Thermodynamics: Lecture 2

Transcription

1 Thermodynamics: Lecture 2 Chris Glosser February 11, OUTLINE I. Heat and Work. (A) Work, Heat and Energy: U = Q + W. (B) Methods of Heat Transport. (C) Infintesimal Work: Exact vs Inexact Differentials II. More Definitions. III. Equations of State. (A) Extensive an Intensive Variables (B) Ex: Ideal Gas. (C) Ex: Van der Wall s Equation. 2 Heat and Work. OK, so we return to what our definition of heat is Heat is the change in internal energy of a system when no work is done on or by the system. This leads us to the first law of thermodynamics, which is precisely this preceding definition 1

2 2.1 Work, Heat and Energy: U = Q + W. That is, The first law aof Thermodynamics states that: 1st Law The change of the internal energy of a system is equal to the sum of the heat transfered to the system plus the work done on or by the system. Note that the idea of a change in heat or a change in Work doesn t really make any sense. Instead, the heat and the work are a function of changes in various variables, as we shall soon see. We will eventually make a transition to the standard notation, which uses inexact differentials 2.2 Methods of Heat Transport. How does heat get transfered to a system? There are basically three ways (the only three that we will consider anyway) I. Conduction: Transfer of heat my molecular contact II. Convection: Transfer of heat by moving stuff around III. Radiation: Transfer of heat by the emination or absorption of light What are some everyday examples of these three methods? 2.3 Infintesimal Work: Exact vs Inexact Differentials Let s review what we mean by an exact differential (vs. an inexact differential)). Let s say we have two degrees of freedom, x and y. We may construct a differential dz, which represents an infintessimal varition. dz = M(x, y)dx + N(x, y)dy (1) Under what circumstances does this satisfy the fundamental theorem of calculus? You should be able to show that the differential dz satisfies the fundamental theorem of calculus provided that the functions M(x,y) and N(x,y) satisfy the following relationship: M y = N x (2) Then, the idea of writing z a z b = dz actually makes sense. Note that you probably think of this as path independence. You will get to work the details on your next assignment. 2

3 Heat ( Q ) and Work ( W ) are inexact differentials. This means that the amount of Work done is dependent on the path, as is the amount of heat transfered. You will sometimes see these defined in terms of d s with dashes through them ( dq, dw ). You should not really think of these as being infintessimal changes, rather, you should think of them as being infinitessimal values. It is almost always the case that an inexact differential may be made exact by multipling it by an integrating factor. This indeed will be the case for Work and Heat, where the integrating factors are 1/P and 1/T, respectively. How exactly can find if an inexact differential has an integrating factor? Well, we can try to multiply it by an arbitrary integrating factor and see what happens. Let dz be an inexact differential: dz = Mdx + Ndy, (3) where I am suppressing th function arguments. Now, let µ be the integrating factor that makes dz exact: dw = µ dz = µmdx + µndy. (4) Now, let s consider the case where µ is a function of x only. We can show that the function; [ ( M exp N 1 y N ) ] dx (5) x satisfys equation 2. 3 Equations of State. An equation of state is basically an emperically or theoretically derived relationship between the three variables, P, V, T (there can be others, but for simplicity s sake, we ignore chimical processes for now). An equation of state presumably be solved for one of the variables, say P: P = f(v, T ). (6) Therefore we can express all equations of state in the form; F (P, V, T ) = P f(v, t) = 0; (7) 3

4 3.1 Extensive an Intensive Variables Let s pause for a moment, for a couple more definitions: Extensive: A state variable that depends on the size of the system in such a way that it is proportonal to the mass. Intensive: A variable that is independent of the size of the system. It should be noted that an extensive variable times an intensive variable is extensive (naturally), and that an extensive variable divided by an extensive variable is intensive. Here are some examples of each kind: Extensive: Volume (V); Internal Energy (U); Mass (m); Number of moles(n); Intensive Temperature (T) Mass density (m/v) Molar density (n/v) Presure (P) Can you think of any other extensive variables? How about intensive ones? Anyway, we see that Work ( dw ), being an extensive (dv ) quantity times an intensive (P ) quantity is itself extensive. Indeed this must be true since it is proportionalo to an extensive quantity, the system energy. Of course this implies that the Heat (however you define it) is itself an extensive quantity. This of course begs the question: can we express the (inexact) differential ( dq) in terms of an intensive quantity and an extensive exact differential in analogy to Work? The answer is yes, and we will get back to that later. 3.2 Ex: Ideal Gas. The familiar equation of state for an ideal gas is; P V = nrt ; (8) 4

5 3.3 Ex: Van der Wall s Equation. Another equation of state that you will run into if you do the homework is Van der Wall s equation; (P + a v 2 ) (v b) = RT (9) (the quantity v is the inverse of the molar density, called the molar volume). 3.4 Expansivity and Compressability This is not actually a specific model, but a definition of two macroscopic parameters in term of their state equations. Let as assume that our equation of state F (P, V, T ) = 0 can be solved for the molar volume, v; Now, taking the differential, we find: dv = v = v(p, T ). (10) ( ) ( ) v v dt + dp. (11) T P P T If we define the expansivity and the compressability by; β = 1 v ( ) v T (12) and; Then we may express this as; κ = 1 v ( ) v, (13) P dv = βvdt κvdp. (14) For your next homework, I am going to ask you to compute β and κ for the ideal gas and the Van der Wall model. 5