Note 2 - Note 2. Energy 2.1 Ideal gases: A simple system to...

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1 Note 2. Energy 2.1 Ideal gases: A simple system to play with Ideal gases are a natural place to start learning about thermodynamics. We have some intuition about how gases work and an “ideal” gas is a simple model of how gases work. In particular, an ideal gas is a great model system to learn and test our understanding of thermodynamics since all of the fundamental thermodynamic properties we will talk about can be found in them and they allow us to study thermodynamics without getting lost in too much math. Later on, we will study more realistic gases and see the nature of the differences. 2.1.1 Basic properties Like many types of matter, we characterize an ideal gas by certain properties: volume ( V ), pressure ( P ), temperature ( T ), and how many moles of atoms are in the gas ( n ). There is a simple expression relating these quantities: PV = nRT This equation is called the equation of state for this system, since it relates the state variables ( P , n , V , and T ) in this case. Actually, many gases behave like ideal gases in certain conditions (this is called “ideal” conditions). There are two types of properties here: 1. Extensive properties are properties which are related to “how much stuff” there is. For example, n and V are extensive properties. If the system is duplicated, these variables get doubled. 2. Intensive properties are independent of the size of the system. For example, P , T , are intensive properties. So is the density ρ = N/V , where N is the total number of atoms. 2.1.2 A brief glimpse of phase transitions: Van der Waals equation When gases cool, they condense to liquids. When liquids cool, they freeze into solids. These are two examples of phase transitions. However, ideal gases cannot have phase transitions. This is what is meant by “ideal.” Thus, far from the phase transition, they are good models, but do not work well near the transition. What makes a phase transition? Interaction between molecules. Gas particles start to stick together at lower temperatures and form a liquid. How can we model this interaction? We can modify the ideal gas equation to include interactions. We’ll do so in two steps:
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2 1. What’s the probability that a gas particle will bump into another one? The density ρ = N/V is a lot like a probability that we’ll find a given particle at a given spot. If the density is high, then there is a high probability that the particle is there. The probability of finding two particles at the same spot goes like the 2 . It’s like what’s the probability of flipping two coins and having them both come up heads: it’s the probability of one coming up heads squared. When two particles get close, they attract each other and lead to a force which brings them together. We can include this force as a modified pressure. This makes the equation of state become (P + a 2 )V = nRT which can be rewritten (P + an 2 /V 2 )V = nRT The constant a has to do with how strong is the attraction between atoms.
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Note 2 - Note 2. Energy 2.1 Ideal gases: A simple system to...

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