## Systems and equilibrium

Thermodynamics is the study of macroscopic or bulk systems of masses and their interrelations under conditions of steady state (no dependence on time). By macroscopic we mean the system contains large numbers of individual molecules (within a few orders of magnitude of a mole1 which contains 6.02 x 1023 molecules). We call these states equilibrium states if they are not only time independent but also stable under small perturbations. Thermodynamic states are describable by a set of dimensional quantities which we refer to as coordinates. Thermodynamics is concerned with the changes in energy-related quantities (certain of the coordinates) when the system undergoes a transition from one state to another. A thermodynamic system is a region of space containing matter with certain internally uniform properties such as pressure and temperature. We will be concerned with the interior of the system and the variables (coordinates) that characterize it. For example, a mass of pure gas (only one chemical species) contained in a vessel may be characterized by the pressure it exerts on the walls of the vessel, the volume of the vessel and the temperature (p, V, T). These comprise the complete set of thermodynamic coordinates for this particular system. For more general situations such as mixtures of species or phases, the coordinates necessary to describe the state have to be determined experimentally. It is important to note that an individual thermodynamic system is uniform in its interior. There are no gradients of pressure or temperature, for example, inside the system.2

1 The mole is an SI unit defined as the number of carbon atoms in a mass of 0.012 kg of pure carbon. The number of moles of a substance is the number of multiples of this number (known as Avogadro's number: Na = 6.02 x 1023). In formulas the unit is designated as "mol."

2 Note that a column of air in the atmosphere is not a simple thermodynamic system because its pressure and temperature vary with altitude. However, it is convenient to consider the column as composed of thin slabs, each of which contains substance with approximately uniform temperature, pressure and composition. Then each individual slab may be considered as a simple thermodynamic system for many purposes.

### 1.3.1 Examples of thermodynamic systems

Gas in a vessel Suppose a container holds a gas of uniform chemical composition. Let the walls of the container be thermally insulating and let the volume be fixed. In a very short time after fixing these conditions the gas will come to values of temperature and pressure that are uniform throughout and independent of the shape of the container. This is the simplest thermodynamic system in a state of equilibrium.

A second case is where the container's walls are held at a fixed temperature and the pressure is allowed to vary. Equilibrium will be established such that the temperature of the gas becomes equal to that of the surrounding walls, the volume is given and the pressure comes to some value that we can estimate.

A third case is where the container has a frictionless movable piston that is pushed upon externally by a fixed pressure (such as the atmospheric pressure). This means that the pressure in the vessel is held fixed along with that of the temperature. The piston will shift in such a way to make the pressure inside equal to that outside, and the volume will change until all these conditions are met.

Our gas might not be homogeneous, but instead it might be composed of a mixture of chemically noninteracting gases, such as those in our atmosphere: nitrogen, oxygen and argon. We still have a thermodynamic system as long as the composition does not vary from location to location or from time to time. In each of the above cases let two of the following be fixed: volume, temperature, or pressure. Then the remaining variable is allowed to find its equilibrium value. Note that once in equilibrium, the variables or coordinates are uniform throughout the vessel.

Two-phase system Suppose we have a liquid of uniform chemical composition such as water in our vessel and vacuum above the liquid surface. Let the temperature and volume be fixed. After a sufficient adjustment time some liquid will have evaporated into the volume above its surface and an equilibrium will be established (the flux of water molecules leaving the surface becomes equal to the flux entering and sticking to the surface). There will be a gas pressure exerted on the walls by the vapor that evaporated from the liquid surface. This is a two-phase system with liquid and gaseous phases, but only one component (water) which depicts the number of distinct chemical species. The pressure throughout will be uniform (ignore the pressure increase as a function of depth due to gravity in the liquid). The temperature will also be uniform throughout both phases of the system. This two-phase configuration is also a thermodynamic system. The system can be made to pass through changes in volume, temperature, etc., to establish new thermodynamic states of equilibrium. Note that the temperature and pressure are uniform throughout but the density varies from one phase to the other. As we shall see in a later chapter there is another quantity that is also uniform in the two-phase system called the specific Gibbs energy (chemical potential in the chemical literature when expressed as molar Gibbs energy). It acts as an intensive variable (see Section 1.5) in such multicomponent systems similarly to pressure or temperature.

