System in Different States of Aggregation

Excerpts

• System in Different States of Aggregation

  In the critical state the compressibility is infinite; so are also the thermal coefficient of expansion and the specific heat at constant pressure; the heat of vaporization is zero.

• System in Different States of Aggregation

  these curves will meet in one point, the *fundamental point*, also called the *triple point*.

• System in Different States of Aggregation

  The existence of a sharp bend in the curve, however, can only be inferred from theory.

• System in Different States of Aggregation

  No off-hand statement can be made with regard to the value of $h_{1}$; even its sign must in the mean time remain uncertain.

• System in Different States of Aggregation

  A geometrical representation may facilitate a general survey of the problem.

• System in Different States of Aggregation

  Watt assumed this to be the case for steam.

• System in Different States of Aggregation

  The total mass $M$, the volume $V$, and the energy $U$ of a system being given, its corresponding state of stable equilibrium is determined by the position of the point $v = \dfrac{V}{M}$, $u = \dfrac{U}{M}$, in the plane of [fig:4]Fig. 4.

• System in Different States of Aggregation

  In this way we may find that ice cannot exist in stable equilibrium at a higher temperature than the fundamental temperature ($0.0074°$ C.), no matter how the pressure may be reduced. Liquid water, on the other hand, may, under suitable pressure, be brought to any temperature without freezing or evaporating.

• System in Different States of Aggregation

  A question which may also be answered directly is the following. Through what stages will a body pass if subjected to a series of definite external changes?

• System in Different States of Aggregation

  It will be seen that their ratio is that of the three triangles, which the point $(v, u)$ makes with the three sides of the fundamental triangle.

• System in Different States of Aggregation

  This quantity is essentially positive, since $M_{12}$, $M_{21}$, as well as $c_{v}$, are always positive, and $\dfrac{\dd p}{\dd v}$ always negative for states of equilibrium.

• System in Different States of Aggregation

  The conditions of stable equilibrium of any substance can thus be found, provided its fundamental triangle, its vaporization, fusion, and sublimation curves have been drawn once for all.

• System in Different States of Aggregation

  The entropy may in general, however, as we shall see, assume several relative maxima, under the given external conditions. Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium.

• System in Different States of Aggregation

  The system in a state of this kind (*e.g.* as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.

• System in Different States of Aggregation

  This was first verified by the measurements of W. Thomson (Lord Kelvin).

• System in Different States of Aggregation

  Experience immediately shows, however, that in any state of equilibrium $\dfrac{\dd p}{\dd v}$ is negative, since the pressure, whether positive or negative, and the volume always change in opposite directions.

• System in Different States of Aggregation

  The equations % [eqn:(99)](99)% might therefore be called the system’s *internal* or *intrinsic* conditions of equilibrium

• System in Different States of Aggregation

  Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium. The system in a state of this kind (*e.g.* as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.

• System in Different States of Aggregation

  It is still very uncertain whether the molecules of liquid water are the same as those of ice.

• System in Different States of Aggregation

  These six equations represent necessary properties of any state, which corresponds to a maximum value of the entropy, *i.e.* of any state of equilibrium. As the first four refer to equality of temperature and pressure, the main interest centres in the last two, which contain the thermodynamical theory of fusion, evaporation, and sublimation.

• System in Different States of Aggregation

  We learn, therefore, from equation % [eqn:(102)](102)% that in every isotherm the pressure, under which two states of aggregation of the substance may be kept in lasting contact, is represented by the ordinate of the straight line parallel to the axis of abscissæ, which intercepts equal areas on both sides of the isotherm.

• System in Different States of Aggregation

  It is the heat which must be added to unit mass of the liquid, in order to completely change it to vapour under the constant pressure of its saturated vapour.

• System in Different States of Aggregation

  By direct observation Regnault found the heat of vaporization of water at $100°$ C. to be $536$.

• System in Different States of Aggregation

  This shows that the external work forms only a small part of the value of the latent heat of vaporization.

• System in Different States of Aggregation

  The melting pressure, therefore, just as the pressure of evaporation, depends on the temperature only. Conversely, a change of pressure produces a change in the melting point:

• System in Different States of Aggregation

  If this point lie within one of the regions $(1)$, $(2)$, or $(3)$, the system behaves as a homogeneous gas, liquid, or solid. If it lie within $(12)$, $(23)$, or $(31)$, the system splits into two different states of aggregation, indicated by the numbers used in the notation of the region.

