Introduction to Thermodynamic States { Gaseschem430a/scans/01.pdf · Introduction to Thermodynamic States {Gases ... and consequences of these statistical rules are clear. Perhaps,

Chapter 1

Introduction toThermodynamic States –Gases

We begin our study in thermodynamics with a survey of the properties of gases.Gases are one of the first things students study in general chemistry. Moststudents have a good intuitive feel for many of the properties of gases which wewill be discussing. Gases are also a convenient entry point to exploring fluids.

1.1 Microscopic versus Macroscopic State of aSubstance

How many variables do you think are needed to completely specify the state ofa system?

Consider a system with only 4 particles. The following are three differentdrawings representing the positions of the 4 particles. Are these three systemsin the same state?

Figure 1.1: Three different systems having 4 particles each.

Obviously, you will say no. Since the particles are in different places in thethree drawgins, the three states are distinguishable and therefore must not be

1

21.1. MICROSCOPIC VERSUS MACROSCOPIC STATE OF A

SUBSTANCE

the same. In 3-dimensions, the position of each particle is specified by 3 coor-dinates; therefore, the “configuration” of a N -particle system must be specifiedby 3N numbers.

Besides its position, a particle can also be in motion. Even if the two sys-tems below have their 4 particles in exactly the same positions, the momentum(indicated by arrow) of each particle is different between the two drawings.Therefore, the two drawings must represent different states because the parti-cles have different momemta. In 3-dimensions, the momentum of each particleis specified by its x-, y- and z-components. So, in addition to the configura-tion of a system, we need an additional 3N numbers to specify the momenta– requiring a total of 6N numbers to uniquely specify the state of a molecularsystem.

Figure 1.2: Two systems with the same particle configuration, but differentmomenta.

The situation is somewhat more complicated if the particles in the system arequantum mechanical. Because the uncertainty principle insists that the positionand momentum of a quantum particle cannot be specified simultaneously, youcan know the position or the momentum precisely but not both. Therefore,for a N -particle quantum system, you will only be able to use 3N numbersto specify its state, not 6N . Since most of this course will deal with classicalsystems, this complication will not concern us much. But in the few caseswhere the quantum mechanics of the particles does make a difference in thethermodynamic outcome, we will deal with the quantal intricacies when wecome upon them later in the course.

Therefore, it seems quite clear that we need to specify on the order of 6Nnumbers, or “variables”, to completely define the “state” of a N -particle system.This kind of state is better referred to as the “microscopic state” of the system,because the descriptor of the state (the 6N variables) allows me to reconstructthe system with all its molecular details (in chemistry, “microscopic” means “onthe molecular level”).

Say you have a 8-oz glass of water. You can easily calculate the number ofwater molecules from its volume, the density and its molecular weight. You willsee that an enormous number of variables (≈ 1023) are needed to specify themicroscopic state of the glass of water you have. Moreover, since the moleculesare always in motion, the system will constantly be moving from one state toanother rapidly. So it would appear that the task of understanding any macro-scopic system with any reasonble precision is going to require an astronomical

Copyright c© 2009 by C.H. Mak

CHAPTER 1. INTRODUCTION TO THERMODYNAMIC STATES –GASES 3

amount of information.Now consider this. Let’s say your research project is to measure a certain

property of water. On Monday, your professor gave you a 8-oz sample of waterand ask you to make the measurement. At the end of the day, you reportedthe measurement to her. To make sure your measurement is good, she gaveyou anther 8-oz sample of water on Tuesday and ask you to make the samemeasurement. At the end of the day, you reported the second measurement toher. Not satisfied, she asked you to make a third measurement with yet anothersample of water on Wednesday. Clearly, the three samples you were given mustbe in vastly different microscopic states, because there is no chance on earth(and in heaven) they can all have the same 6N variables, N being 1023. But ifeach of these measurement gives you a different answer, you will be repeatingthe same experiment forever. The experiment you are doing will have very littlevalue indeed. In fact, if the measurement does yield a different number everytime the system is in a different microscopic state there would be no purposeto chemistry at all, because what measurement you get just depends on whichmicroscopic state the system happens to be in at the moment, yet there areclose to an infinity number of possible microscopic states.

