HERMODYNAMICS Lecture 5: Second Law of hermodynamics Pierwsza strona
Second Law of hermodynamics In the course of discussions on the First Law of hermodynamics we concluded that all kinds of energy are equally useful, none of them is a preferred one, but in the case of isolated system it has to be conserved. We can however discern different kinds of energy such as mechanical work and internal energy, which can be changed through applied work. We discern also the heat, which is defined by internal energy and work. δ + δw 0 hermodynamics From the above we can conclude that both kinds of energy are equivalent. Second Law of hermodynamics assumes the fact that heat and work are not equivalent and provides a series of relations, supplementary to the First Law of hermodynamics in investigations of thermodynamical systems.
hermodynamics
Let s remind ourselves four formulations of the second law and scrutinise first and fourth. 3. A battery will discharge through the resistor releasing during such process some heat, but the reverse process will not be possible. 4. It is not possible to construct a continuously operating machine which would cool one reservoir and perfom an equivalent amount of work. he first statement is reduced to a point that a cup of water in a fridge will not boil, despite it would be possible from the point of view of energy conservation. he flow of heat is only in one direction, which is not a result of energy conservation. hermodynamics. Heat flows from higher temperature to lower temperature and not the other way around!. A hot body will cool in a contact with a colder body and also not the other way around!.. wo gases in an isolated compartment will mixe homogeneously in the tank and will not spontaneously separate
Statement number 4 says that construction of perpetuum mobile of the second kind is not possible. Perpetuum mobile of the first kind would be a device, which produces energy without consideration of the I Law of hermodynamics. Summarising we can say that II Law of hermodynamics assumes a one directional flow of heat and some pre-determined types of energy conversion. We will attempt to formulate the II Law by providing analytical relations, basing on macroscopic arguments and assuming the fourth formulation as an experimental axiomat. hermodynamics
Second Law of hermodynamics 850 Clausius statement: It is impossible for any system to operate in such a way that the sole result would be an energy transfer by heat from a cooler to a hotter body. > hermodynamics
Second Law of hermodynamics 85 Kelvin-Planck statement: It is impossible for any system to operate in a thermodynamics cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir. W Entropy statement: W Itisimpossiblefor anysystem to operateina waythatentropyisdestroyed. hermodynamics
Reversible processes and cycles Let s define first the quasistatic process. hat is a process which takes place so slowly that all the time the system is arbitrarily close to the equilibrium. When the process can be named a reversible? If we consider a portion of gas in a cylinder then we could notice the a quasistatic expansion of that gas is related to performing work and removal of heat. If we can return to initial conditions by adding the same amount of heat and performing same amount of work then the gas in a cylinder can be compressed to the initial conditions. Hence it can be said that a quasistatic process is a reversible process. In other words the reversible process it is such a process after the occurence of which the initial conditions can be recovered only by imposing a constraint condition removed at the beginning of the process. hermodynamics
And yet another formulation: If we are capable of carrying out the process in such a way that the Second Law is not infringed then we can say that the process is reversible.. Anirreversibleprocessissucha processwhichisnot reversible. A reversible cycle is a sequence of constituting reversible processes such that a system returns to the initial conditions in a periodical way. hermodynamics
W W > Conversion of work into heat is a reversible process. An inverse situation infringes the formulation of Second Law due to Kelvin-Planck Heat transport caused by a finite temperature difference is an irreversible process. An inverse situation infringes the formulation of Second Law due to Kelvin-Planck hermodynamics
We ought also define the power cycle, i.e. the cycle delivering work as a result of supplied heat, and refrigeration cycle, which is accomplished as a result of supplied work. A reversible power cycle can be converted to a reversible refrigeration cycle by inversion of the rate of heat and work. he identity of Clausius statement and a Kelvin-Planck formulation of the Second Law can be shown in a following way. Let s assume that possible is a transfer of heat contrary to the Clausius statement, i.e. directly from a lower temperature reservoir to a higher temperature reservoir. Such picture can be supplemented by a reversible engine, as in the middle of picture. A result of adding of such processes is development of an engine which operates only using one heat source. Such process is excluded by the Kelvin-Planck formulation of the Second Law hermodynamics
+ W - W - - We have shown that violation of the Clausius formulation causes violation of the Kelvin-Planck formulation of the Second Law of hermodynamics. hermodynamics
For the sake of completness let s formulate also the refrigeration/heat pump cycle. W - If we want to sort out the sign convention of heat and work then: Energy added to the system Energy accumulated in the system + Energy removed from the system. hermodynamics
Let s consider that we have at our disposal two reversible thermal machines operating in a cyclic manner: A B W A W B A B hermal efficiency η t of a cyclic machine is defined in a following way: hermodynamics
useful energy obtained work η t energy input used heat In a considered case that would be: W η t. (6.) (6.3) Cycles A and B can be constructed in a different manner. Let s assume that the efficiency of cycle A is greater then that of cycle B, and A B hermodynamics In such case W A > W B and A < B. Dueto thefactthatbothmachinesarereversiblethemachineb can be inversed and be connected with machine A. In such way we obtain a situation depicted in the next slide.