• System in Different States of Aggregation

  The masses of these three portions may then be determined by the equations % [eqn:(121a)](121a)%. It will be seen that their ratio is that of the three triangles, which the point $(v, u)$ makes with the three sides of the fundamental triangle.

• System in Different States of Aggregation

  For instance, the behaviour of a body of mass $M$, when cooled or heated at constant volume $V$, may be known by observing the line $v = \dfrac{V}{M}$ parallel to the axis of ordinates. The regions which this line traverses show the states through which the body passes, *e.g.* whether the substance melts during the process, or whether it sublimes, etc.

• System in Different States of Aggregation

  It follows that the plane area $\phi''$ rises everywhere above the surface $\phi'$, and that $\phi'' - \phi'$ is never negative. This proves that the third solution within its region of validity (the fundamental triangle of the substance) represents stable equilibrium.

• System in Different States of Aggregation

  By direct measurement, Regnault found the mean specific heat of steam under atmospheric pressure for temperatures somewhat higher than $100°$ C. to be $0.48$.

• System in Different States of Aggregation

  For, if during a rise of temperature of $1°$ the vapour is to remain just saturated, it must evidently be compressed while being heated, since the specific volume of the saturated vapour decreases as the temperature rises. This compression, however, generates heat, and the question is, whether the latter is so considerable that it must be in part withdrawn by conduction, so as not to superheat the vapour.

• System in Different States of Aggregation

  Water vapour at $100°$ C. represents the first of the cases described above, *i.e.* saturated water vapour at $100°$ is superheated by adiabatic compression. Conversely, saturated water vapour at $100°$ becomes supersaturated by adiabatic expansion.

• System in Different States of Aggregation

  Then the two states which are in contact with one another are identical. Such a value of $\theta$ is called a *critical temperature* of the substance.

• System in Different States of Aggregation

  According to equations % [eqn:(120)](120)%, the fundamental temperature is characterized by the condition that at it the pressure of the saturated vapour is equal to the pressure of fusion. It necessarily follows, by addition of the last two equations, that this pressure is also equal to the pressure of sublimation.

• System in Different States of Aggregation

  Let us determine, *e.g.*, the fundamental state of water. $0°$ C. is not its fundamental temperature, for at $0°$ C. the maximum vapour pressure of water is $4.62~\Unit{mm.}$, but the melting pressure of ice is $760~\Unit{mm}$.

• System in Different States of Aggregation

  *i.e.* at $1°$ C. the maximum vapour pressure of ice is $0.045~\Unit{mm.}$ less than that of water. This has been verified by experiment. The existence of a sharp bend in the curve, however, can only be inferred from theory.

• System in Different States of Aggregation

  If, as a rough approximation, we assume this same ratio to hold for much lower temperatures, the latent heat of fusion would be zero at about $-120°$ C., and this would be the critical point of the fusion curve. The pressure here would be about $17,000$ atmospheres, and water and ice would become identical. We might imagine this to be the result of a considerable increase in the viscosity of water and in the plasticity of ice, as they both approach this state.

Equations

• System in Different States of Aggregation

  v_{1} = \frac{R\theta}{mp_{1}}

  For the vapour, if the perfect-gas characteristic equation is applied, its specific volume is R times the temperature divided by m times its pressure.

• System in Different States of Aggregation

  \left(\frac{\dd v_{1}}{\dd \theta}\right)_{p} = \frac{R}{mp_{1}}

  Under the perfect-gas approximation, the rate of change of the vapour's specific volume with temperature at constant pressure is R divided by m times its pressure.

• System in Different States of Aggregation

  (c_{p})_{1} - (c_{p})_{2} = \frac{dL}{d\theta}

  With the perfect-gas approximation the difference between the specific heats at constant pressure of vapour and liquid equals the rate of change of the latent heat with temperature.

• System in Different States of Aggregation

  \frac{dL}{d\theta} = (c_{p})_{1} - (c_{p})_{2} + \frac{L}{\theta} - \frac{L}{v_{1} - v_{2}} \left[\left(\frac{\dd v_{1}}{\dd \theta}\right)_{p} - \left(\frac{\dd v_{2}}{\dd \theta}\right)_{p}\right]

  The rate of change of the latent heat with temperature is expressed through the specific heats, the latent heat, the temperature, and the specific volumes and their thermal expansion rates at constant pressure.

