CHAPTER 7 Materials and Models Ideal systems and models: the ideal gas Bouncing molecules: the microscopic point of view The macroscopic point of view Pressure Archimedes’ principle Bernoulli’s equation Absolute temperature The surprising bridge to temperature Internal energy and heat capacity Heating Work The first law of thermodynamics Other systems: adding pieces from reality Molecules Phase changes Condensed matter Metals Chemical energy Quantum theory Back to heat capacities Diatomic gases We have applied Newton’s laws of motion to the objects they were meant for: objects that are large enough to see or feel. In this chapter we use them to describe the mechanics of atoms and molecules. As we do that we have to keep in mind that Newton’s laws and classical mechanics apply in the microscopic realm only to a limited extent, and that the ideas and methods of quantum mechanics may be necessary. So far we have applied Newton’s laws to “objects” without considering the material that they are made of or their internal structure.To treat them in terms of their atoms and molecules takes us from a few simple particles and unchanging objects to composite objects that consist of very large numbers of particles.The real systems are so complex that we approximate them with models, invented, imagined systems that, however, retain the essential properties of the systems that we seek to understand. 136 / Materials and Models We begin this chapter by introducing the model called the ideal gas.Itisan enormously fruitful model, which describes some of the most important proper- ties of real gases, and, to some extent, also those of other materials. It also provides a bridge to a seemingly entirely different subject, that of heat and temperature. In addition, it shows the correspondence between two very different aspects of the world around us. On the one hand there are the large-scale objects of our direct sensory experience. Our instruments show us that underlying this macroscopic world is a microscopic world that we are normally not aware of, the world of moving, vibrating, rotating, colliding atoms and molecules, absorbing, emitting, and exchanging energy. 7.1 Ideal systems and models: Bouncing molecules: the the ideal gas microscopic point of view How can we get started talking about the vastly The basic feature of the model that we call the complex real world? We know that we need to ideal gas is that its components are always in talk about one that is simpler, a model system, motion and have only kinetic energy. They have sometimes called an ideal system. Let’s review no internal structure, no internal motion, and the rules of the game. We decide what the model no internal energy. They exert no forces on each is and how it works. It is an invented system other except when they touch, so that there is no that shares some of the characteristics of the mutual potential energy. Since the model requires corresponding system in the real world. that these elementary components are point par- The words ideal and model describe a sys- ticles with no internal structure, it doesn’t matter tem with properties that we make up and laws whether we call them atoms or molecules. (We’ll that we prescribe. We set the rules that we need call them molecules.) The figure shows a portion to calculate the properties and behavior of our of a gas with just four molecules. model system. We can then compare the model and the real material to see to what extent their properties overlap. We can also ask what the model predicts for behavior under conditions that were not orig- inally considered, and to see how the ideal and the actual systems then compare. Here is the model that we will look at in some detail: it’s a gas whose constituents (its atoms or molecules) are particles, i.e., they have no size or internal structure. They exert forces When the molecules collide with each other and experience forces only when they touch each they can interchange energy and momentum in other or the walls of their container. They move accord with the laws of conservation of energy in accord with the laws of Newtonian mechanics. and momentum. The total momentum of the sys- This model is called the ideal gas. tem of which they are parts does not change. A real gas is very different. Its smallest units Neither does the total kinetic energy, since we may be molecules, composed of two or more have decided that in this model there is no other atoms, or single atoms, consisting of nuclei and kind of energy. electrons. They are not zero-size particles, and When we put the gas in a box or other con- they exert forces on each other even when they tainer the molecules bounce off the walls. At each don’t touch. Nevertheless, the ideal-gas model bounce the wall exerts a force on a molecule. describes and predicts the behavior of real gases From Newton’s third law we know that there is at very well under many circumstances. the same time a force by the molecule on the wall. 7.1 Ideal systems and models: the ideal gas / 137 Our aim is to relate what the molecules To find the force exerted by the molecule on are doing, as described by their microscopic the wall we use Newton’s second law of motion. variables, namely their velocities and kinetic Instead of Fnet = ma we use the momentum form. = Δv energies, to the macroscopic variables that we Since a Δt (the change in the velocity divided observe and measure directly, namely the pres- by the time it takes for the velocity to change), Δ(mv) sure (equal to the average force on the walls ma is equal to Δt , i.e., the force is equal to divided by the area of the walls) and the volume the change in momentum divided by the time of the container. during which the momentum changes. For our We start with a single molecule moving back single molecule the magnitude of the momentum and forth in a cubical box whose sides have change, Δ(mv), is 2mvx at each collision, and this length L and area A = L2, with volume V = L3. happens once in every time interval Δt, on one The molecule moves with an initial velocity vi wall and then on the other, so that the average Δ 2 in the x direction toward a wall that is at right force is (mv) = 2mvx = 2mvx . Δt L / vx L angles to the x-direction. We assume that we can The molecule travels back and forth and neglect the effect of the gravitational force on contributes to the pressure on both walls. To its path. find the pressure we need to divide the force by Like a ball hitting a solid wall at right angles 2mv2 mv2 the area of both walls, 2L2, to get x or x . to it, the molecule will bounce back along the 2L3 V This is the contribution to the pressure of a sin- same line, with its kinetic energy unchanged, gle molecule bouncing back and forth. For a gas and the direction of its momentum reversed. 2 of N molecules it is Nmvx . We can also take into The kinetic energy is a scalar quantity that does V account that the velocities are not all the same, not depend on direction, but the velocity and and use the average of mv2, which we write as the momentum are vector quantities. The initial x 2 2 = Nmvx momentum vector mvi (where m is the mass of mvx, so that P V , which can be rewritten = 2 a molecule) and the momentum vector after the as PV Nmvx. collision, mvf, differ by the vector m v, where This is really the end of the derivation. But + = vi v vf. we can also look at what happens if the molecules do not just move along the x direction. In that v case there are the additional velocity components f vy and vz. The velocity v is related to them 2 = 2 + 2 + 2 v by v vx vy vz . With a large number of molecules, randomly moving in all possible direc- v v v f i tions, all components will be equally represented, i 2 2 2 and their averages vx, vy , and vz will be equal. 2 2 v is then equal to 3vx. This allows us to rewrite = 1 2 our result as PV Nmv , which we can also Since all the velocities and momenta are 3 write as PV = 2 N 1 mv2 . along the x direction, we can call the initial 3 2 momentum mvx and the final momentum −mvx. We have achieved our goal. We now have The change in momentum is −2mvx, with a mag- a relation between the macroscopic variables, P nitude of 2mvx. Between a collision with one and V, quantities that apply to the whole con- wall and a collision with the opposite wall the tainer with the gas in it, on the left, and the micro- molecule moves a distance L in a time that we’ll scopic variables that refer to the molecules, i.e., Δ Δ = Δ = L their number and their average kinetic energy, on call t, so that vx t L,or t v . x the right. EXAMPLE 1 mvx Joe and Jane are practicing tennis by hitting balls –mvx against a wall. The mass of a ball is 0.15 kg.