0–2 friendly defeat to Cape Verde in Estoril.


title: Bernardo Silva

text: On 27 June 2015, Silva scored Portugal U21s' first goal in a 5–0 win against Germany U21 in the semi-finals of the European Under-21 Championship in Czech Republic. He was left out of the Portuguese squad for the finals of Euro 2016 due to injury.


title: Bernardo Silva

text: Silva was selected for the Portugal squad for the 2017 Confederations Cup which was hosted in Russia. He scored the second goal of the match in the group stage of the tournament, a 4–0 win over New Zealand. The national team reached the semi-finals before losing out to Chile in a penalty shootout after a 0–0 draw at the end of regulation time. He missed the third place play-off in which Portugal defeated Mexico 2–1 after extra time.


title: Bernardo Silva

text: Silva was named in Portugal's 23-man squad for the 2018 World Cup in Russia. He featured in all 4 matches with Portugal, but was knocked out of the tournament after a 1–2 loss to Uruguay.


title: Bernardo Silva

text: Silva was selected for three matches in the 2018–19 UEFA Nations League group stage, scoring a goal in a 3–2 victory against Poland. In the UEFA Nations League Finals, Silva played both matches and registered an assist in each as Portugal won the trophy on home soil, as he went on to be named the tournament's best player.


title: Bernardo Silva

text: Silva was named in Portugal's final squad for the delayed UEFA Euro 2020 tournament, appearing in all games of the eventual round of 16-exit to Belgium.


title: Bernardo Silva

text: In October 2022, he was named in Portugal's preliminary 55-man squad for the 2022 FIFA World Cup in Qatar, being included in the final 26-man squad for the tournament.


title: Bernardo Silva

text: Silva is a diminutive, elegant, and creative left-footed playmaker, with a slender build, who is capable of playing in several attacking and midfield positions. Throughout his career, he has been fielded in a variety of roles, including as a winger on either flank (although he favours the right side of the pitch), as a central midfielder, as a deep-lying playmaker, as an attacking midfielder, or even as a second striker; he has also been deployed in a false 9 role, or even as wing-back on occasion. He is mainly known for his technique, acceleration, agility, passing, vision, and dribbling skills, as well as his energy and defensive work-rate, which enables him to cover ground, press opponents, and start attacks after winning back possession. Although he is known for his ability to carry the ball forward, run at defences, and retain possession, he is also known to be an intelligent team player, with a strong personality, who prefers to participate in the build-up of offensive plays rather than undertaking individual dribbling runs. Despite primarily being a creative midfielder, he is also capable of scoring goals himself.


title: Bernardo Silva

text: Silva's English has been described as "perfect" by the media, partly as a result of his parents sending him to an English-speaking school in Lisbon when he was six years old. In addition to English and his native Portuguese, Silva speaks French and Spanish. His cousin Matilde Fidalgo, also born in 1994, has represented Portugal and Manchester City's women's teams. Silva is a supporter of Portuguese club S.L. Benfica.


title: Bernardo Silva

text: He announced his engagement to girlfriend and model Ines Tomaz in May 2022, with both of them sharing a photo together of her accepting his ring and proposal.


title: Zelda Williams

text: Zelda Rae Williams (born July 31, 1989) is an American actress, director, producer, and writer. She is the daughter of the late actor and comedian Robin Williams and film producer and philanthropist Marsha Garces Williams. She voiced Kuvira in the Nickelodeon cartoon "The Legend of Korra".


title: Zelda Williams

text: Williams was born in New York City to Robin and Marsha Garces Williams. She was born 10 days after her father's 38th birthday. Her father stated that he named her after Princess Zelda from "The Legend of Zelda" video game series. Her mother is of Filipino and Finnish descent. Zelda, the older of Williams's children by his second wife, has a younger brother, Cody, and an older half-brother, Zachary Pym "Zak" Williams.


title: Zelda Williams

text: She made her acting debut at age 5. At 15, Williams acted in the 2004 film "House of D" opposite her father and actor Anton Yelchin as the latter's young first love, Melissa Loggia.


title: Zelda Williams

text: In June 2011, both she and her father were featured in a television commercial for the Nintendo 3DS game "". On October 25, 2011, she was the special guest host at the London Zelda Symphony Concert at the Hammersmith Apollo, marking the 25th anniversary of the Zelda Series. On November 17, 2011, subscribers to Nintendo emails received an email of her and her father playing the new game "" for the Wii console. She appears in an interview with Nintendo representatives in which she says that one of her favorite things to do in her free time is play "Super Smash Bros. Melee" with her friends, always playing as the character Sheik.


title: Zelda Williams

text: She appeared in "People" magazine's 100 Most Beautiful People issue for 2007. She appears in the Wynter Gordon music video for "Buy My Love" and has a cameo in the Cobra Starship music video "You Make Me Feel", alongside her father.