• System in Different States of Aggregation

  c = \frac{du}{d\theta} + p\, \frac{dv}{d\theta}

  The specific heat under any heating condition equals the rate of change of internal energy with temperature plus the pressure times the rate of change of specific volume (first law applied to heating).

• System in Different States of Aggregation

  h_{1} = \frac{du_{1}}{d\theta} + p_{1}\, \frac{dv_{1}}{d\theta}

  The specific heat of the saturated vapour, defined for the process that keeps the vapour saturated, is the internal-energy rate plus pressure times volume rate for the vapour.

• System in Different States of Aggregation

  h_{2} = \frac{du_{2}}{d\theta} + p_{2}\, \frac{dv_{2}}{d\theta}

  The specific heat of the liquid kept under the pressure of its saturated vapour is the internal-energy rate plus pressure times volume rate for the liquid.

• System in Different States of Aggregation

  h_{2} = (c_{p})_{2}

  Because external pressure has no appreciable effect on a liquid unless it is many atmospheres, the liquid's saturation specific heat practically equals its specific heat at constant pressure.

• System in Different States of Aggregation

  h_{1} = (c_{p})_{2} + \frac{dL}{d\theta} - \frac{L}{\theta}

  The specific heat of saturated vapour equals the liquid's specific heat at constant pressure plus the rate of change of latent heat with temperature minus the latent heat divided by the temperature.

• System in Different States of Aggregation

  \left(\frac{\dd p}{\dd v}\right)_{2} = 0

  One of the two conditions of the critical state: the pressure's derivative with respect to specific volume vanishes at the critical point.

• System in Different States of Aggregation

  \left(\frac{\dd^{2} p}{\dd v^{2}}\right)_{2} = 0

  The second of the two conditions of the critical state: the second derivative of pressure with respect to specific volume vanishes at the critical point.

• System in Different States of Aggregation

  p = p_{2} + \left(\frac{\dd p}{\dd v}\right)_{2} (v - v_{2}) + \tfrac{1}{2} \left(\frac{\dd^{2} p}{\dd v^{2}}\right)_{2} (v - v_{2})^{2}\Add{,}

  For a small volume difference, Taylor's theorem expresses the pressure at any intermediate volume through its value and first two derivatives at state 2.

• System in Different States of Aggregation

  p_{1} = p_{2} = p_{3}\Add{,}

  In the three-state equilibrium the pressures of the vapour, liquid and solid are all equal.

• System in Different States of Aggregation

  \phi_{1} - \phi_{2} = \frac{u_{1} - u_{2} + p_{1}(v_{1} - v_{2})}{\theta}\Add{,}

  In three-state coexistence, the difference of specific entropies of the gaseous and liquid states equals the internal-energy difference plus pressure times volume difference, divided by the temperature.

• System in Different States of Aggregation

  \phi_{2} - \phi_{3} = \frac{u_{2} - u_{3} + p_{1}(v_{2} - v_{3})}{\theta}\Add{.}

  The corresponding relation between the liquid and solid states. Note: the book writes p_1 here where the pattern of the previous equation would give p_2; since all three pressures are equal (the line above) the statement is unaffected, but the subscript is as printed.

• System in Different States of Aggregation

  M_{1} + (M_{2} + M_{3}) = M\Add{,}

  The three portion masses sum to the total mass of the substance.

• System in Different States of Aggregation

  M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V\Add{,}

  The total volume is the sum of each portion's mass times its specific volume.

• System in Different States of Aggregation

  M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U\Add{,}

  The total energy is the sum of each portion's mass times its specific energy.

• System in Different States of Aggregation

  p_{12} = p_{21}

  The saturated vapour pressure is the same whether the vapour is referred to as in contact with the liquid or the liquid is referred to as in contact with the vapour.

• System in Different States of Aggregation

  \frac{dp_{12}}{d\theta} = \frac{L_{12}}{\theta (v_{1} - v_{2})}

  The slope of the vaporization pressure curve at the fundamental point equals the latent heat divided by temperature times the volume difference of the two phases.