However, from your experience, you know this is not true. In fact, if youmake the same measurement on different macroscopic samples of water, you doexpect to get the same answer (to within experimental accuracy). So while thesystem can exist in one of an almost infinity number of microscopic states, themeasurement consistently gives you the same number. In fact, our experiencefrom experimentation tells us that only a handful of variables are usually neededin order to uniquely specify the macroscopic state of a system. That is, iftwo samples of water have the same set of these variables, then all the othermacroscopic observables are expected to be identical for these two samples, eventhough they are in vastly different microscopic state.

This hugh reduction of information that is required to specify the precisestate of the system is clearly to our advantage. Armed with this empiricalexperience, scientists can have hope in understanding systems with large numberof particles. The enormous reduction of information required, from 1023 to justa handful, is truly remarkable and it happens only for systems with a largeenough number of particles. Somehow, systems with large N seem to obeycertain statistical rules. These rules form the basis of the study of “statisticalmechanics”, which is a microscopic theory of thermodynamics. Unfortunately,we will not have time to study statistical mechanics in this course, but we will useour microscopic knowledge of chemistry to try to understand thermodynamics.

You may ask: Where do these statistical rules come from? As it is often truewith most of the rest of nature, we don’t actually know. But the implicationsand consequences of these statistical rules are clear. Perhaps, you can appreciatethis by drawing an analogy with the distribution of exam scores in a large collegeclass. Certainly, you are aware that the distribution of scores in any class is oftenclose to a Gaussian. In fact, the largely the number of students, the more closelythe distribution will approximate a Gaussian. If it is in fact Gaussian, then onlytwo variables – the mean and the standard deviation – are needed to specifying


4 1.2. STATE VARIABLES

the grades distribution of the entire class, instead of the individual score of eachstudent. In probability theory, this is known as the “central limit theorem”.

Macroscopic systems in which a few variables, instead of ≈ 1023, are neededto specify their macroscopic states are said to be in an “equilibrium” state. Noneof the properties of a system in an equilibrium state can be dependent on thehistory of the system (otherwise, it would have required more than just a fewvariables to completely specify the state of the system, because in that case thestate will depend on what happens to it a minute ago, an hour ago, yesterday,last week, etc.) Systems in equilibrium obey the laws of thermodynamics, whichwill be our focus for the rest for rest of this course. Not all systems are in anequilibrium state. Sometime a system that was original in an equilibrium statecould be driven out of equilibrium, and during this time, all 6N microscopicvariables will indeed be needed in order to completely specify the state of thesystem.

1.2 State Variables

We have shown that the macroscopic state of a system in equilibrium is oftencompletely specified by just a handful of variables. Precisely how many cannotbe determined from first principle (e.g. you can’t predict it using quantummechanics). Only experience can tell.

Let’s take an example. If we trap a fixed amount of gas inside a balloon,we can compress it, i.e. make its volume smaller, by applying a slightly largerpressure from the outside. So the state of the air inside the balloon has acertain volume, which will shrink in response to the increased pressure fromthe outside until the pressure of the gas inside is equal to the outside. If thisprocess is carried out at a fixed temperature, then we would be able to determinethe relationship between the volume and pressure of the gas. If we continueto compress the gas, at some sufficiently high pressure, the gas would beginto condense into a liquid, which is obviously in a different macroscopic statecompared to the original gas. Therefore, we would expect that as the state ofthe system changes, the volume (V ) and the pressure (P ) change, and P and Vcould possibly be used to characterize the macroscopic state of the system.

