Posts Tagged ‘thermodynamics’

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Before we begin

Here’s some news. My January 2014 blogpost “Carathéodory: the forgotten pioneer” has been translated into Greek by Giorgos Vachtanidis, and can be seen here.

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Two years on …

Despite the somewhat esoteric nature of Carathéodory’s axiomatic approach to thermodynamics via the geometric behavior of Pfaffians – or perhaps even because of it – my blogpost “Carathéodory: the forgotten pioneer” has received a surprisingly large number of hits, with plenty of brave individuals willing to click on the link to the English version of Carathéodory’s original paper published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics].

Carathéodory’s second axiom “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes”, and the associated theorem giving the condition for dQ to be an integrable differential, constitute the real novelty of his approach.

My original post described Carathéodory’s theorem without going into the proof, since it is rather abstruse and would have appealed only to more avid students of his work. Two years on however, the statistics for this blogpost reveal that there are plenty of avid students wanting to know more. So as a supplementary post, here is a proof of Carathéodory’s theorem, due to Pierre Perrot. Enjoy.

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Carathéodory’s theorem

“If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.”

In the following, use is made of a classical result known as the Clausius inequality

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Proof

Cases n = 1 and n = 2 are trivial because a differential function of only one variable is necessarily total whereas a differential function of two variables is necessarily integrable. All points accessible to a given point P form a continuous domain around P. In an n-dimensional space (n≥3) around P, this domain fills a volume [n dimensions], or a surface [(n-l) dimensions], or a curve [≤ (n-2) dimensions].

The first possibility is excluded because it contradicts the hypothesis that around P there are points which are inaccessible. The third possibility is also excluded because the expression dQ = 0 already defines a surface element containing only points accessible to P. Therefore, points close to P and accessible to P define only a surface. If we now consider a point P’ on that surface, it is impossible to go from P to P’ by a curve satisfying the condition ∫dQ = 0 and not situated on this surface, otherwise every point situated within the immediate proximity of P would be accessible, which contradicts the hypothesis.

From a point P1 it is possible to define a surface S1, upon which all points are accessible to P1. Also, from a point P2 not situated on S1, it is possible to define a surface S2. Surfaces S1 and S2 have no common point between them, otherwise it would be possible to go from P1 to P2 by a path such that ∫dQ = 0. Therefore there is a family of surfaces where σ(x1, x2,…, xn) is constant, filling the space and having no common point among them. For this one-parameter family, dσ = 0 implies dQ = 0, from which, between dQ and dσ, there exists a relation of the type:

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where, because dQ = ΣXidxi

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Naturally, the family of surfaces for which σ is constant may also be expressed by S(σ) = constant, where S(σ) is an arbitrary function of σ:

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Hence

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(l/T) is the integrating factor. If a differential dQ has one integrating factor, it has an infinity, S being an arbitrary function of σ.

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Summary

Carathéodory’s theorem shows that if a differential dQ is integrable, the equation dQ = 0 characterizes in a space a family of surfaces sharing no common point. For any point P on one of these surfaces, it is always possible to find, immediately near that point, points which do not belong to the surface and which are therefore inaccessible by a curve solution of the equation dQ = 0. On the other hand, if dQ is not integrable, the equation dQ = 0 does not define any surface in the space and it will always be possible to link any two points with a curve solution of the equation dQ = 0.

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If the man who almost single-handedly invented chemical thermodynamics – the American mathematical physicist Josiah Willard Gibbs – had owned an automobile, he would have had no trouble figuring out the action of antifreeze.

“The problem reduces to consideration of a binary solution in equilibrium with solid solvent,” I can hear old Josiah saying. “Such a thermodynamic system has two degrees of freedom, so at constant pressure there must be a relation between temperature and composition.”

And indeed there is. The relation corresponds to the observed depression of the freezing point of a solvent by a solute. What’s more, its exact form confirms how antifreeze really works.

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Computing chemical potential

We have Josiah Willard Gibbs to thank for introducing the concept of chemical potential (μ) as a sort of generalized force driving the flow of chemical components between coexistent phases.