A B W A + W B W A -W B A B B A We can see that we would obtain a cycle where W A -W B B A, violates the Kelvin-Planck formulation of second Law. herefore our assumption that η A > η B was incorrect. hermodynamics
herefore we can conclude, that: all reversible thermal machines operating in the same temperature range have the same efficiency. η t W (6.4) We can also conclude that / is a function of these temperatures. herefore we would obtain a relation: We can show that, where F is a new function. f (, ) (6.5) F( ) (6.6) F( ) hermodynamics
Relation (6.6) can be obeyed by several functions F. Kelvin suggested to assume the simplest form of that function, i.e. (6.7) And at the same time regard that equation as a definition of absolute thermodynamical temperature. Efficiency of reversible thermal machine operating between two heat reservoirs with temperatures L lower and H higher, is given by an expression; η t L H (6.8) hermodynamics
Clausius inequality H Let s consider a device which receives the heat d H from a tank with constant temperature H and transports that heat to the reversible machine Z producing work in the amount d W Z. Reversible machine Cyclic machine d H Z d W Z d C L d W C he heat rejected by the machine Z feeds the cyclic machine C producing work in the amount d W C. Considering these two machines as a single system, the total work accomplished is equal to: d W d W Z + d W C Basing on the efficiency of reversible engine Z, we can write: hermodynamics
d' W Z d' W C d' ( H d' H H ) d' ( ) i.e. H d' W d' ( + ) H d' In thecaseofa fullcycleequation(6.9) assumesa form (6.9) d ' W H d '. (6.0) he machine presented in the schematic in the diagram cannot perform work as that is a contradictory process to the Kelvin- Planck statement. he device can only work with a cyclic input of work and cyclic transfer of heat to the tank. hermodynamics
Mathematically that means d 'W 0 (6.) where d W is a resulting work. We can also write that d ' (6.) 0 he latter is named the Clausius inequality. Up till now we did not considered the fact that the engine C can be reversible. Let s assume that it is so and that d'w < 0 If C is a reversible engine then we get, hermodynamics
hermodynamics d'w > 0 hat is not possible as we would develop a perpetuum mobile of the second kind. Itresultsfromherethatincaseofreversibleprocessesin equation (6.) the following relation must hold d' ( ) odwr 0 (6.3) Macroscopic definition of entropy and the principle of entropy increase In equation (6.3) the expression inside the integral must be an absolute differential of some function of state. Hence we can write d ' ds ) ( odwr (6.4) hat function S is entropy
hat equation presents the macroscopic definition of entropy. Entropy is defined only for reversible processes and the change of entropy can be calculated from the relation; d' ΔS S S ( ) odwr (6.5) Let s consider two arbitrary points of state of our system. Irreversible process According to (6.) CycleN+O Reversible process d' d' N + d' O < 0 he inequality sign has been used as the whole cycle is irreversible. hermodynamics
hermodynamics Recalling that ' S S d O he last equation can be written as: > < + ' 0 ' d S S or S S d N N In a general case we can write: ' d S S (6.6) he inequality sign is important in the case of irreversible processes and the equal sign in case of reversible equations
In case of adiabatic process d 0, hence S S 0. If that is going to be an adiabatic process then the change of entropy will be equalzerp. Suchprocessisanisentropic process. We can say that none of the real processes is reversible. If the process is irreversoble and adiabatic then entropy must increase. In case of isolated system, Δ S izol 0. (6.7) Basing on equation (6.4) we can find that for an arbitrary reversible isothermal process odwr izoterm Δ S. (6.8) In - S system of coordinates the adiabatic process can be presented as reversible and irreversible. hermodynamics
Adiabatic reversible process hegreatertheincreaseof entropy the more irreversible process is. he reason for a smaller or greater irreversibility of processes are various kinds of friction (stirring of the soup). Adiabatic irreversible process S ΔS irreversible adiabatic. Entropy of a pure substance We have shown that entropy is a property of thermodynamical system, an extensive property. It is the same property as total energy, internal energy and enthalpy. It can be calculated from the specific entropy. hermodynamics
S m s (6.9) In case of pure substances the entropy can be tabulated in a same way as enthalpy, specific volume or other thermodynamic property. A twofold type of diagrams is usually presented, relation of temperature and entropy or enthalpy and entropy. he latter is named the Molier diagram (h-s). hermodynamics Entropy of ideal gas Basing on already derived relations, du dh c c As well as information that in case of reversible process d ds and assuming that the ideal gas is a compressible liquid we can write: V p d d
i.e. hermodynamics d ' du + p dv ds du p ds + dv. Using the ideal gas equation we obtain i.e. p R v d ds cv + dv R v. In case of c V const we obtain the expression describing the change of entropy between two states of ideal gas s V + s c ln R ln v v (6.0)
Such equation can also be written basing on the relations d ' ds dh c p d vdp R ds dp p, as s p s c ln Rln p p (6.) Both in equations (6.0) and (6.) the change of entropy is calculated between two states of thermodynamical system (p,v, ) and (p,v, ). As the entropy is a function of state, its change is not dependent upon the course of the process. hermodynamics