Introduction to Thermodynamic States Gases

Chapter 1 Introduction to Thermodynamic 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. Most students have a good intuitive feel for many of the properties of gases which we will be discussing. Gases are also a convenient entry point to exploring fluids. 1.1 Microscopic versus Macroscopic State of a Substance How many variables do you think are needed to completely specify the state of a system? Consider a system with only 4 particles. The following are three different drawings representing the positions of the 4 particles. Are these three systems in 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 the three drawgins, the three states are distinguishable and therefore must not be 1

2 1.1. MICROSCOPIC VERSUS MACROSCOPIC STATE OF A SUBSTANCE the same. In 3-dimensions, the position of each particle is specified by 3 coordinates; therefore, the configuration of a N-particle system must be specified by 3N numbers. Besides its position, a particle can also be in motion. Even if the two systems 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 particles have different momemta. In 3-dimensions, the momentum of each particle is specified by its x-, y- and z-components. So, in addition to the configuration 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 molecular system. Figure 1.2: Two systems with the same particle configuration, but different momenta. The situation is somewhat more complicated if the particles in the system are quantum mechanical. Because the uncertainty principle insists that the position and momentum of a quantum particle cannot be specified simultaneously, you can 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 numbers to specify its state, not 6N. Since most of this course will deal with classical systems, this complication will not concern us much. But in the few cases where the quantum mechanics of the particles does make a difference in the thermodynamic outcome, we will deal with the quantal intricacies when we come upon them later in the course. Therefore, it seems quite clear that we need to specify on the order of 6N numbers, 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 reconstruct the system with all its molecular details (in chemistry, microscopic means on the molecular level ). Say you have a 8-oz glass of water. You can easily calculate the number of water molecules from its volume, the density and its molecular weight. You will see that an enormous number of variables ( 10 23 ) are needed to specify the microscopic state of the glass of water you have. Moreover, since the molecules are always in motion, the system will constantly be moving from one state to another rapidly. So it would appear that the task of understanding any macroscopic system with any reasonble precision is going to require an astronomical

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 water and ask you to make the measurement. At the end of the day, you reported the measurement to her. To make sure your measurement is good, she gave you anther 8-oz sample of water on Tuesday and ask you to make the same measurement. At the end of the day, you reported the second measurement to her. Not satisfied, she asked you to make a third measurement with yet another sample of water on Wednesday. Clearly, the three samples you were given must be 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 10 23. But if each of these measurement gives you a different answer, you will be repeating the same experiment forever. The experiment you are doing will have very little value indeed. In fact, if the measurement does yield a different number every time the system is in a different microscopic state there would be no purpose to chemistry at all, because what measurement you get just depends on which microscopic state the system happens to be in at the moment, yet there are close to an infinity number of possible microscopic states. However, from your experience, you know this is not true. In fact, if you make the same measurement on different macroscopic samples of water, you do expect to get the same answer (to within experimental accuracy). So while the system can exist in one of an almost infinity number of microscopic states, the measurement consistently gives you the same number. In fact, our experience from experimentation tells us that only a handful of variables are usually needed in order to uniquely specify the macroscopic state of a system. That is, if two samples of water have the same set of these variables, then all the other macroscopic observables are expected to be identical for these two samples, even though they are in vastly different microscopic state. This hugh reduction of information that is required to specify the precise state of the system is clearly to our advantage. Armed with this empirical experience, scientists can have hope in understanding systems with large number of particles. The enormous reduction of information required, from 10 23 to just a handful, is truly remarkable and it happens only for systems with a large enough number of particles. Somehow, systems with large N seem to obey certain statistical rules. These rules form the basis of the study of statistical mechanics, which is a microscopic theory of thermodynamics. Unfortunately, we will not have time to study statistical mechanics in this course, but we will use our microscopic knowledge of chemistry to try to understand thermodynamics. You may ask: Where do these statistical rules come from? As it is often true with most of the rest of nature, we don t actually know. But the implications and consequences of these statistical rules are clear. Perhaps, you can appreciate this by drawing an analogy with the distribution of exam scores in a large college class. Certainly, you are aware that the distribution of scores in any class is often close to a Gaussian. In fact, the largely the number of students, the more closely the distribution will approximate a Gaussian. If it is in fact Gaussian, then only two 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 each student. In probability theory, this is known as the central limit theorem. Macroscopic systems in which a few variables, instead of 10 23, are needed to specify their macroscopic states are said to be in an equilibrium state. None of the properties of a system in an equilibrium state can be dependent on the history of the system (otherwise, it would have required more than just a few variables to completely specify the state of the system, because in that case the state 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, which will be our focus for the rest for rest of this course. Not all systems are in an equilibrium state. Sometime a system that was original in an equilibrium state could be driven out of equilibrium, and during this time, all 6N microscopic variables will indeed be needed in order to completely specify the state of the system. 1.2 State Variables We have shown that the macroscopic state of a system in equilibrium is often completely specified by just a handful of variables. Precisely how many cannot be determined from first principle (e.g. you can t predict it using quantum mechanics). 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 larger pressure from the outside. So the state of the air inside the balloon has a certain volume, which will shrink in response to the increased pressure from the outside until the pressure of the gas inside is equal to the outside. If this process is carried out at a fixed temperature, then we would be able to determine the relationship between the volume and pressure of the gas. If we continue to compress the gas, at some sufficiently high pressure, the gas would begin to condense into a liquid, which is obviously in a different macroscopic state compared to the original gas. Therefore, we would expect that as the state of the system changes, the volume (V ) and the pressure (P ) change, and P and V could possibly be used to characterize the macroscopic state of the system. In fact, this is not new to most chemistry students, because back in general chemistry, we learned about the gas laws. The ideal gas law is an equation which relate several macroscopic properties of an ideal gas with each other. These properties are pressure P, volume V, number of moles n and temperature T : P V = nrt, where R is the gas constant. In fact, P, V, n and T are the most common variables used to characterize the macroscopic state of a pure substance. These properties are called state variables (because they specify the macroscopic state of the substance). The ideal gas law suggests that these four state variables are related to each other by a single equation, so not all four of them are independent. We can consider any three of them to be independent variables, while the fourth one is necessarily depedent on them, i.e. it can be expressed as a function of the other three state variables.

