203 LECTURE 14. STATES OF MATTER: AN INTRODUCTION TO THE IDEAL GAS LAW Summary. Talk about a change of pace. After half a semester on the theoretical underpinnings for the chemical bond, we get decidedly practical and consider the simple idea that matter can be fairly simply categorized by whether it is a gas, liquid or solid. The next several lectures will explore the theories that give rise to the empirical observations we can make about matter in its various states. The most intresting thing is that we will find when we explore condensed matter (liquids and solids) that the concepts we developed in bonding can be applied to explain, and even predict, many of the properties that chemical compounds exhibit. But first, we consider gases. We will find that the simple models that explain most of gas behavior are actually founded on the idea that gas molecules are all the same, and in fact that everything we learned about the chemical bond is irrelevant to the study of gases. Yes, gases are both boring and easy. The first lecture on gases is an elementary introduction in two parts: Part 1. Nomenclature: A brief bit of history on the scientists who investigated gases and the words used to allow us to discuss the nature of gases in a more sophisticated manner. Part 2. Calculations: PV = nRT static system Å Type 1 V1/ T1 = V2/T2 change in state Å Type 2 The second lecture on gases develops somewhat more sophisticated ideas about gases. Part 3: Theory: Kinetic Molecular Theory (kmt) is used to derive ideal gas law PV =nRT and to used to calculate gas velocity, diffusion, and effusion Part 4: Ideal vs Non- ideal gases: the idea that PV = nRT ideally but that PV ≠ nRT in reality, because of gas attraction and size, which violate k.m.t.204 Useful variables for gases In the lectures on thermodynamics we will learn a great deal about chemical systems: a chemical system is the environment in which our chemical species are found. It is characterized by variables, like pressure or volume, that are state functions. Consider a system with the state functions P1, V1, T1, n1 are state functions; (there are many others in thermo). We will learn how to quantify the relationship between these state functions using the ideal gas law. It will often be the case that we will perturb our system somehow to change the values of the state functions, P2,V2,T2,n2, and that the change in these state functions, ∆P, ∆V, ∆T, ∆n, will be information that we want to know. Let’s get to know our state functions: n is the moles of a gas. The amount of gas is most easily defined by the mole and can be related through stoichiometry and other unit factors to other system parameters like n = g/MW; n = MV; ρ(density) = g/V. Note that the moles of gas are typically much smaller in number because gas density is much smaller (about 1000 times less dense.) V is the volume of the gas and has units like ml, l, gallons, ounces, etc. The volume is defined by the space in which gases travel. So it becomes very large in open environments. P is the pressure of the gas with units like atm. It reflects number of times gas molecules hit the surface of the container.205 T is the temperature of the system and has units like K, ° C, ° F and is directly proportional to energy in the system. Kinetic molecular theory (kmt) Scientists sit around thinking really hard about a model for gases. They decide the following is a good way to think of them given that they seem to be rather boring chemically and are very low density*: • gases are hard spheres • gases are infinity small in volume • gases have no attraction for each other or surface of system, therefore collisions are elastic • the energy in a gas system is constant and is determined by temperature Temp → T α E ← energy • the velocity of gas particles is determined by E = ½ mv² * these are “ideal” notions and are used to derive the ideal gas law. Note that gas particles are all tje same according to kmt regardless of type. For example, He is an infinitely small particle that undergoes elastic collisions. So are CO2, N2, or NH3, or H2O.206 Famous experiments with gases Boyle: Investigated the pressure- volume relationship and found an inverse relationship. P goes up ↑, V goes down ↓ Mathematically Pα 1/V inverse relationship or P = k (1/V) as we create an equation with a constant, k. Thus when Dr. Laude sat on a balloon he was repeating Boyle’s experiments from centuries ago that showed:207 Charles: Investigated the volume – temperature relationship and found a direct relationship. As T goes down, V goes down. Mathematically V α T or V = kT Avogadro: Investigated the mole- volume relationship. He discovered the extraordinary idea that the more moles of gas you have, the bigger the balloon. Mathematically, n is directly proportional to volume n α V or n = kV Combining Three Laws Gives Us the Ideal Gas Law. Using a little algebra to combine these three relationships and consolidating the 3 constants into 1, we have ideal gas equation PV = nRT where R is the ideal gas constant. Note the three experimental relationships are all present in this one equation. Also note this equation is true only under ideal conditions (kinetic molecular theory conditions)204 So what kinds of problems can we work: Calculations for a Static System For a system that is not perturbed, we can perform simple calculations with the ideal gas law equation, P1V1 = n1 RT1. These are classic plug and chug calculations in which we typically hold 3 values constant and solve for a 4th that is unknown. Example: What is the volume when 1 mole of ideal gas is held at 1 atm and 298K. P = 1 atm N = 1 mole T = 298K R = 0.08206 latm/ Kmole Rearrange and V = nRT / R = (1) (.082) (298) / 1 atm = 22.4 l ** ** this tells us that at standard (room) temp and pressure, an ideal gas fills 22.4 l of volume. Hint. Two things to note for problems in which there is no change of state 1. You can find an R for any combination of units, but I will give one that you will use a lot. R = .082 l atm/Kmol 2. The only trick is to convert all P,V, T and n values into units consistent with R. That is really the only complicating feature of this kind of problem. And usually the biggest hassle is
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