• System in Different States of Aggregation

  \frac{dp_{23}}{d\theta} = \frac{L_{23}}{\theta (v_{2} - v_{3})}

  The slope of the fusion pressure curve at the fundamental point equals the latent heat of fusion divided by temperature times the volume difference of liquid and solid.

• System in Different States of Aggregation

  \frac{dp_{31}}{d\theta} = \frac{L_{31}}{\theta (v_{3} - v_{1})}

  The slope of the sublimation pressure curve at the fundamental point equals the latent heat of sublimation divided by temperature times the volume difference of solid and vapour.

• This equation is in Molecular Weight (Molecular Weight)

• System in Different States of Aggregation

  u = \dfrac{U}{M}

  The mean specific energy of the system is its total energy divided by its total mass.

• System in Different States of Aggregation

  \Phi = M\phi

  For a single homogeneous state the total entropy is the total mass times the specific entropy.

• System in Different States of Aggregation

  \Phi' = M\phi' = M_{12} \phi_{12} + M_{21} \phi_{21}

  For the vapour-liquid solution the total entropy is the sum of the entropies of the two portions, and its specific form is M times phi-prime.

• System in Different States of Aggregation

  \Phi'' = M\phi'' = M_{1} \phi_{1} + M_{2} \phi_{2} + M_{3} \phi_{3}\Add{.}

  For the three-state coexistence the total entropy is the sum of the entropies of the three portions.

• System in Different States of Aggregation

  \phi'' > \phi' > \phi

  For any system with all partial masses positive, the specific entropy of the three-state solution exceeds that of the two-state solution, which exceeds that of the single-state solution (the claim is stated as shown for positive partial masses).

• System in Different States of Aggregation

  \phi_{12} - \phi_{21} = \frac{u_{12} - u_{21} + p_{12} (v_{12} - v_{21})}{\theta_{12}}

  On the vaporization curve, the entropy difference of the vapour and liquid at corresponding points equals the internal-energy difference plus pressure times volume difference, divided by temperature.

• System in Different States of Aggregation

  \frac{du_{12}}{dv_{12}} = \left(\frac{\dd u}{\dd v}\right)_{12} + \left(\frac{\dd u}{\dd \theta}\right)_{12} \frac{d\theta_{12}}{dv_{12}}

  Chain rule: along the vaporization curve the slope du/dv equals the partial derivatives at constant temperature and constant volume, weighted by the change of saturation temperature with volume.

• System in Different States of Aggregation

  \frac{du_{12}}{dv_{12}} = \theta_{12} \left(\frac{\dd p}{\dd \theta}\right)_{12} - p_{12} + (c_{v})_{12}\, \frac{d\theta_{12}}{dv_{12}}

  Along the vaporization-curve branch the slope du/dv is given by temperature times the pressure's thermal derivative minus pressure plus specific heat at constant volume times the saturation-temperature change with volume.

• System in Different States of Aggregation

  \frac{du_{21}}{dv_{21}} = \theta_{21} \left(\frac{\dd p}{\dd \theta}\right)_{21} - p_{12} + (c_{v})_{21}\, \frac{d\theta_{12}}{dv_{21}}

  The corresponding relation for the liquid branch of the vaporization curve. Note: the book writes p_{12} and d\theta_{12}/dv_{21} here; the text states p_{21}=p_{12} and \theta_{21}=\theta_{12}, so the printed subscripts are equivalent by those identities, but are reproduced exactly as printed.

• System in Different States of Aggregation

  \frac{u_{12} - u_{21}}{v_{12} - v_{21}} = \theta_{12}\, \frac{dp_{12}}{d\theta_{12}} - p_{12}

  The slope of the chord joining corresponding vapour and liquid points equals temperature times the pressure derivative along saturation minus the pressure.

• System in Different States of Aggregation

  \frac{\dd u}{\dd v} = \theta\, \frac{dp}{d\theta} - p

  At constant temperature the partial derivative of specific energy with respect to specific volume equals temperature times the thermal pressure derivative minus pressure.

• System in Different States of Aggregation

  \frac{dp_{12}}{d\theta_{12}} = \frac{\dd p}{\dd \theta} + \frac{\dd p}{\dd v} · \frac{dv_{12}}{d\theta_{12}}

  Total derivative of the saturation pressure with respect to saturation temperature: the thermal derivative plus the volume derivative times the volume change along saturation.