When the phases are in equilibrium at constant temperature and pressure, the chemical potential of any component has the same value in each phase

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The key point to note here is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

dce12 …(1)

With pressure and temperature fixed, this equation has a single variable (xi), from which we can draw the conclusion that the variation in chemical potential of a component in an ideal solution is determined solely by its own mole fraction.

The significance of this fact can be appreciated by considering the following diagrams

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Here is water in equilibrium with ice at 273K. The chemical potentials of the solid and liquid phases are equal; there is no net driving force in either direction. Now consider the effect of adding an antifreeze agent to the liquid phase

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Assuming the temperature held constant at 273K, the addition of antifreeze reduces the mole fraction of water, lowering its chemical potential in accordance with equation 1. The coexistent solid phase now has a higher potential, providing the driving force to transform ice into water. Since the temperature is held constant, this equates to the lowering of the freezing point of water in the mixture.

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Deducing a formula for freezing-point depression

To obtain a formula for the freezing point of water in a solution containing antifreeze, we start with the equilibrium relation

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where the zero superscript indicates a standard potential, i.e. that the solid phase consists of pure ice whose mole fraction x is unity. Substituting the left hand side with

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we obtain

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which after differentiation with respect to temperature at constant pressure and subsequent integration yields the formula for the freezing point of water in a solution containing antifreeze at 1 atmosphere pressure:

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The terms on the right are the molar enthalpy of fusion of water (ΔHf0), the freezing point of pure water (Tf0), the gas constant R and the mole fraction of water (xH2O) in the solution containing antifreeze.

The latter is the only variable, confirming that the freezing point of water in a solution containing antifreeze is determined solely by the mole fraction of water in the mixture – in other words the extent to which the water is diluted by the antifreeze agent.

This is how antifreeze works. There is nothing active about its action. It exerts its effect passively by being miscible and thereby reducing the mole fraction of water in the liquid mixture. There’s really nothing more to it than that.

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Using the formula

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Values for constants

Enthalpy of fusion of water ΔHf0 = 6.02 kJmol-1
Freezing point of pure water Tf0 = 273.15 K
Gas constant R = 0.008314 kJmol-1K-1

Example

651 grams of the antifreeze agent ethylene glycol (molecular weight 62.07) are added to 1.5 kg of water (molecular weight 18.02). What is the freezing point of water in this solution?

Strategy

1. Calculate the mole fraction of water in the solution

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Number of moles of water = 1500/18.02 = 83.2
Number of moles of ethylene glycol = 651/62.07 = 10.5
Mole fraction of water = 83.2/(83.2 + 10.5) = 0.89

2. Calculate the freezing point of water in the solution

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The solution will give antifreeze protection down to 261.65K or –11.5°C

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gibbs

It was the American mathematical physicist Josiah Willard Gibbs who introduced the concepts of phase and chemical potential in his milestone monograph On the Equilibrium of Heterogeneous Substances (1876-1878) with which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter” Gibbs mentions – in passing – that for a system of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential μ of any component must have the same value in every phase.

This simple statement turns out to have considerable practical value as we shall see. But first, let’s go through the formal proof of Gibbs’ assertion.

An important result

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Consider a system of two phases, each containing the same components, in equilibrium at constant temperature and pressure. Suppose a small quantity dni moles of any component i is transferred from phase A in which its chemical potential is μ’i to phase B in which its chemical potential is μ”i. The Gibbs free energy of phase A changes by –μ’idni while that of phase B changes by +μ”idni. Since the system is in equilibrium at constant temperature and pressure, the net change in Gibbs free energy for this process is zero and we can write

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hence

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This result can be generalized for any number of phases: for a system in equilibrium at constant temperature and pressure, the chemical potential of any given component has the same value in every phase.

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Visualizing variance

The equality of pressure P, temperature T and component chemical potentials μn between coexistent phases in equilibrium provides a convenient way to visualize variance, or the number of degrees of freedom a system possesses. For example, the triple point of a single component system can be visualized as the array

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where the solid, liquid and vapor phases are indicated by one, two and three primes respectively.

