Posts Tagged ‘state function’

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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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P Mander March 2015

photo credit: theguardian.com

photo credit: theguardian.com

From the search term phrases that show up on this blog’s stats, CarnotCycle detects that a significant segment of visitors are studying foundation level thermodynamics  at colleges and universities around the world. So what better than a post that tackles that favorite test topic – exact and inexact differentials.

When I was an undergraduate, back in the time of Noah, we were first taught the visual approach to these things. Later we dispensed with diagrams and got our answers purely through the operations of calculus, but either approach is equally instructive. CarnotCycle herewith presents them both.

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The visual approach

Ok, let’s start off down the visual track by contemplating the following pair of pressure-volume diagrams:

visual track

The points A and B have identical coordinates on both diagrams, with A and B respectively representing the initial and final states of a closed PVT system, such as an ideal gas, whose state can be defined in terms of any two of the three variables p,v,t, since the third will be determined by the equation of state. In this case, the pressure P and the volume V have been selected as the independent variables.

The paths taken between A and B are visibly different. In path I on the left, the volume first increases at constant pressure (ΔV>0, external work done), then the pressure decreases at constant volume (ΔV=0, no work done). In path II on the right, these operations are performed in reverse order.

Now the PV diagram has the property that the area under the curve represents the work done during the transition from initial state to final state. This is represented by the shaded area in the diagrams; it is evident that the work done in passing from A to B is greater for path I than it is for path II.

The conclusion can thus be drawn that unless the path is specified, the work done in the process A→B cannot be determined.

Now let’s apply the first law of thermodynamics to the process A→B. For an appreciable (i.e. measurable) change

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where q is the heat supplied to the system and w is the work done by the system. The internal energy U is a state function – this is an axiom of thermodynamics – and for a closed system it means that ΔU has the same value for all paths between state A and state B. Since the above equation does not specify a path, and we know that under these circumstances w is indeterminate, q must also be indeterminate.

For a PVT system undergoing a reversible process, we can write the differential equation which expresses the first law as follows:

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The same considerations applying to appreciable differences also apply to infinitesimals on integration. For the process A→B, the values of the integrals ∫ABdqrev and ∫ABdwrev are path-dependent.

Now the mathematics of multivariate functions teaches us that if the differential of a function Y, where Y = Y(X1, X2,…, Xn), can be set equal to a differential expression ΣCidXi where Xi are independent variables and the coefficients Ci are functions of the Xi, then dY is said to be an exact differential.

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Importantly for thermodynamics, dY then also has the characteristic that the value of the integral ∫ABdY = ΔY is independent of the path followed from state A to state B. As we have seen, dU comes into this category.

If a differential does not have this characteristic, i.e. if the value of the integral is path-dependent, then it is said to be an inexact differential. And as we have seen, dqrev and dwrev come into this category.

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The math approach

Right then, let’s see how we can set about supplying the answer as to why dqrev is inexact by taking the purely mathematical route. We can write the differential equation which expresses the first law for a PVT system undergoing a reversible process in the following way:

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Actually, when you look at this equation it does seem reasonable to think that dqrev is exact, since both the variables on the right are exact. But appearances can be deceptive.

Now since P,V and T are the thermodynamic variables which determine the state of a PVT system, and U is a state function, we can consider U a function of any two of them – the third will be determined by the equation of state. So let’s choose V and T (makes the calculus a bit easier) and write the partial differential equation:

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Substituting for dU in the first law equation gives:

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At this point, it is convenient to recall the property of exact differentials that if Y and its derivatives are continuous, then for any pair of independent variables X1 and X2 there is the mathematical requirement that

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We can use this to test the exactness of the property relation we have just derived for dqrev. Differentiating M wrt T and N wrt V gives:

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Since we know that U is a function of T and V, it follows that (∂2U/∂T∂V) and (∂2U/∂V∂T) must be equal. (∂M/∂T)V and (∂N/∂V)T are therefore not equal, and since (∂P/∂T)V is generally non-zero (i.e. not a horizontal line on a PT plot), we must conclude that the test fails to establish exactness, and that dqrev – and therefore dwrev – are inexact differentials.