• System in Different States of Aggregation

  p_{12} (v - v_{12}) + (u - u_{12}) - \theta_{12} (\phi - \phi_{12}) = 0

  The plane in (v, u, phi) space through a corresponding pair of points on the vaporization curve; it contains both points and the line joining them.

• System in Different States of Aggregation

  v = \frac{\lambda v_{12} + \mu v_{21}}{\lambda + \mu}

  Points on the straight line joining corresponding points lie at weighted averages of the two corresponding specific volumes, with positive weights lambda and mu.

• System in Different States of Aggregation

  \delta \phi' = \frac{\delta u + p_{12}\, \delta v}{\theta_{12}}

  The variation of the two-state specific entropy equals the variation of specific energy plus pressure times variation of specific volume, divided by the saturation temperature.

• System in Different States of Aggregation

  \delta (\phi' - \phi) = \left(\frac{1}{\theta_{12}} - \frac{1}{\theta}\right) \delta u + \left(\frac{p_{12}}{\theta_{12}} - \frac{p}{\theta}\right) \delta v

  The variation of the difference between the two entropy surfaces is a linear form in the variations of u and v, which vanishes along the curve of contact.

• System in Different States of Aggregation

  \theta\, \delta^{2} (\phi' - \phi) = (\delta \theta - \delta \theta_{12})\, \delta \phi + (\delta p_{12} - \delta p)\, \delta v

  At points of contact the second variation of the entropy difference, scaled by temperature, is expressed through the variations of temperature, saturation temperature, pressure and entropy.

• System in Different States of Aggregation

  \delta \phi = \frac{c_{v}}{\theta}\, \delta \theta + \frac{\dd p}{\dd \theta}\, \delta v

  The variation of specific entropy in terms of the variations of temperature and specific volume, with specific heat at constant volume (the book cites Eq. 81 for this).

• System in Different States of Aggregation

  \delta p = \frac{\dd p}{\dd \theta}\, \delta \theta + \frac{\dd p}{\dd v}\, \delta v

  The variation of pressure is the sum of its thermal and volume derivatives times the variations of temperature and specific volume.

• System in Different States of Aggregation

  \delta \theta_{12} = \frac{c_{v}\, \delta \theta - \theta\, \dfrac{\dd p}{\dd v} · \dfrac{dv_{12}}{d\theta_{12}}\, \delta v}{c_{v} - \theta\, \dfrac{\dd p}{\dd v} \left(\dfrac{dv_{12}}{d\theta_{12}}\right)^{2}}

  The variation of saturation temperature expressed through the variations of temperature and specific volume of the state.

• System in Different States of Aggregation

  \delta^{2} (\phi' - \phi) = -\frac{\dd p}{\dd v} · \frac{c_{v}}{\theta} · \frac{\left(\dfrac{dv_{12}}{d\theta_{12}}\, \delta \theta - \delta v\right)^{2}}{c_{v} - \theta\, \dfrac{\dd p}{\dd v} \left(\dfrac{dv_{12}}{d\theta_{12}}\right)^{2}}

  The second variation of the entropy difference is essentially positive, since c_v is positive and dp/dv is negative for stable states, so the two-state surface rises above the single-state surface, establishing stability of the two-state solution within its region.

• System in Different States of Aggregation

  v &= \frac{\lambda v_{1} + \mu v_{2} + \nu v_{3}}{\lambda + \mu + \nu}

  The specific volume of a point on the plane spanned by the three corner points is the weighted average of the corner volumes v1, v2, v3, with weights lambda, mu, nu.

• System in Different States of Aggregation

  u &= \frac{\lambda u_{1} + \mu u_{2} + \nu u_{3}}{\lambda + \mu + \nu}

  The specific energy of a point on the plane spanned by the three corner points is the weighted average of the corner energies u1, u2, u3, with weights lambda, mu, nu.

• System in Different States of Aggregation

  M\, \delta\phi'' = \phi_{1}\, \delta M_{1} + \phi_{2}\, \delta M_{2} + \phi_{3}\, \delta M_{3}

  The variation of the total entropy of the mixed system equals the sum of the entropies of the three portions weighted by the variations of their masses.

