imported>Paul Wormer |
imported>Paul Wormer |
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| ==Entropy== | | ==Parabolic mirror== |
| Clausius was able to give a mathematical expression of the second law of thermodynamics. To that end he needed a totally new thermodynamic concept, one that had no mechanical analogy and that had no intuitive meaning like temperature. He called the new thermodynamic property [[entropy]] from the classical Greek έν + τροπη (tropè = change, en = at). Following in his footsteps entropy will be introduced in this section.
| | {{Image|Refl parab.png|right|350px|Fig. 2. Reflection in a parabolic mirror}} |
| | Parabolic mirrors concentrate incoming vertical light beams in their focus. We show this. |
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| The state of a [[thermodynamic system]] (a point in state space) is characterized by a number of variables, such as [[pressure]] ''p'', [[temperature]] ''T'', amount of substance ''n'', volume ''V'', etc. Any thermodynamic parameter can be seen as a function of an arbitrary independent set of other thermodynamic variables, hence the terms "property", "parameter", "variable" and "function" are used interchangeably. The number of ''independent'' thermodynamic variables of a system is equal to the number of energy contacts of the system with its surroundings.
| | Consider in figure 2 the arbitrary vertical light beam (blue, parallel to the ''y''-axis) that enters the parabola and hits it at point ''P'' = (''x''<sub>1</sub>, ''y''<sub>1</sub>). The parabola (red) has focus in point ''F''. The incoming beam is reflected at ''P'' obeying the well-known law: incidence angle is angle of reflection. The angles involved are with the line ''APT'' which is tangent to the parabola at point ''P''. It will be shown that the reflected beam passes through ''F''. |
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| An example of a reversible (quasi-static) energy contact is offered by the prototype thermodynamical system, a gas-filled cylinder with piston. Such a cylinder can perform work on its surroundings,
| | Clearly ∠''BPT'' = ∠''QPA'' (they are vertically opposite angles). Further ∠''APQ'' = ∠''FPA'' because the triangles ''FPA'' and ''QPA'' are congruent and hence ∠''FPA'' = ∠''BPT''. |
| :<math>
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| DW = pdV, \quad dV > 0,
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| </math>
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| where ''dV'' stands for a small increment of the volume ''V'' of the cylinder, ''p'' is the pressure inside the cylinder and ''DW'' stands for a small amount of work. Work by expansion is a form of energy contact between the cylinder and its surroundings. This process can be reverted, the volume of the cylinder can be decreased, the gas is compressed and the surroundings perform work ''DW'' = ''pdV'' ''on'' the cylinder.
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| The small amount of work is indicated by ''D'', and not by ''d'', because ''DW'' is not necessarily a differential of a function. However, when we divide ''DW'' by ''p'' the quantity ''DW''/''p'' becomes obviously equal to the differential ''dV'' of the differentiable state function ''V''. State functions depend only on the actual values of the thermodynamic parameters (they are local), and ''not'' on the path along which the state was reached (the history of the state). Mathematically this means that integration from point 1 to point 2 along path I in state space is equal to integration along a different path II,
| | We prove the congruence of the triangles: By the definition of the parabola the line segments ''FP'' and ''QP'' are of equal length, because the length of the latter segment is the distance of ''P'' to the directrix and the length of ''FP'' is the distance of ''P'' to the focus. The point ''F'' has the coordinates (0,''f'') and the point ''Q'' has the coordinates (''x''<sub>1</sub>, −''f''). The line segment ''FQ'' has the equation |
| :<math>
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| V_2 - V_1 = {\int\limits_1\limits^2}_{{\!\!}^{(I)}} dV
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| = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} dV
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| \;\Longrightarrow\; {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DW}{p} =
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| {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DW}{p}
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| </math>
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| The amount of work (divided by ''p'') performed along path I is equal to the amount of work (divided by ''p'') along path II. This condition is necessary and sufficient that ''DW''/''p'' is a differentiable state function. So, although ''DW'' is not a differential, the quotient ''DW''/''p'' is one.