In fact, this is not new to most chemistry students, because back in generalchemistry, we learned about the gas laws. The ideal gas law is an equationwhich relate several macroscopic properties of an ideal gas with each other.These properties are pressure P , volume V , number of moles n and temperatureT : PV = nRT , where R is the gas constant. In fact, P , V , n and T are themost common variables used to characterize the macroscopic state of a puresubstance. These properties are called “state variables” (because they specifythe macroscopic state of the substance). The ideal gas law suggests that thesefour state variables are related to each other by a single equation, so not all fourof them are independent. We can consider any three of them to be independentvariables, while the fourth one is necessarily depedent on them, i.e. it can beexpressed as a function of the other three state variables.



Of course, the ideal gas law is valid only when the gas is under certain (i.e.“ideal”) conditions. Nonetheless, the fact that three of the state variables maybe chosen independently while the fourth one is necessarily dependent on themremains true even when the gas is not ideal. In fact, this is true not only forgases but for liquids and solids too. In those cases the ideal gas equation mustbe replaced by another equation that is valid under non-ideal conditions. Theequation that expresses the dependent state variable as a function of the otherthree independent variable for a pure substance is called an “equation of state”(or more appropriately “equation of state variables”).

You may ask: ”Why do we have to use P , V , n and T as the state vairalbes?Can’t we use, for example, T , V , n and say the compressibility as the fourthvariable? Or perhaps the viscosity, or the heat capacity?” In fact, it is possibleto replace any of the state variables in your descriptor of the macroscopic stateby any other macroscopic property of the system, as long as it is not a variablederived from the first three. For example, you cannot choose P , V , n andthe molar volume V = V/n, because V is not really a new variable. But byfar, P , V , n and T are the most common choice for the state variables for apure substance, and the thermodynamic properties of macroscopic systems areusually formulated as a function of these. In the case where you have a mixtureof two or more substances, then the number of moles of each component isneeded in addition to P , V and T for the descriptor of the macroscopic state ofthe mixture.

Before we proceed to describing the general thermodynamic properties offluids, I should point out that the list P , V , n and T are not always enough.For example, if the substance has a permanent magnetic moment, and there isan external magnetic field, the orientation of the system relative to the externalfield will matter. Therefore, the orientation as well as the external field strengthmust be added to the descriptor of the macroscopic state. As another example,if the substance has a permanent electric dipole, the orientation of it relativeto an external electric field must also be included in the descriptor of the state.Depending of the substance and the experimental conditions, the list of statevariables may need to be expanded. We will explore this further when we discussthe first law of thermodynamics in the next chapter.

1.3 Temperature and the Zeroth Law of Ther-modynamics

While temperature is a familiar concept, its definition is not so straightforward.The temperature is intrinsically different from all the other state variables. Thetemperature of a system is defined by the direction of heat flow when it is placedin thermal contact with another system at a different temperature. We will seelater in the course that temperature is intimately related to heat and entropy.

The difficulty associated with finding a precise definition of temperaturewas not fully realized until a good part of thermodynamics had already been


6 1.4. NONIDEALITY OF GASES

developed. When two bodies in thermal contact have the same temperature,they are in thermal equilibirum, i.e. no net heat will transfer between them.The relationship between temperature and thermal equilibrium is summarizedby the zeroth law of thermodynamics, which states that if two bodies A and Bare in thermal equilibrium, and B and another body C are in turn in thermalequilibrium, then A and C must also be in thermal equilibrium.

1.4 Nonideality of Gases

A gas is approximately ideal when distance between individual gas particles islarge compared to the range of interactions between molecules. You can estimatethe average distance between particles in a typical gas at normal temperatureand pressure, and you will realize that the average separation is between 30 to40 A.

Now try this calculation. We know that in the liquid state, the moleculesare much closer together, but how close are they actually? Compare the densityof liquid water, for example, with that of air and estimate the average distancebetween two water molecules in the liquid state. You will find that they areapproximately 3 to 4 Aapart in the liquid, much closer than they are in the gas.