Aqueous solutions Imagine a vessel filled with water (at a fixed temperature and pressure) and some salt is placed in the liquid. If we continue to put more salt into the water eventually some salt will remain in crystal form sinking to the bottom (but ignore gravity otherwise). We will have established an equilibrium between the saturated saline solution and the precipitated crystalline salt. A change in temperature will result in a new equilibrium state with a different concentration of salt in solution (concentration of a species in solution is another thermodynamic coordinate). This is an example of a thermodynamic system. Variations on this include allowing the water vapor above the liquid to be in equilibrium with the saline solution. The presence of salt in solution will alter the vapor pressure over the liquid surface (as well as the freezing temperature). As the temperature changes the vapor pressure will change, etc.

Chemical equilibrium Imagine a gaseous mixture in our vessel at fixed temperature and pressure composed of O and O2. There will be a reaction where M is a background molecule used to carry away momentum (e.g., O2, N2 or Ar in the atmosphere).3 Some ozone will decay and after a while there will be an equilibrium established and the reaction can be written:

The amount of reactants (the left-hand side) may be more than the amount of products (right-hand side) for a given temperature. But as the temperature is changed the balance may shift. This is a thermodynamic system. The ratio of O2 to O3 is now a thermodynamic coordinate along with T, p, V, Mtotal.

Of course, there are many other types of thermodynamic systems, and we will encounter several of them in due course.

Everything outside the system which may affect the system's behavior is called the surroundings. In atmospheric science, we can often approximate an infinitesimal volume of gas embedded in the natural atmosphere as having uniform interior properties. When appropriate, such an infinitesimal volume element can be considered as a thermodynamic system. In many cases the "infinitesimal volume element" might be as big as a classroom or sometimes as small as a cubic centimeter depending on the application.

A thermodynamic system composed of a very large mass is called a reservoir and is characterized by a temperature, TR. If a finite system is brought into contact with the reservoir through a diathermal membrane (one which allows the passage

3 Energy and momentum cannot be conserved simultaneously when two bodies go to one with a release of energy.

A third body in the collision can provide the means of conserving both.

p(hPa) 1000

Isotherms for 1 kg of air

Isotherms for 1 kg of air p(hPa) 1000 Figure 1.1 Isotherms for 1kg of dry air taken as an ideal gas. The vertical coordinate is pressure in hPa, the abscissa is volume in m3. Upper curve, 300 K; lower curve, 200 K.

Figure 1.1 Isotherms for 1kg of dry air taken as an ideal gas. The vertical coordinate is pressure in hPa, the abscissa is volume in m3. Upper curve, 300 K; lower curve, 200 K.

of thermal energy,4 but not mass), the smaller system will adjust the values of its coordinates (for a gas, p, V, T) to new values, while the reservoir does not change its state appreciably (this actually defines how massive the reservoir has to be). The system is said to come into thermal equilibrium with the reservoir (its temperature approaches that of the reservoir). In the case of a gaseous system, experiments have shown that there is a locus of pairs of values (V, p) for which the system is in equilibrium with a given reservoir - in other words, a curve p = pT (V) in the V-p plane. To put it another way, if our system has a certain fixed volume, then when it is brought into contact with the reservoir of temperature T, the pressure will always come to the same value,p = pT(V). As we do the experiment with different control volumes we can sweep out the locus of points in the V-p plane. This curve is called the isotherm of the system for that reservoir temperature (Figure 1.1). The isotherm represents a series of equilibrium states that can occur while the system is in contact with the reservoir (of fixed temperature). For example, the volume might be forced to alter by a change in the wall dimension (e.g., a piston can have different positions in a cylinder which contains the system in question). In this case the pressure will change as a function of volume along the isotherm. While we could invent an algorithm based upon a series of reservoirs of different temperatures to build a temperature scale, it will suffice for our present purposes simply to use the familiar thermometer.

