The second principle of thermodynamics (also known under the name of second law of thermodynamics ) establishes the irreversibility phenomena Physique S, in particular during heat exchange. It is a principle of evolution which was stated for the first time by Sadi Carnot in 1824. It has since fact the object of many generalizations and successive formulations by Clapeyron (1834), Clausius (1850), Lord Kelvin, Ludwig Boltzmann in 1873 and max Planck (see Histoire of thermodynamics and statistical mechanics), throughout the 19th century and beyond.

It introduces the function of state Entropie: S , usually comparable with the concept of disorder.

Statement of the law

All transformation of a thermodynamic Système is carried out with increase in the total Entropie including the entropy of the system and the external medium. It is still said that there is creation of entropy .

the function of state entropy: S, was regarded as a measurement of the disorder.

$\backslash \; Delta\; total\; S\_\; \{\}\; =\; S\_\; \{\}\; creation\; =\; \backslash \; Delta\; S\_\; \{syst\}\; +\; \backslash \; Delta\; S\_\; \{ext.\}\; \backslash \; geq0~$

In the case of a reversible transformation , the creation of entropy is null.

Remarks

• the entropy of a system insulated can only increase or remain constant since there is no exchange of heat with the external medium.

• the entropy of a system can decrease but that means that the entropy of the external medium increases in a more important way; the positive or null entropic assessment being if the transformation is reversible.

• the expression “degree of disorder of the system” introduced by Boltzmann can appear ambiguous and subjective. Indeed one can also define the entropy as a measurement of the homogeneity of the system considered. The entropy of a thermal system is maximum when the temperature is identical in any point. In the same way, if one pours a liquid coloring in water glass, the entropy of the coloured system will be maximum when, following the mixture, the color of the contents becomes uniform. Any isolated system, seat of a random agitation, spontaneously tends to be homogenized in an irreversible way what intuitively seems contrary with an increase in the disorder.

the second principle is a principle of evolution which stipulates that any real transformation is carried out with creation of entropy.

Concept of reversibility

A reversible transformation is a transformation Quasistatique likely to be reversed following a progressive modification of external pressures, while making it possible the system to find the successive former states. Fact that reverts passing film of the transformation to back! If this film appears ridiculous it is that the transformation is not reversible. Actually, all the real transformations are irreversible. A reversible transformation represents indeed, the borderline case of a real transformation, led in a way infinitely slow, made up of a succession of infinitely close states of balance and characterized by null dissipative phenomena. It is thus an ideal model of transformation.

One can count several causes of irreversibility (nonexhaustive list):

• inhomogeneousness (source of diffusion): molecular density, temperature, pressure,…

• dissipative phenomenon: fluid frictions and solids
• spontaneous reorganization of the matter: chemical reaction.

Formulations of the second principle

The second principle introduces the function of state extensive S, called Entropie . The variation of entropy of a system, at the time of an unspecified transformation, can be described as the sum of a term of exchange and a term of creation:

$\backslash \; Delta\; S\_\; \{syst\}\; =\; \backslash \; Delta\; S\_\; \{\}\; exchange\; +\; S\_\; \{creation\}\; ~$

• the term of creation, always positive or null, imposes the direction of the evolution of the transformation, $S\_\; \{creation\}\; \backslash \; geq0~$; the equality takes place only for one reversible transformation .

• the term of exchange in the case of a system closed exchanging the quantity of heat Q with the medium external with the temperature T is equal to $\backslash \; Delta\; S\_\; \{\}\; exchange\; =\; \backslash \; frac\; \{Q\}\; \{T\}\; ~$.

Another formulation is possible as we saw previously, by considering the entropy of the system and the entropy of the external medium. This formulation is completely compatible with the preceding one.

$\backslash \; Delta\; total\; S\_\; \{\}\; =\; S\_\; \{\}\; creation\; =\; \backslash \; Delta\; S\_\; \{syst\}\; +\; \backslash \; Delta\; S\_\; \{ext.\}\; ~$

Indeed

$\backslash \; Delta\; S\_\; \{exchange\}\; ~$ corresponds to the entropy exchanged by the system with the external medium. If one places side of the external medium the sign is reversed and thus:

$\backslash \; Delta\; S\_\; \{ext.\}\; =\; -\; \backslash \; Delta\; S\_\; \{exchange\}\; ~$

It follows

$\backslash \; Delta\; S\_\; \{syst\}\; =\; -\; \backslash \; Delta\; S\_\; \{ext.\}\; +\; S\_\; \{creation\}\; ~$

From where

$S\_\; \{\}\; creation\; =\; \backslash \; Delta\; S\_\; \{syst\}\; +\; \backslash \; Delta\; S\_\; \{ext.\}\; ~$

The variation of total entropy corresponds to the entropy created and is equal to the sum of the variations of entropy of the system and the external medium. It is always positive in the case of the real transformations irreversible. On the other hand in the ideal case of the reversible transformations it is null.

Let us consider a transformation carried out either in a reversible way or in an irreversible way, at the temperature T. the entropy being a function of state its variation will be the same one for the two ways considered. On the other hand heat will depend on the followed way because it is not a function of state.