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| Reversible absorption of a small amount of heat ''DQ'' is another energy contact of a system with its surroundings; ''DQ'' is again not a differential of a certain function. In a completely analogous manner to ''DW''/''p'', the following result can be shown for the heat ''DQ'' (divided by ''T'') absorbed by the system along two different paths (along both paths the absorption is reversible):
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| <div style="text-align: right;" > | |
| <div style="float: left; margin-left: 35px;" >
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| <math>{\int\limits_1\limits^2}_{{\!\!}^{(I)}}\frac{DQ}{T} = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DQ}{T} .
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| </math>
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| </div>
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| <span id="(1)" style="margin-right: 200px; vertical-align: -40px; ">(1)</span>
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| </div>
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| <br><br>
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| Hence the quantity ''dS'' defined by
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| :<math>
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| dS \;\stackrel{\mathrm{def}}{=}\; \frac{DQ}{T}
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| </math>
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| is the differential of a state variable ''S'', the ''entropy'' of the system. In a later subsection equation (1) will be proved from the Clausius/Kelvin principle. Observe that this definition of entropy only fixes entropy differences:
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| :<math>
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| S_2-S_1 \equiv \int_1^2 dS = \int_1^2 \frac{DQ}{T}
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| </math>
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| Note further that entropy has the dimension energy per degree temperature (joule per degree kelvin) and recalling the [[first law of thermodynamics]] (the differential ''dU'' of the [[internal energy]] satisfies ''dU'' = ''DQ'' − ''DW''), it follows that
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| :<math> | | :<math> |
| dU = TdS - pdV.\,
| | \lambda\begin{pmatrix}0\\ f\end{pmatrix} + (1-\lambda)\begin{pmatrix}x_1\\ -f\end{pmatrix}, \quad 0\le\lambda\le 1. |
| </math> | | </math> |
| (For convenience sake only a single work term was considered here, namely ''DW'' = ''pdV'', work done ''by'' the system).
| | The midpoint ''A'' of ''FQ'' has coordinates (λ = ½): |
| The internal energy is an extensive quantity, that is, when the system is doubled, ''U'' is doubled too. The temperature ''T'' is an intensive property, independent of the size of the system. The entropy ''S'', then, is an extensive property. In that sense the entropy resembles the volume of the system. | |
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| An important difference between ''V'' and ''S'' is that the former is a state function with a well-defined mechanical meaning, whereas entropy is introduced by analogy and is not easily visualized. Indeed, as is shown in the next subsection, it requires a fairly elaborate reasoning to prove that ''S'' is a state function, i.e., equation [[#(1)|(1)]] to hold.
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| ===Proof that entropy is a state function===
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| When equation [[#(1)|(1)]] has been proven, the entropy ''S'' is shown to be a state function. The standard proof, as given now, is physical, by means of [[Carnot cycle]]s, and is based on the Clausius/Kelvin formulation of the second law given in the introduction.
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| {{Image|Entropy.png|right|350px|Fig. 1. ''T'' > ''T''<sub>0</sub>. (I): Carnot engine E moves heat from heat reservoir R to "condensor" C and needs input of work DW<sub>in</sub>. (II): E generates work DW<sub>out</sub> from the heat flow from C to R. }} An alternative, more mathematical proof, postulates the existence of a state variable ''S'' with certain properties and derives the existence of [[thermodynamical temperature]] and the second law from these properties.
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| In figure 1 a finite heat bath C ("condensor")<ref>Because of a certain similarity of C with the condensor of a steam engine C is referred as "condensor". The quotes are used to remind us that nothing condenses, unlike the steam engine where steam condenses to water</ref> of constant volume and variable temperature ''T'' is shown. It is connected to an infinite heat reservoir R through a reversible Carnot engine E. Because R is infinite its temperature ''T''<sub>0</sub> is constant, addition or extraction of heat does not change ''T''<sub>0</sub>. It is assumed that always ''T'' ≥ ''T''<sub>0</sub>. One may think of the system E-plus-C as a ship and the heat reservoir R as the sea. The following argument then deals with an attempt of extracting energy from the sea in order to move the ship, i.e., with an attempt to let E perform net outgoing work in a cyclic (i.e., along a closed path in the state space of C) process.