4 Thermal energy refers to the microscopic motion of molecules in the system. When in diathermal contact, the thermal energy of molecules from one system can pass from the system to its neighbor through collisions. In time the thermal energies of the two systems will equalize. More on this in later chapters. The transfer of thermal energy is loosely referred to as heat transfer.

1000 Figure 1.2 Isotherm and adiabat for 1 kg of dry air taken as an ideal gas. The upper curve (solid line) is the 300 K isotherm and the dashed curve is the adiabat passing through the 300 K isotherm at V = 1m3. The vertical coordinate is pressure in hPa, the abscissa is volume in m3.

Figure 1.2 Isotherm and adiabat for 1 kg of dry air taken as an ideal gas. The upper curve (solid line) is the 300 K isotherm and the dashed curve is the adiabat passing through the 300 K isotherm at V = 1m3. The vertical coordinate is pressure in hPa, the abscissa is volume in m3.

A system can also be in equilibrium when isolated (no mass or thermal energy flows into or out of the system) from other systems. We call this an isolated system. It can have coordinates just as in the case of a system in contact with a reservoir. We call the locus of values of pressure in the isolated system for different volumes of the system adiabats (Figure 1.2). We could find the temperature of the isolated system at fixed values of p and V by bringing it into contact with different reservoirs until we find one which does not cause the coordinates of the system to change. The system has the same temperature as this reservoir. In this way we could map out the locus of points defining the isotherm which crosses the adiabat at the point in question. As a simpler alternative, we could insert a thermometer, whose mass is so small that it will come to equilibrium with the system (which now acts as a reservoir with respect to the tiny thermometer) without disturbing the state of the system appreciably.

States of thermodynamic equilibrium must not involve time. They are steady and only require a knowledge of the thermodynamic coordinates such as temperature, pressure and volume. When the "states" traversed by a system involve the time we cannot use thermodynamic equilibrium states to describe them. Conventional thermodynamics cannot be used to describe what goes on in states that are not in equilibrium.

Certain changes of a system can be made to occur through a sequence of infinitesimally nearby equilibrium states. For example, we might bring the system into contact one at a time with a series of reservoirs of infinitesimally differing temperatures, and at each step we wait for equilibrium to be established. We call this a quasi-static process. Such quasi-static processes can be approximated in the laboratory. From a molecular point of view the gas in the interior of the system has to have time during each infinitesimal shift of the constraints to adjust to a new equilibrium with its surroundings. In a gas this is roughly the time for a typical molecule to make a few hundred collisions, but over a finite sized volume it might be more appropriate to use the time for sound waves to traverse the volume several hundred times. This multiple pass traversal time works for pressure, but other properties might take considerably longer. For example, temperature and species concentrations smooth out much more slowly because these differences are smoothed out by diffusive processes such as thermal conduction. Stirring due to turbulence can speed up the homogenization but even then the adjustment is slower than for pressure differences. At each infinitesimal step (waiting for these adjustments) along such a system path, we could reverse direction and retrace the same steps. This is a reversible process.

Note that a system may go from one thermodynamic state to another by a path which does not involve such a sequence of thermodynamic states. We call this an irreversible change in state. An example of an irreversible process is the case of a system which goes from state A to state B spontaneously, but not from B to A. A concrete example is if two bricks, one hot and one cold, are brought into contact, the result is two warm bricks. This is an irreversible process. Note that it never happens that when we bring two warm bricks into contact we end up with a warm brick and a cold brick (even though energy is conserved).

Reversible processes do not actually occur in nature. So why study them? The reasons are pretty simple. First of all, irreversible processes are nearly impossible to treat theoretically. Secondly, experience has shown that approximating the nearly quasi-static processes that do occur in nature works reasonably well in many cases when we treat them as exactly quasi-static. We proceed then to adopt the philosophy used by practitioners for many years: we will freely approximate many processes in the real atmosphere by idealized reversible analogies in order to obtain numerical results that can be used in practical situations.

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