Each row represents a single variable, so the number of rows equates to the total number of variables. Each column lists the variables in a single phase. All but one of these may be independently varied; the last is determined by the Gibbs-Duhem relation

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There are one of these for each phase, so the number of columns equates to the number of relations (=constraints) to which the system variables are subject. The variance, or number of degrees of freedom (f) of the system is defined

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For arrays of the kind presented above, this transposes into

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For the triple point of a single component system, there are three rows and three columns, so f =0. With zero degrees of freedom, the triple point is not subject to independent variation and is represented by a fixed point in the PT plane.

The above rule implies that a system of coexistent phases in equilibrium cannot have more phases than intensive system variables.

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Generating useful equations

For a component present in any pair of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential of that component has the same value in both phases

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From this general relation, equations may be deduced for computing various properties of thermodynamic systems such as ideal solutions, for example the elevation of boiling point, the depression of freezing point, and the variation of the solubility of a solute with temperature.

The key point to grasp is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

dce12 … (1)

where for a given phase, μi is the arbitrary chemical potential of i in the mixture, μ°i is the chemical potential of the pure substance, and xi is the mole fraction of the component.

As an example, let us take the relation

pe08 …(2)

where the chemical potential of the solid solvent is necessarily the standard potential because the mole fraction x is unity. The above relation will generate an equation for the depression of the solvent freezing point in a solution at a fixed pressure (p).

Substituting (1) for the liquid phase in (2) gives

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where by convention the subscript 1 refers to the solvent. Differentiating with respect to T at constant pressure

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using the quotient rule for ΔG/T gives

pe12 … (3)

Now since

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equation (3) simplifies to

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Integrating from the pure solvent state, where the mole fraction x1=1 and T0fus is the freezing point of the pure solvent, to the solution state where the mole fraction x1= x1 and Tfus is the freezing point of the solvent in the solution

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yields the equation for the depression of the solvent freezing point in a solution at a fixed pressure (p)

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Since x1<1 in a solution, the logarithm is negative and therefore the freezing point of the solvent in the solution must be lower than the freezing point of the pure solvent.

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Ok, so maybe that wasn’t the simplest procedure for generating a useful thermodynamic equation. But the point to be made here is that the same procedure applies in the other cases, so you only need to understand the principle once.

For example, the equation for elevation of solvent boiling point in solution with a non-volatile solute at a fixed pressure (p) is

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The similarity to the previous equation is evident.

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Reversible change is a key concept in classical thermodynamics. It is important to understand what is meant by the term as it is closely allied to other important concepts such as equilibrium and entropy. But reversible change is not an easy idea to grasp – it helps to be able to visualize it.

Reversibility and mechanical systems

The simple mechanical system pictured above provides a useful starting point. The aim of the experiment is to see how much weight can be lifted by the fixed weight M1. Experience tells us that if a small weight M2 is attached – as shown on the left – then M1 will fall fast while M2 is pulled upwards at the same speed.

Experience also tells us that as the weight of M2 is increased, the lifting speed will decrease until a limit is reached when the weight difference between M2 and M1 becomes vanishingly small and the pulley moves infinitely slowly, as shown on the right.

We now ask the question – Under what circumstances does M1 do the maximum lifting work? Clearly the answer is visualized on the right, where the lifted weight M2 is as close as we can imagine to the weight of M1. In this situation the pulley moves infinitely slowly (like a nanometer in a zillion years!) and is indistinguishable from being at rest.

This state of being as close to equilibrium as we can possibly imagine is the condition of reversible change, where the infinitely slow lifting motion could be reversed by an infinitely small nudge in the opposite direction.

From this simple mechanical experiment we can draw an important conclusion: the work done under reversible conditions is the maximum work that the system can do.

Any other conditions i.e. when the pulley moves with finite, observable speed, are irreversible and the work done is less than the maximum work.

The irreversibility is explained by the fact that observable change inevitably involves some dissipation of energy, making it impossible to reverse the change and exactly restore the initial state of the system and surroundings.