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| A Carnot engine performs reversible cycles (in the state space of E, not be confused with cycles in the state space of C) and per cycle either generates work ''DW''<sub>out</sub> when heat is transported from high temperature to low temperature (II), or needs work ''DW''<sub>in</sub> when heat is transported from low to high temperature (I), in accordance with the Clausius/Kelvin formulation of the second law.
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| The definition of [[thermodynamical temperature]] (a positive quantity) is such that for II,
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| :<math>
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| \frac{DW_\mathrm{out}}{DQ} = \frac{T-T_0}{T},
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| </math>
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| while for I
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| :<math>
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| \frac{DW_\mathrm{in}}{DQ_0} = \frac{T-T_0}{T_0}.
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| </math>
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| The first law of thermodynamics states for I and II, respectively,
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| :<math>
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| -DW_\mathrm{in} -DQ_0 + DQ=0\quad\hbox{and}\quad DW_\mathrm{out} + DQ_0-DQ=0
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| </math>
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| {{Image|Cycle entropy.png|right|150px|Fig. 1. Two paths in the state space of the "condensor" C.}}
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| For I,
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| :<math>
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| \begin{align}
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| \frac{DW_\mathrm{in}}{DQ_0} &= \frac{DQ- DQ_0}{DQ_0} = \frac{DQ}{DQ_0} -1 \\
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| &=\frac{T-T_0}{T_0} = \frac{T}{T_0} - 1 \; | |
| \Longrightarrow DQ_0 = T_0 \left(\frac{DQ}{T}\right)
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| \end{align}
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| </math>
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| For II we find the same result,
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| :<math>
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| \begin{align}
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| \frac{DW_\mathrm{out}}{DQ} &= \frac{DQ- DQ_0}{DQ} = 1- \frac{DQ_0}{DQ} \\
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| &=\frac{T-T_0}{T} = 1- \frac{T_0}{T}
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| \;\Longrightarrow DQ_0 = T_0 \left(\frac{DQ}{T}\right)
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| \end{align}
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| </math>
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| In figure 2 the state diagram of the "condensor" C is shown. Along path I the Carnot engine needs input of work to transport heat from the colder reservoir R to the hotter C and the absorption of heat by C raises its temperature and pressure. Integration of ''DW''<sub>in</sub> = ''DQ'' − ''DQ''<sub>0</sub> (that is, summation over many cycles of the engine E) along path I gives
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| :<math> | |
| W_\mathrm{in} = Q_\mathrm{in} - T_0 {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DQ}{T} \quad\hbox{with}\quad Q_\mathrm{in} \equiv {\int\limits_1\limits^2}_{{\!\!}^{(I)}} DQ.
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| </math>
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| Along path II the Carnot engine delivers work while transporting heat from C to R. Integration of ''DW''<sub>out</sub> = ''DQ'' − ''DQ''<sub>0</sub> along path II gives
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| :<math>
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| W_\mathrm{out} = Q_\mathrm{out} - T_0 {\int\limits_2\limits^1}_{{\!\!}^{(II)}} \frac{DQ}{T}
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| \quad\hbox{with}\quad Q_\mathrm{out} \equiv {\int\limits_2\limits^1}_{{\!\!}^{(II)}} DQ
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| </math>
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| Assume now that the amount of heat ''Q''<sub>out</sub> extracted (along path II) from C and the heat ''Q''<sub>in</sub> delivered (along I) to C are the same in absolute value. In other words, after having gone along a closed path in the state diagram of figure 2, the condensor C has not gained or lost heat. That is,
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| :<math> | | :<math> |
| Q_\mathrm{in} + Q_\mathrm{out} = 0, \,
| | \frac{1}{2}\begin{pmatrix}0\\ f\end{pmatrix} + \frac{1}{2}\begin{pmatrix}x_1\\ -f\end{pmatrix} = |
| | \begin{pmatrix}\frac{1}{2} x_1\\ 0\end{pmatrix}. |
| </math> | | </math> |
| then
| | Hence ''A'' lies on the ''x''-axis. |
| | The parabola has equation, |
| :<math> | | :<math> |
| W_\mathrm{in} + W_\mathrm{out} = - T_0 {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DQ}{T}
| | y = \frac{1}{4f} x^2. |
| - T_0 {\int\limits_2\limits^1}_{{\!\!}^{(II)}} \frac{DQ}{T}.