CHAPTER 1. INTRODUCTION TO THERMODYNAMIC STATES GASES 5 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 may be chosen independently while the fourth one is necessarily dependent on them remains true even when the gas is not ideal. In fact, this is true not only for gases but for liquids and solids too. In those cases the ideal gas equation must be replaced by another equation that is valid under non-ideal conditions. The equation that expresses the dependent state variable as a function of the other three 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 fourth variable? Or perhaps the viscosity, or the heat capacity? In fact, it is possible to replace any of the state variables in your descriptor of the macroscopic state by any other macroscopic property of the system, as long as it is not a variable derived from the first three. For example, you cannot choose P, V, n and the molar volume V = V/n, because V is not really a new variable. But by far, P, V, n and T are the most common choice for the state variables for a pure substance, and the thermodynamic properties of macroscopic systems are usually formulated as a function of these. In the case where you have a mixture of two or more substances, then the number of moles of each component is needed in addition to P, V and T for the descriptor of the macroscopic state of the mixture. Before we proceed to describing the general thermodynamic properties of fluids, 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 is an external magnetic field, the orientation of the system relative to the external field will matter. Therefore, the orientation as well as the external field strength must be added to the descriptor of the macroscopic state. As another example, if the substance has a permanent electric dipole, the orientation of it relative to 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 state variables may need to be expanded. We will explore this further when we discuss the first law of thermodynamics in the next chapter. 1.3 Temperature and the Zeroth Law of Thermodynamics While temperature is a familiar concept, its definition is not so straightforward. The temperature is intrinsically different from all the other state variables. The temperature of a system is defined by the direction of heat flow when it is placed in thermal contact with another system at a different temperature. We will see later in the course that temperature is intimately related to heat and entropy. The difficulty associated with finding a precise definition of temperature was 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 summarized by the zeroth law of thermodynamics, which states that if two bodies A and B are in thermal equilibrium, and B and another body C are in turn in thermal equilibrium, 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 is large compared to the range of interactions between molecules. You can estimate the average distance between particles in a typical gas at normal temperature and pressure, and you will realize that the average separation is between 30 to 40 Å. Now try this calculation. We know that in the liquid state, the molecules are much closer together, but how close are they actually? Compare the density of liquid water, for example, with that of air and estimate the average distance between two water molecules in the liquid state. You will find that they are approximately 3 to 4 Åapart in the liquid, much closer than they are in the gas. The interaction of two molecule with each other are dictated by their interaction potential energy. For two neutral nondoplar molecules, the interaction potential energy between them has the typical dependence on separation distance depicted in the following drawing. The closer the particles are, the more they will interact with each other. Therefore, at very low density, the gas particles don t see each other, and the behavior of the gas becomes ideal, i.e. they behave as if no other particles are there. Figure 1.3: Typical interparticle potential energy curve as a function of separation between two neutral nonpolar molecules. Of course, it is the nonideality of the gas that makes things complicated and interesting at the same time. The nonideality of the gas becomes increasingly obvious when the average distance between the gas particles is reduced, e.g. by compressing it. The compressibility factor, defined as Z = P V /RT, reveals the nonideality of a gas in a simple way when plotted against either 1/V or P. At infinitely low pressure, the gas is ideal and Z = 1. Any deviation from ideality