• System in Different States of Aggregation

  \delta M_{1} + \delta M_{2} + \delta M_{3} &= 0\Add{,}

  The variations of the three portion masses sum to zero, so the total mass is unchanged.

• System in Different States of Aggregation

  v_{1}\, \delta M_{1} + v_{2}\, \delta M_{2} + v_{3}\, \delta M_{3} &= M\, \delta v\Add{,}

  The variation of the total volume equals the sum of the volume variations of the three portions.

• System in Different States of Aggregation

  u_{1}\, \delta M_{1} + u_{2}\, \delta M_{2} + u_{3}\, \delta M_{3} &= M\, \delta u\Add{.}

  The variation of the total internal energy equals the sum of the energy variations of the three portions.

• System in Different States of Aggregation

  \delta \phi'' = \frac{\delta u + p_{1}\, \delta v}{\theta_{1}}

  The variation of the mean specific entropy of the mixed state equals the variation of specific energy plus p1 times the variation of specific volume, divided by the temperature theta_1.

• System in Different States of Aggregation

  \theta_{1}^{2}\, \delta^{2} (\phi'' - \phi') = \left[\delta u - \left(\theta_{1}\, \frac{dp_{12}}{d\theta_{12}} - p_{1}\right) \delta v\right] \delta \theta_{12}.

  The second variation of the entropy difference along the sheet (12) is proportional to the variation of temperature theta_12 times a linear combination of the variations of u and v; its sign decides stability.

• System in Different States of Aggregation

  \frac{M_{12}\, \delta v_{12} + M_{21}\, \delta v_{21} - M\, \delta v}{v_{12} - v_{21}} = \frac{M_{12}\, \delta u_{12} + M_{21}\, \delta u_{21} - M\, \delta u}{u_{12} - u_{21}}

  Eliminating the masses M_12 and M_21 gives a relation between the variations of volume and energy for the two-phase states along the coexistence line.

• System in Different States of Aggregation

  M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U

  The energies of the portions add up to the given energy of the system.

• System in Different States of Aggregation

  M_{1} + M_{2} + M_{3} = M

  The masses of the solid, liquid and gaseous portions add up to the total mass of the system.

• System in Different States of Aggregation

  M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V

  The volumes of the portions, each mass times its specific volume, add up to the given volume of the system.

• System in Different States of Aggregation

  \Phi = M_{1} \phi_{1} + M_{2} \phi_{2} + M_{3} \phi_{3}

  The entropy of the system is the sum of the masses times the specific entropies of the portions.

• System in Different States of Aggregation

  \delta \phi = \frac{\delta u + p\, \delta v}{\theta}

  For an infinitesimal change of state, the change in specific entropy equals the change in specific energy plus pressure times change in specific volume, divided by the temperature.

• System in Different States of Aggregation

  \theta_{1} = \theta_{2} = \theta_{3} (= \theta)

  In equilibrium the temperatures of the three portions are all equal to a common temperature.

• System in Different States of Aggregation

  p_{1} = p_{2} = p_{3}

  In equilibrium the pressures of the three portions are equal.

• System in Different States of Aggregation

  \phi_{1} - \phi_{2} = \frac{(u_{1} - u_{2}) + p_{1}(v_{1} - v_{2})}{\theta}

  In equilibrium between the first and second portions, their difference in specific entropy is fixed by their energies, pressure and volumes at the common temperature.

• System in Different States of Aggregation

  \phi_{2} - \phi_{3} = \frac{(u_{2} - u_{3}) + p_{2}(v_{2} - v_{3})}{\theta}

  In equilibrium between the second and third portions, their difference in specific entropy is fixed by their energies, pressure and volumes at the common temperature.

• System in Different States of Aggregation

  \theta\, \delta^{2} \Phi = -\tsum M_{1} \left(\frac{(c_{v})_{1}}{\theta}\, \delta \theta_{1}^{2} - \left(\frac{\dd p_{1}}{\dd v}\right)_{\theta} \delta v_{1}^{2}\right)

  The second variation of the entropy, multiplied by the temperature, is given in independent variables; it is negative, and the entropy is a maximum, when specific heat is positive and the isothermal change of pressure with volume is negative.