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| </math> | | </math> |
| If the total net work ''W''<sub>in</sub> + ''W''<sub>out</sub> is positive (outgoing), this work is done by heat obtained from R, which is not possible because of the Clausius/Kelvin principle. If the total net work ''W''<sub>in</sub> + ''W''<sub>out</sub> is negative, then by inverting all reversible processes, i.e., by going down path I and going up along II, the net work changes sign and becomes positive (outgoing). Again the Clausius/Kelvin principle is violated. The conclusion is that the net work is zero and that
| | The equation of the tangent at ''P'' is |
| :<math> | | :<math> |
| T_0 {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DQ}{T} +
| | y = y_1 + \frac{x_1}{2f} (x-x_1)\quad \hbox{with}\quad y_1 = \frac{x_1^2}{4f}. |
| T_0 {\int\limits_2\limits^1}_{{\!\!}^{(II)}} \frac{DQ}{T} = 0
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| \;\Longrightarrow\; {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DQ}{T} = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DQ}{T}. | |
| </math> | | </math> |
| From this independence of path it is concluded that
| | This line intersects the ''x''-axis at ''y'' = 0, |
| :<math> | | :<math> |
| dS \equiv \frac{DQ}{T}
| | 0 = \frac{x_1^2}{4f} - \frac{x_1^2}{2f} + \frac{x_1}{2f} x |
| | \Longrightarrow \frac{x_1}{2f} x = \frac{x_1^2}{4f} \longrightarrow x = \tfrac{1}{2}x_1. |
| </math> | | </math> |
| is a state (local) variable. | | The intersection of the tangent with the ''x''-axis is the point ''A'' = (½''x''<sub>1</sub>, 0) that lies on the midpoint of ''FQ''. The corresponding sides of the triangles ''FPA'' and ''QPA'' are of equal length and hence the triangles are congruent. |
Parabolic mirror
PD Image Fig. 2. Reflection in a parabolic mirror
Parabolic mirrors concentrate incoming vertical light beams in their focus. We show this.
Consider in figure 2 the arbitrary vertical light beam (blue, parallel to the y-axis) that enters the parabola and hits it at point P = (x1, y1). The parabola (red) has focus in point F. The incoming beam is reflected at P obeying the well-known law: incidence angle is angle of reflection. The angles involved are with the line APT which is tangent to the parabola at point P. It will be shown that the reflected beam passes through F.
Clearly ∠BPT = ∠QPA (they are vertically opposite angles). Further ∠APQ = ∠FPA because the triangles FPA and QPA are congruent and hence ∠FPA = ∠BPT.
We prove the congruence of the triangles: By the definition of the parabola the line segments FP and QP are of equal length, because the length of the latter segment is the distance of P to the directrix and the length of FP is the distance of P to the focus. The point F has the coordinates (0,f) and the point Q has the coordinates (x1, −f). The line segment FQ has the equation
The midpoint A of FQ has coordinates (λ = ½):
Hence A lies on the x-axis.
The parabola has equation,
The equation of the tangent at P is
This line intersects the x-axis at y = 0,
The intersection of the tangent with the x-axis is the point A = (½x1, 0) that lies on the midpoint of FQ. The corresponding sides of the triangles FPA and QPA are of equal length and hence the triangles are congruent.