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Reversibility and thermodynamic systems

The work-producing system so far considered has been purely mechanical – a pulley and weights. Thermodynamic systems produce work through different means such as temperature and pressure differences, but however the work is produced, the work done under reversible conditions is always the maximum work that a system can do.

In thermodynamic systems, heat q and work w are connected by the first law relationship

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What this equation tells us is that for a given change in internal energy (ΔU), both the heat absorbed and the work done in a reversible change are the maximum possible. The corresponding irreversible process absorbs less heat and does less work.

It helps to think of this in simple numbers. U is a state function and therefore ΔU is a fixed amount regardless of the way the change is carried out. Say ΔU = 2 units and the reversible work w = 4 units. The heat q absorbed in this reversible change is therefore 6 units. These must be the maximum values of w and q, because ΔU is fixed at 2; for any other change than reversible change, w is less than 4 and so q is less than 6.

For an infinitesimal change, the inequality in relation to q can be written

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and so for a change at temperature T

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The term on the left defines the change in the state function entropy

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Since reversible conditions equate to equilibrium and irreversible conditions equate to observable change, it follows that

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These criteria are fundamental. They are true for all thermodynamic processes, subject only to the restriction that the system is a closed one i.e. there is no mass transfer between system and surroundings. It is from these expressions that the conclusion can be drawn – as famously stated by Clausius – that entropy increases towards a maximum in isolated systems.

Rudolf Clausius (1822-1888)

Rudolf Clausius (1822-1888)

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Die Entropie der Welt strebt einem Maximum zu

Consider an adiabatic change in a closed system: dq = 0 so the above criteria for equilibrium and observable change become dS = 0 and dS > 0 respectively. If the volume is also kept constant during the change, it follows from the first law that dU = 0. In other words the volume and internal energy of the system are constant and so the system is isolated, with no energy or mass transfer between system and surroundings.

Under these circumstances the direction of observable change is such that entropy increases towards a maximum; when there is equilibrium, the entropy is constant. The criteria for these conditions may be expressed as follows

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Note:
The assertion that entropy increases towards a maximum is true only under the restricted conditions of constant U and V. Such statements as “the entropy of the universe tends to a maximum” therefore depend on assumptions, such as a non-expanding universe, that are not known to be fulfilled.

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Imagine a perfect gas contained by a rigid-walled cylinder equipped with a frictionless piston held in position by a removable external agency such as a magnet. There are finite differences in the pressure (P1>P2) and volume (V2>V1) of the gas in the two compartments, while the temperature can be regarded as constant.

If the constraint on the piston is removed, will the piston move? And if so, in which direction?

Common sense, otherwise known as dimensional analysis, tells us that differences in volume (dimensions L3) cannot give rise to a force. But differences in pressure (dimensions ML-1T-2) certainly can. There will be a net force of P1–P2 per unit area of piston, driving it to the right.

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The driving force

In thermodynamics, there exists a set of variables which act as “generalised forces” driving a system from one state to another. Pressure is one such variable, temperature is another and chemical potential is yet a third. What they have in common is that they are all intensive variables – those which are independent of the quantity of a phase.

Each intensive variable has a conjugate extensive variable – those which are dependent on the quantity of a phase – and together they form a generalised force-displacement pair which has the dimensions of work (= energy ML2T-2). Examples include pressure × volume, temperature × entropy, and voltage × charge.

But back to those intensive variables. Experience confirms that it is these generalised forces which are the agents of change. And spontaneous change results when there are finite differences in intensive variables between one system and another – the direction of change being determined by their relative values.

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The result is work

In the illustrated example above, the freed piston moves spontaneously to the right because of finite differences in pressure (P1>P2). As a result, the system consisting of the left hand compartment does PV work on the system consisting of the right hand compartment. Likewise, finite differences in other intensive variables such as temperature and chemical potential will act through their respective force-displacement pairs to perform work.

We can thus conclude that finite differences in intensive variables drive spontaneous change, and that due to the dimensionality of their respective conjugate extensive variables, spontaneous change results in the performance of work. It should however be noted that this work is not always useful work. The spontaneous diffusion of two gases into each other is a classic case, where it is difficult to imagine how the work could be usefully obtained.

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