The interaction of two molecule with each other are dictated by their inter-action potential energy. For two neutral nondoplar molecules, the interactionpotential energy between them has the typical dependence on separation dis-tance depicted in the following drawing. The closer the particles are, the morethey will interact with each other. Therefore, at very low density, the gas parti-cles don’t see each other, and the behavior of the gas becomes “ideal”, i.e. theybehave as if no other particles are there.

Figure 1.3: Typical interparticle potential energy curve as a function of separa-tion between two neutral nonpolar molecules.

Of course, it is the nonideality of the gas that makes things complicated – andinteresting at the same time. The nonideality of the gas becomes increasinglyobvious when the average distance between the gas particles is reduced, e.g. bycompressing it. The compressibility factor, defined as Z = PV /RT , reveals thenonideality of a gas in a simple way when plotted against either 1/V or P . Atinfinitely low pressure, the gas is ideal and Z = 1. Any deviation from ideality



will result in Z 6= 1.The figure below shows what Z may look like for a real gas at different

temperatures. The curve can be Taylor expanded around P = 0 as follows:

Z =PV

RT= 1 + B′P + C ′P 2 + . . . (1.1)

In this equation, B′ is called the second virial coefficient of the gas, and C ′ isthe third virial coefficient, etc. Clearly, the virial coefficients are functions of T .The coefficient B′ of the first-order term in the Taylor series is just the slope ofthe curve at P=0.

Figure 1.4: The compressibility factor for a real gas at different temperatures.

Similarly, Z can also be plotted as a function of 1/V . Notice that it is notplotted against just V , because the gas is ideal when 1/V → 0. The vivialexpansion in this case looks like:

Z =PV

RT= 1 +

B

V+

C

V2 + . . . (1.2)

The two sets of virial coefficients {B, C, . . .} and {B′, C ′, . . .} are not indepen-dent but are related to each other via the equation of state.

Notice that in the Z vs. P curve Z can be above or below 1. When Z > 1, Vis greater than the ideal volume. An ideal gas is assumed to have no volume evenwhen it is under high compression (i.e. large P ). Therefore Z > 1 correspondsto the real gas having a non-zero volume. On the other hand, when Z < 1, V issmaller than the ideal volume. This is due to the attraction between molecules,which holds them closer together than in the ideal gas. An ideal gas has nointer-particle attraction.

1.5 Phase Transitions

A gas, when it is far from ideal conditions, can condense into a liquid. Conden-sation is an example of a phase transition. Let’s follow the behavior of a gas


8 1.5. PHASE TRANSITIONS

when it condenses by looking at the pressure as a function of the molar volumeat different temperatures.

Figure 1.5: Typical pressure-volume isotherms.

Each of the curves in the figure represents how P behaves when the volume ofthe gas is compressed at a constant temperature T . For this reason, the curvesare sometimes called P -V isotherms. At high temperatures (black curve), agas should pretty much follow the ideal gas law. If we start at the high Vside (right), V is large and P should be small. When the gas is compressed Vdecreases and so P increases, travelling from right to left. Because the gas isalmost ideal at high T , P ∼ 1/V and the curve resembles a hyperbola. At thenext highest T , the curve is still a hyperbola, but now lower because T is smallthan in the first curve.

When we continue to lower T , the gas will begin to condense at some lowenough temperature. At the lowest T (red curve), the gas begins to condenseat a certain value of V . At this point, some of the gas begins to form dropletsof liquid. A little change in P causes a very large contraction in V . Therefore,in this “two-phase” region, the P -V curves looks flat. Continue moving tothe left, the curve stays flat until all the gas is finally condensed to liquid.Inside the two-phase region, the liquid-gas mixture is infinitely stretchable andcompressible and the same time. Once all the gas has been converted to liquid,it is much harder to compress it further (you probably know by experience howmuch tougher it is to compress liquid water than air). Thus, the curve is muchsteeper on the left (liquid) side of the two-phase region than on the right (gas)side.