• System in Different States of Aggregation

  \int_{v_{2}}^{v_{1}} p\, dv = p_{1} (v_{1} - v_{2})

  For two states of aggregation in contact, the integral of pressure along the isotherm between the two specific volumes equals the constant pressure times the difference of volumes.

• System in Different States of Aggregation

  \int_{v_{3}}^{v_{2}} p\, dv = p_{2}(v_{2} - v_{3})

  The corresponding integral condition holds between the second and third portions in contact.

• System in Different States of Aggregation

  \phi_{1} - \phi_{2} = \frac{u_{1} - u_{2}}{\theta} + \frac{1}{\theta} \int_{v_{2}}^{v_{1}} p\, dv

  The entropy difference between two portions is obtained by integrating along an isotherm: the energy difference over temperature plus the integral of pressure over volume divided by temperature.

• System in Different States of Aggregation

  \frac{R\theta}{v_{1} - a} - \frac{c}{\theta (v_{1} + b)^{2}} = \frac{R\theta}{v_{2} - a} - \frac{c}{\theta (v_{2} + b)^{2}}

  Under Clausius' equation of state, the saturated vapour and the liquid in contact have equal pressure at the same temperature.

• System in Different States of Aggregation

  R\theta \log \frac{v_{1} - a}{v_{2} - a} - \frac{c}{\theta} \left(\frac{1}{v_{2} + b} - \frac{1}{v_{1} + b}\right) = (v_{1} - v_{2}) \left(\frac{R\theta}{v_{1} - a} - \frac{c}{\theta (v_{1} + b)^{2}}\right)

  Under Clausius' equation of state, the equal-area condition on the isotherm (the integral of pressure between the two volumes equals the coexistence pressure times the volume difference) takes this explicit form.

• System in Different States of Aggregation

  u - \theta \phi = f

  The free energy per unit mass is defined as specific energy minus temperature times specific entropy.

• System in Different States of Aggregation

  f_{2} - f_{1} = p_{1} (v_{1} - v_{2})

  In equilibrium between vapour and liquid, the difference of free energies per unit mass equals the coexistence pressure times the volume difference.

• System in Different States of Aggregation

  \left(\frac{\dd f}{\dd \theta}\right)_{v} = -\phi

  At constant specific volume, the rate of change of free energy with temperature equals minus the specific entropy.

• System in Different States of Aggregation

  \left(\frac{\dd f}{\dd v}\right)_{\theta} = -p

  At constant temperature, the rate of change of free energy with specific volume equals minus the pressure.

• System in Different States of Aggregation

  (u_{1} - u_{2}) + p_{1} (v_{1} - v_{2}) = \theta (v_{1} - v_{2})\, \frac{dp_{1}}{d\theta}

  The latent heat of vaporization equals the temperature times the volume change times the slope of the vapour-pressure curve (Clapeyron's relation).

• System in Different States of Aggregation

  \phi_{1} - \phi_{2} = (v_{1} - v_{2})\, \frac{dp_{1}}{d\theta}

  The difference of specific entropies of vapour and liquid equals their volume difference times the slope of the vapour-pressure curve.

• System in Different States of Aggregation

  L = u_{1} - u_{2} + p_{1}(v_{1} - v_{2})

  The latent heat of vaporization is the change of energy plus the external work done against the saturated vapour pressure.

• System in Different States of Aggregation

  W = -p_{1}(v_{1} - v_{2})

  The external work performed during vaporization at constant pressure equals minus the pressure times the volume increase.

• System in Different States of Aggregation

  L = \theta (v_{1} - v_{2})

  The heat of vaporization equals the absolute temperature times the difference of specific volumes of vapour and liquid; this relation was deduced by Clapeyron from Carnot's theory and rigorously proved by Clausius.

• System in Different States of Aggregation

  u_{1} = c_{v} \theta + \const

  For a perfect gas the specific energy is a linear function of temperature with the specific heat at constant volume as slope.

• System in Different States of Aggregation

  \frac{d\theta}{dp_{1}} = \frac{\theta (v_{1} - v_{2})}{L}

  The change of melting point (or boiling point) with pressure is the temperature times the volume change divided by the latent heat.

• System in Different States of Aggregation

  \frac{L}{\theta} = \phi_{1} - \phi_{2}

  The heat of phase change divided by the temperature equals the difference of specific entropies of the two states.

Problems

No exercises in this chapter.