• reversible Transformation:

$\backslash \; Delta\; S\_\; \{syst\}\; =\; \backslash \; Delta\; S\_\; \{\}\; exchange\; =\; \backslash \; frac\; \{Q\; (rev)\}\{T\}\; ~$ since the entropy created is null.

• irreversible Transformation:

$\backslash \; Delta\; S\_\; \{syst\}\; =\; \backslash \; Delta\; S\_\; \{\}\; exchange\; +\; S\_\; \{creation\}\; ~$

$\backslash \; Delta\; S\_\; \{syst\}\; =\; \backslash \; frac\; \{Q\; (irrev)\}\{T\}\; +\; S\_\; \{creation\}\; ~$

It follows that $\backslash \; Delta\; S\_\; \{syst\}\; >\; \backslash \; frac\; \{Q\; (irrev)\}\{T\}\; ~$ since the entropy created is positive.

the expression thus obtained was formulated by Clausius. It is still called inequality of Clausius. It is another way of expressing the second principle.

• Consequence on the transfer of heat :

Intuitively it is known that heat passes from a hot body to a colder body. The second principle makes it possible to show it. Let us consider an isolated system made up of two subsystems, syst1 and syst2 from which the respective temperatures T1 and T2 are different.

The heat exchanged by syst1 is Q1 and that exchanged by syst2 is Q2 . As the system is isolated the heat exchanged with the external medium is null, therefore Q1 + Q2 = 0 . From where Q2 = - Q1 .

Let us apply the second principle

S_créée = ΔS_syst + ΔS_ext > 0

however ΔS_syst = ΔS_syst1 + ΔS_syst2 and ΔS_ext = 0 since the system is isolated.

It follows:

S_créée = ΔS_syst1 + ΔS_syst2

ΔS_syst1 = Q1/T1

ΔS_syst2 = Q2/T2 = - Q1/T2

Thus S_créée = Q1/T1 - Q1/T2

S_créée = Q1 (1/T1 - 1/T2)

As the transformation is irreversible:

S_créée = Q1 (1/T1 - 1/T2) > 0

If T1 is higher than T2, it is necessary that Q1 is negative so that the entropic assessment is positive. According to the rule of the signs, that means that the syst1 provides heat to the syst2 which receives it and thus that heat passes from the cold heat.

Into any rigor, the temperature does not change brutally between the two subsystems because in the vicinity of the border, the temperature varies gradually between T1 and T2. It is said that there is a Gradient temperature; phenomenon closely related to the concept of irreversibility. Nevertheless this phenomenon is not opposed to the preceding demonstration showing the direction of the transit of heat. If the temperatures T1 and T2 are very close one to the other, one can consider that the transformation approaches a reversible transformation (small imbalance of the variable temperature) and one notes whereas ΔS_créée tends towards zero.

• Consequence on the useful work provided by a system :

The work as well as the heat are not functions of state and their value depends on the nature of the transformation affecting the system.

Let us consider a transformation carried out either in a reversible way or in an irreversible way at the temperature T . The variation of entropy will be the same one because the entropy is a function of state. On the other hand, W (rév) ≠ W (irrév) and Q (rév) ≠ Q (irrév) .

ΔS (syst) = Q (rév) /T

ΔS (syst) > Q (irrév) /T

Thus Q (rév) > Q (irrév)

Now let us apply the First principle

ΔU = W (rév) + Q (rév) = W (irrév) + Q (irrév)

It results from it that: W (rév) < W (irrév)

However for a driving system providing of work, work is counted negatively according to the selected rule of the signs in thermodynamics. What is important it is the absolute value of useful work. From where:

|W (rév)| > |W (irrév)|

the useful work provided by a driving system is more important if the transformation is reversible .

Frictions being the leading cause of irreversibility it is included/understood why one tries to minimize them by lubrication.

History of the law

The origin of the second law of thermodynamics goes up with 1824 and is due to the Physicien French Sadi Carnot, wire of Lazare Carnot. It is him which, in the treaty " Reflections on the horse-power of fire and the machines suitable to develop this puissance" (Sadi Carnot used the term of machine with fire to indicate the heat engines), was the first to establish that the output of such a machine depended on the difference in temperature between the hot source and the cold source. Although using the concept exceeded of the heating which considered that the heat, by analogy with a fluid, was a material substance which could be either added, or removed, or transferred from one body to the other, it succeeds, by a Expérience of thought, to imagine the following principle: the best performance $\backslash \; eta$ of a driving ditherme functioning with an hot source of temperature $T\_1$ and a cold source of temperature $T\_2$ is worth: $\backslash \; eta\; =\; 1\; -\; \backslash \; frac\; \{T\_2\}\; \{T\_1\}$

Demonstration This expression or output of Carnot corresponds to cyclic operation and Réversible of a machine ditherme. During the cycle, the hot source with T₁ provides the quantity of heat Q₁ to the driving system. This one provides a work W and restores a quantity of heat Q₂ with the cold source with T₂ .