CHAPTER 1. INTRODUCTION TO THERMODYNAMIC STATES GASES 7 will result in Z 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 = P V RT = 1 + B P + C P 2 +... (1.1) In this equation, B is called the second virial coefficient of the gas, and C is the 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 of the 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 not plotted against just V, because the gas is ideal when 1/V 0. The vivial expansion in this case looks like: Z = P V RT = 1 + B V + C V 2 +... (1.2) The two sets of virial coefficients {B, C,...} and {B, C,...} are not independent 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, V is greater than the ideal volume. An ideal gas is assumed to have no volume even when it is under high compression (i.e. large P ). Therefore Z > 1 corresponds to the real gas having a non-zero volume. On the other hand, when Z < 1, V is smaller 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 no inter-particle attraction. 1.5 Phase Transitions A gas, when it is far from ideal conditions, can condense into a liquid. Condensation 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 volume at different temperatures. Figure 1.5: Typical pressure-volume isotherms. Each of the curves in the figure represents how P behaves when the volume of the gas is compressed at a constant temperature T. For this reason, the curves are sometimes called P -V isotherms. At high temperatures (black curve), a gas should pretty much follow the ideal gas law. If we start at the high V side (right), V is large and P should be small. When the gas is compressed V decreases and so P increases, travelling from right to left. Because the gas is almost ideal at high T, P 1/V and the curve resembles a hyperbola. At the next highest T, the curve is still a hyperbola, but now lower because T is small than in the first curve. When we continue to lower T, the gas will begin to condense at some low enough temperature. At the lowest T (red curve), the gas begins to condense at a certain value of V. At this point, some of the gas begins to form droplets of 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 to the 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 and compressible 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 how much tougher it is to compress liquid water than air). Thus, the curve is much steeper 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 to condense later and comes out of the two-phase region sooner, making the twophase region narrower. The two-phase region gets narrower the higher we go in T, until it shrinks to a point at some critical temperature, T c. For T > T c, the gas simply remains as a gas no matter how much it is compressed, i.e. it never condenses. Sometimes, these curves are also called phase diagrams, because they tell you which phase exists under what conditions. Notice that these P -V isotherms are nothing but slices of the equation of

CHAPTER 1. INTRODUCTION TO THERMODYNAMIC STATES GASES 9 state at constant T. To reconstruct the surface corresponding to equation of state in three dimensions, i.e. P -V -T, we can stretch these P -V isotherms out along 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 thermodynamics, the hypersurface that relates the state variables of a system to each other is often called an equilibrium manifold or the equation-of-state manifold. We see that on the liquid side, the manifold is steeper than on the gas side. Between the liquid and the gas regions, the two-phase region resembles a shear 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 is shown 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 point is called the critical point. We have already mentioned that the temperature there is called the critical temperature T c. The pressure and the volume at the critical point are naturally called the critical pressure P c and the critial volume V c, respectively. A large body of scientific work has been focused on just the critical point itself, because critical phenomena can be explained by elegent mathematical theories, leading to a few Nobel prizes. We won t spend much time on it in this course, except to point out that at the critical point, both the first and the second derivatives, ( P/ V ) T and ( 2 P/ 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 intensive task. Representing it by a simple mathematical form is also almost impossible. But we often need the equation of state in thermodynamics to perform certain calculations or derivations. Therefore, an approximate equation of state, expressing the relationship among P, V and T, is useful. A common approximate expression is the van der Waals equation: ( P + a V 2 ) (V b) = RT. (1.3) In this equation, a and b are empirical parameters, which can be determined experimentally for each gas. The parameter b corrects for the nonzero volume of the real gas, and a corrects the pressure in the presence of intermolecular attraction in the real gas. We can determine the critical point of the van der Waals equation. Substituting P c, V c and T c into the van der Waals equation for P, V and T yields one equation. Then using the conditions ( P/ V ) T = 0 and ( 2 P/ V 2 ) T = 0, we get two other equations. Solving these three equations for the three unknowns P c, V c and T c yields: V c = 3b, P c = a 27b 2, T c = 8a 27bR. (1.4) Since the van der Waals equation is an approximation, it is of course not going to give the correct critical values for any real gas. Also, by plotting the P -V isotherms of the van der Waals equation at temperature below T c, you will see 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 in thermodynamics, not only for gases but also for solutions and even solids and

CHAPTER 1. INTRODUCTION TO THERMODYNAMIC STATES GASES 11 other complex systems. Often, the assumption of ideality yields an exactly solvable model system. Then the non-ideal parts of the system can either be introduced as perturbations to the ideal model, or the particle interactions can be treated approximately in a background given by the ideal system using what are called mean field theories. We will revisit the concept of ideality when we get to dilute solutions.