title: Zelda Williams

text: In 2018, Williams wrote and directed the short film "Shrimp", which she also starred in alongside Conor Leslie, Paulina Singer, Frances Fisher, and Jake Abel. The film depicts the lives of dominatrices in a Los Angeles BDSM den. Williams later entered into a deal with Gunpowder & Sky to develop the short film into a half-hour series.


title: Zelda Williams

text: Williams has described herself as an avid athlete and video game player, and a fan of her namesake "The Legend of Zelda" series. She has stated that "" is her favorite game and had voiced support of its release on the 3DS by supporting .


title: Blood

text: Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the circulatory system is also known as "peripheral blood", and the blood cells it carries, "peripheral blood cells".


title: Blood

text: Blood is composed of blood cells suspended in blood plasma. Plasma, which constitutes 55% of blood fluid, is mostly water (92% by volume), and contains proteins, glucose, mineral ions, hormones, carbon dioxide (plasma being the main medium for excretory product transportation), and blood cells themselves. Albumin is the main protein in plasma, and it functions to regulate the colloidal osmotic pressure of blood. The blood cells are mainly red blood cells (also called RBCs or erythrocytes), white blood cells (also called WBCs or leukocytes) and platelets (also called thrombocytes). The most abundant cells in vertebrate blood are red blood cells. These contain hemoglobin, an iron-containing protein, which facilitates oxygen transport by reversibly binding to this respiratory gas thereby increasing its solubility in blood. In contrast, carbon dioxide is mostly transported extracellularly as bicarbonate ion transported in plasma.


title: Blood

text: Vertebrate blood is bright red when its hemoglobin is oxygenated and dark red when it is deoxygenated.


title: Blood

text: Some animals, such as crustaceans and mollusks, use hemocyanin to carry oxygen, instead of hemoglobin. Insects and some mollusks use a fluid called hemolymph instead of blood, the difference being that hemolymph is not contained in a closed circulatory system. In most insects, this "blood" does not contain oxygen-carrying molecules such as hemoglobin because their bodies are small enough for their tracheal system to suffice for supplying oxygen.


title: Blood

text: Jawed vertebrates have an adaptive immune system, based largely on white blood cells. White blood cells help to resist infections and parasites. Platelets are important in the clotting of blood. Arthropods, using hemolymph, have hemocytes as part of their immune system.


title: Blood

text: Blood is circulated around the body through blood vessels by the pumping action of the heart. In animals with lungs, arterial blood carries oxygen from inhaled air to the tissues of the body, and venous blood carries carbon dioxide, a waste product of metabolism produced by cells, from the tissues to the lungs to be exhaled.


title: Blood

text: Medical terms related to blood often begin with hemo- or hemato- (also spelled haemo- and haemato-) from the Greek word ("haima") for "blood". In terms of anatomy and histology, blood is considered a specialized form of connective tissue, given its origin in the bones and the presence of potential molecular fibers in the form of fibrinogen.


title: Blood

text: Blood accounts for 7% of the human body weight, with an average density around 1060 kg/m, very close to pure water's density of 1000 kg/m. The average adult has a blood volume of roughly or 1.3 gallons, which is composed of plasma and "formed elements". The formed elements are the two types of blood cell or "corpuscle" – the red blood cells, (erythrocytes) and white blood cells (leukocytes), and the cell fragments called platelets that are involved in clotting. By volume, the red blood cells constitute about 45% of whole blood, the plasma about 54.3%, and white cells about 0.7%.


title: Blood

text: About 55% of blood is blood plasma, a fluid that is the blood's liquid medium, which by itself is straw-yellow in color. The blood plasma volume totals of 2.7–3.0 liters (2.8–3.2 quarts) in an average human. It is essentially an aqueous solution containing 92% water, 8% blood plasma proteins, and trace amounts of other materials. Plasma circulates dissolved nutrients, such as glucose, amino acids, and fatty acids (dissolved in the blood or bound to plasma proteins), and removes waste products, such as carbon dioxide, urea, and lactic acid.


title: Blood

text: The term serum refers to plasma from which the clotting proteins have been removed. Most of the proteins remaining are albumin and immunoglobulins.


title: Blood

text: Blood pH is regulated to stay within the narrow range of 7.35 to 7.45, making it slightly basic (compensation). Extra-cellular fluid in blood that has a pH below 7.35 is too acidic, whereas blood pH above 7.45 is too basic. A pH below 6.9 or above 7.8 is usually lethal. Blood pH, partial pressure of oxygen (pO), partial pressure of carbon dioxide (pCO), and bicarbonate (HCO) are carefully regulated by a number of homeostatic mechanisms, which exert their influence principally through the respiratory system and the urinary system to control the acid–base balance and respiration, which is called compensation. An arterial blood gas test measures these. Plasma also circulates hormones transmitting their messages to various tissues. The list of normal reference ranges for various blood electrolytes is extensive.