As operation is cyclic, the final state is identical at the initial state, therefore the energy interns system remains constant because it is a function of state: ΔU = 0 .

• Let us apply the First principle: ΔU = Q₁ + Q₂ + W = 0

from where W = - (Q₁ + Q₂)

• Let us apply the second principle: ΔS (syst) + ΔS (ext.) = 0 if the cycle is reversible.

Q₁/T₁ + Q₂/T₂ = 0 and Q₂/Q₁ = - T₂/T₁

The output of the engine corresponds to the report/ratio of the work provided (in absolute value) on the heat which it received:

η = |W| /Q₁ = |- (Q₁ + Q₂)|/Q₁ = 1 + Q₂/Q₁.

from where η = 1 - T₂/T₁

It goes without saying this output corresponds to the reversible Cycle of Carnot. It is the maximum theoretical yield which will never be reached at the time of a real cycle.

Remarks

• This relation shows that a thermal engine monotherme cannot provide mechanical work since if T₁ = T₂, the output is null. In fact, so that there is work one needs a transfer of heat; this one not being possible that if there is a difference in temperature.

• In the case of the steam engine the maximum theoretical yield can be calculated. If T₁ = 373K and T₂ = 298K, one finds η = 0,2.

Other interpretations and consequences of the second principle

Transfer of extensity

Another interpretation, more “physics” of the second principle can be formulated. Indeed, let us imagine a hollow roll closed hermetically at the two ends. Let us imagine also a free piston to move in this cylinder. If one moves the piston towards the left, the left part sees its pressure increasing and its volume to decrease and, vice versa, the right part sees its pressure falling and its volume to increase. If the piston is slackened, it spontaneously will move towards the line, towards its initial position of balance. Displacement is thus made part with high pressure, which sees its volume increasing, towards the part with low pressure which sees its volume falling. If one remembers that the intensive Grandeur is the pressure here and that the extensive Grandeur is volume here, this example illustrates the statement according to correspondent with another formulation of the second principle:

energy always runs out of the high intensity towards the low intensity by a transfer of extensity .

In this case: δW = - p.dV

If one puts in contact two objects at different electrostatic potentials, energy will go from the most potential (intensive size) towards low by a transfer of charge (extensive size): of = v.dq .

In the same way, if one puts in contact two sources at different temperatures, heat will run out of the source at high temperature towards that at low temperature by transfer of entropy. The entropy is thus the extensity associated with the energy form called heat: δQ = T.dS .

Second principle and chaos

See also: Formula of Boltzmann, Entropy

Boltzmann studied the second principle under its microscopic aspect what revolutionized physics, putting an end to the hopes Laplace founded on a integral Déterminisme.

In the statistics of Maxwell-Boltzmann, one reasons indeed on a great number of indistinguishable, independent and identical particles. In this case, the entropy of a macro-state Ω is defined (in a statistical way) by the formula of Boltzmann: S = k_b.ln Ω .

Ω corresponds to the number of microphone-state S different observable in a given macro-state.

Cycles of Poincaré

See also: Theorem of recurrence

The famous mathematician Henri Poincaré showed in 1890 an extremely general theorem, whose physical statement is: “Any macroscopic system also passes by again an infinity of time close which one wants of his initial state. ” This “theorem of recurrence” was opposed to the second principle, because it implies that any macroscopic evolution is reversible. To counter this apparently unattackable theorem, Boltzmann calculated the time necessary with 100 gas cm³ to return in his initial state. He found 10^E10 years!!! As much to say that, if the problem of the cycles of Poincaré remains, it is not of an extreme urgency (see in the same order of idea, the example of pack of 52 cards, disordered in Entropie).

The box of Maxwell

That is to say a box circular punt, horizontal, separated by a diameter in 2 equal compartments, and containing NR white metal discs and NR black metal discs, of the same ray R, slipping without frictions on this bottom. To open the diameter of a size higher than 2r, to allow the passage of the pawns. To shake, then to immobilize the box: it is rather intuitive that the state generally carried out is that for which there will be N/2 white metal discs and N/2 black metal discs in compartment 1; this with immense fluctuations, all the more large in absolute value that the box will be large and that NR will be large: these fluctuations grow like $\backslash \; sqrt\; \{NR\}$. But the larger NR is, plus these fluctuations will be negligible in front of NR and the distribution will approach N/2 for each color of metal disc in each compartment. One notices here another aspect of the second principle which shows that the spontaneous evolution of a system always goes towards the homogeneity.

See too

Related articles

• Thermodynamic

• Entropy
• Formula of Boltzmann
• Theorem H
• consequences of the second principle on the Irreversibility
• the perestroika of the formulas of thermodynamic induced by the transformations of Legendre on the Functions characteristic
• the introduction of the functions Exergie, alias Functions of Gouy, like Potential thermodynamic
• the First principle of thermodynamics and the Third principle of thermodynamics

• Calculations of the entropy of a pure substance

Random links:

Cingalais | Europe (the moon) | County of Muan | Automobile Grand Prix of France 1971 | Emmanuel Chukwudi Eze