As we move up to the second lowest T (purple curve), the gas begins tocondense later and comes out of the two-phase region sooner, making the two-phase region narrower. The two-phase region gets narrower the higher we go inT , until it shrinks to a point at some “critical temperature”, Tc. For T > Tc, thegas simply remains as a gas no matter how much it is compressed, i.e. it nevercondenses. Sometimes, these curves are also called “phase diagrams”, becausethey tell you which phase exists under what conditions.

Notice that these P -V isotherms are nothing but slices of the equation of



state at constant T . To reconstruct the surface corresponding to equation ofstate in three dimensions, i.e. P -V -T , we can stretch these P -V isotherms outalong the T -axis and create a 3-dimensional picture of the equation of state.Such a surface is sometimes called a manifold in mathematics. In thermody-namics, the hypersurface that relates the state variables of a system to eachother is often called an “equilibrium manifold” or the equation-of-state mani-fold. We see that on the liquid side, the manifold is steeper than on the gasside. Between the liquid and the gas regions, the two-phase region resembles ashear face of a mountain, much like the half dome in Yosemite Park, California(see photo). A more accurate rendering of the P -V -T equilibrium manifold isshown below.

Figure 1.6: The Half Dome in Yosemite Park (photo from www.nps.gov).

Figure 1.7: Equation of state on the P -V -T surface.


10 1.6. THE CRITICAL POINT

1.6 The Critical Point

The place on the phase diagram at which the two-phase region shrinks to a pointis called the “critical point”. We have already mentioned that the temperaturethere is called the critical temperature Tc. The pressure and the volume at thecritical point are naturally called the critical pressure Pc and the critial volumeVc, respectively.

A large body of scientific work has been focused on just the critical pointitself, because critical phenomena can be explained by elegent mathematicaltheories, leading to a few Nobel prizes. We won’t spend much time on it inthis course, except to point out that at the critical point, both the first and thesecond derivatives, (∂P/∂V )T and (∂2P/∂V 2)T become zero.

1.7 The van der Waals Equation of State

Determining the equation of state of a real gas is obviously a labor intensivetask. Representing it by a simple mathematical form is also almost impossible.But we often need the equation of state in thermodynamics to perform certaincalculations or derivations. Therefore, an approximate equation of state, ex-pressing the relationship among P , V and T , is useful. A common approximateexpression is the van der Waals equation:(

P +a

V2

)(V − b) = RT. (1.3)

In this equation, a and b are empirical parameters, which can be determinedexperimentally for each gas. The parameter b corrects for the nonzero volumeof the real gas, and a corrects the pressure in the presence of intermolecularattraction in the real gas.

We can determine the critical point of the van der Waals equation. Substi-tuting Pc, Vc and Tc into the van der Waals equation for P , V and T yields oneequation. Then using the conditions (∂P/∂V )T = 0 and (∂2P/∂V 2)T = 0, weget two other equations. Solving these three equations for the three unknownsPc, Vc and Tc yields:

Vc = 3b, Pc =a

27b2, Tc =

8a

27bR. (1.4)

Since the van der Waals equation is an approximation, it is of course notgoing to give the correct critical values for any real gas. Also, by plotting theP -V isotherms of the van der Waals equation at temperature below Tc, you willsee that the two-phase region is not flat like it should be for a real gas.

1.8 The Concept of Ideality

Even though no gas is really ideal, the concept of ideality is very useful inthermodynamics, not only for gases but also for solutions and even solids and



other complex systems. Often, the assumption of ideality yields an exactlysolvable model system. Then the non-ideal parts of the system can either beintroduced as perturbations to the ideal model, or the particle interactions canbe treated approximately in a background given by the ideal system using whatare called “mean field theories”. We will revisit the concept of ideality when weget to dilute solutions.


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Introduction to Thermodynamic States { Gaseschem430a/scans/01.pdf · Introduction to Thermodynamic States {Gases ... and consequences of these statistical rules are clear. Perhaps,