title: Blood

text: Human blood is typical of that of mammals, although the precise details concerning cell numbers, size, protein structure, and so on, vary somewhat between species. In non-mammalian vertebrates, however, there are some key differences:


title: Blood

text: Blood is circulated around the body through blood vessels by the pumping action of the heart. In humans, blood is pumped from the strong left ventricle of the heart through arteries to peripheral tissues and returns to the right atrium of the heart through veins. It then enters the right ventricle and is pumped through the pulmonary artery to the lungs and returns to the left atrium through the pulmonary veins. Blood then enters the left ventricle to be circulated again. Arterial blood carries oxygen from inhaled air to all of the cells of the body, and venous blood carries carbon dioxide, a waste product of metabolism by cells, to the lungs to be exhaled. However, one exception includes pulmonary arteries, which contain the most deoxygenated blood in the body, while the pulmonary veins contain oxygenated blood.


title: Blood

text: Additional return flow may be generated by the movement of skeletal muscles, which can compress veins and push blood through the valves in veins toward the right atrium.


title: Blood

text: In vertebrates, the various cells of blood are made in the bone marrow in a process called hematopoiesis, which includes erythropoiesis, the production of red blood cells; and myelopoiesis, the production of white blood cells and platelets. During childhood, almost every human bone produces red blood cells; as adults, red blood cell production is limited to the larger bones: the bodies of the vertebrae, the breastbone (sternum), the ribcage, the pelvic bones, and the bones of the upper arms and legs. In addition, during childhood, the thymus gland, found in the mediastinum, is an important source of T lymphocytes.


title: Blood

text: The proteinaceous component of blood (including clotting proteins) is produced predominantly by the liver, while hormones are produced by the endocrine glands and the watery fraction is regulated by the hypothalamus and maintained by the kidney.


title: Blood

text: Healthy erythrocytes have a plasma life of about 120 days before they are degraded by the spleen, and the Kupffer cells in the liver. The liver also clears some proteins, lipids, and amino acids. The kidney actively secretes waste products into the urine.


title: Blood

text: About 98.5% of the oxygen in a sample of arterial blood in a healthy human breathing air at sea-level pressure is chemically combined with the hemoglobin. About 1.5% is physically dissolved in the other blood liquids and not connected to hemoglobin. The hemoglobin molecule is the primary transporter of oxygen in mammals and many other species (for exceptions, see below). Hemoglobin has an oxygen binding capacity between 1.36 and 1.40 ml O per gram hemoglobin, which increases the total blood oxygen capacity seventyfold, compared to if oxygen solely were carried by its solubility of 0.03 ml O per liter blood per mm Hg partial pressure of oxygen (about 100 mm Hg in arteries).


title: Blood

text: With the exception of pulmonary and umbilical arteries and their corresponding veins, arteries carry oxygenated blood away from the heart and deliver it to the body via arterioles and capillaries, where the oxygen is consumed; afterwards, venules and veins carry deoxygenated blood back to the heart.


title: Blood

text: Under normal conditions in adult humans at rest, hemoglobin in blood leaving the lungs is about 98–99% saturated with oxygen, achieving an oxygen delivery between 950 and 1150 ml/min to the body. In a healthy adult at rest, oxygen consumption is approximately 200–250 ml/min, and deoxygenated blood returning to the lungs is still roughly 75% (70 to 78%) saturated. Increased oxygen consumption during sustained exercise reduces the oxygen saturation of venous blood, which can reach less than 15% in a trained athlete; although breathing rate and blood flow increase to compensate, oxygen saturation in arterial blood can drop to 95% or less under these conditions. Oxygen saturation this low is considered dangerous in an individual at rest (for instance, during surgery under anesthesia). Sustained hypoxia (oxygenation less than 90%), is dangerous to health, and severe hypoxia (saturations less than 30%) may be rapidly fatal.


title: Blood

text: A fetus, receiving oxygen via the placenta, is exposed to much lower oxygen pressures (about 21% of the level found in an adult's lungs), so fetuses produce another form of hemoglobin with a much higher affinity for oxygen (hemoglobin F) to function under these conditions.


title: Blood

text: CO is carried in blood in three different ways. (The exact percentages vary depending whether it is arterial or venous blood). Most of it (about 70%) is converted to bicarbonate ions by the enzyme carbonic anhydrase in the red blood cells by the reaction ; about 7% is dissolved in the plasma; and about 23% is bound to hemoglobin as carbamino compounds.