1 / 35

Chapter 2. EQUATIONS OF STATE

Chapter 2. EQUATIONS OF STATE. In Chapter 1 we introduced concepts and terms that will be used throughout the course. The zeroth law of thermodynamics was discussed. The difficulties in measuring the central variable of thermodynamics, the temperature, were considered.

iain
Télécharger la présentation

Chapter 2. EQUATIONS OF STATE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2. EQUATIONS OF STATE

  2. In Chapter 1 we introduced concepts and terms that will be used throughout the course. The zeroth law of thermodynamics was discussed. The difficulties in measuring the central variable of thermodynamics, the temperature, were considered. We now continue with a discussion of materials, in particular gases. Additional quantities will be defined that can be measured and for which tabulations exist.

  3. Although thermodynamics applies to all forms of matter it is easiest to consider a gas or vapor. The equation of state is of the form f(P,V,T)=0. The mechanical variables (P,V) occur as “canonically conjugate pairs”, the extensive variable V and the intensive variable P. Question: Is there some extensive state variable that is conjugate to the intensive variable T? We discuss this important question much later. NOTE: The term “canonically conjugate” comes from Lagrangian mechanics. The close association of these variables will become clear as we develop the formalism.

  4. Ideal Gas If P, V and T measurements are made on a sample of a real gas it is found that, if the results are plotted graphically, we obtain: (N=number of molecules) High Temp k Low Temp P

  5. As the pressure decreases approaches a fixed value. Furthermore this fixed value is found to be the same for all gases. This experimental no. is called Boltzmann’s Constant and the measured value is: Thus, at low pressure, PV  N kT By definition, an ideal gas, obeys this equation of state exactly. PV = NkT and a real gas behaves like an ideal gas when its pressure is low.

  6. By definition, the universal gas constant R is ideal gas PV = n RT

  7. ideal gas isotherms P = nRT/V (hyperbolae)

  8. van der Waals equation of state for a real gas. At higher densities, the equation PV = n RT does not work well for a gas. Many empirical equations have been put forward. The most common one is van der Waal’s eqn.: Let comes from interactions among gas molecules. b comes from volume of the molecules themselves. a and b are different for different gases. We draw the isotherms for this equation.

  9. Van der waals Gas van der Waals gas isotherms isotherms Tc CP

  10. One isotherm, has a point of inflection called the critical point. At the point of inflection: Example: Isotherms below the critical isotherm have a maximum. Consider an isothermal compression starting at point a. According to the van der Waals equation as v is decreased the path will be along the isotherm shown. However this does not happen. At point b a change of phase begins to take place. Liquid starts to form. At this point, as v is further decreased, P remains constant. At point d the substance has been completely converted to a liquid and as the volume continues to be decreased the pressure rises steeply. With a liquid, a small decrease in volume requires a large increase in pressure. Above the critical T there is no maximum on the isotherms. It is impossible to liquefy the gas regardless of the applied P.

  11. Comment: The van der Waals equation is a cubic equation in v so there are three possible roots for v. We can write in which the are the roots. In the region above the critical isotherm, there is only one real root. In the region below the critical isotherm, are three real roots, all different. At the critical temperature there are three identical real roots and we have

  12. Comment: You will be doing HW problems involving the van der Waals Equation. The thermodynamic variables are written in terms of the critical values: The equation then becomes: Notice that this equation does not contain the quantities a or b which characterize a particular substance. Law of corresponding states: The equation of state when expressed in terms of the reduced variables is a universal equation valid for all substances. Since the van der Waals Equation is not exactly true, the law of corresponding states is only approximately true.

  13. P-V-T surfaces for real gases. We consider a fixed amount of a real substance. The thermodynamic variables are P, V and T, two of which are independent. Equations, such as that for an ideal gas or the van der Waals equation, describe the system over a limited range of these variables, but no single equation is adequate for extended regions. Thus, P-V-T data are often given in tables. A convenient way to visualize the data is with a PVT surface. A cartesian coordinate system with P, V and T as the axes is used to represent the system. If we think as T and V as fixed, then P is determined. Changing T and V changes P and in this fashion a surface is generated. A quasistatic process may be represented by a path on this surface. Let us consider the PVT surface of carbon dioxide, which is characteristic of many pure substances. The surface shown is qualitatively correct, but not quantitatively.

  14. f Carbon Dioxide S-L coexistance regions P G e d S Critical point L a L-V e c T d b f 1 2 triple point line a V T(crit) S-V V

  15. S=solid G=gas L=liquid V=vapor 1 2 triple point line (green) • The following are important aspects of the drawing: • The coexistance regions are ruled surfaces. A line parallel to the V- axis which touches the surface at one point, touches it at all points. • The line 1-2 is called the triple point line. At this unique temperature and pressure (T = 216.6 K, P = 5.18×105 Pa for CO2) the substance can exist in all three phases. • For carbon dioxide: Tc = 304.2 K, Pc = 73.0×105Pa

  16. A line drawn on the surface such that T is constant is called an isotherm. The isotherm passing through the critical point is called the critical isotherm. When the system is in a fluid phase at a T greater than the critical temperature it is called a gas. The fluid phases at T’s lower than critical are called liquid and vapor. Projections onto the P-T plane are useful (phase diagram). Two such diagrams are shown on the next slide. The one on the left is for a substance that contracts upon freezing and that on the right is for a substance that expands (rare) upon freezing. Note that the ruled surfaces become lines in this projection.

  17. S-L S-L CP CP liquid liquid solid solid P P L-V L-V TP TP S-V S-V vapor vapor T T contracts on freezing expands on freezing L-V saturated vapor or boiling point (or vapor pressure) curve S-L freezing point curve S-V sublimation curve It should be emphasized that, with a change of phase, P and T remain constant, but V changes. Most substances contract on freezing. Water and a few other substances expand on freezing.

  18. Expansivity and compressibility We now consider a simple system that can be described by assigning P,V, and T. The equation of state is f(P,V,T) = 0 Two quantities are found to be independent. We can, for example, solve the equation of state for V. V = V(T,P)

  19. Consider an infinitesimal change from one equilibrium state to another equilibrium state. The temperature and pressure, the two variables chosen to be independent, will generally change and we can then write, for the infinitesimal change in V. Each partial can be a function of P and T. We cannot integrate this equation to obtain the change in volume when there are temperature and pressure changes because the partial derivatives are unknown. One way to proceed is to try to obtain expressions for the partials by creating a model of the system. However the approach in macroscopic thermodynamics is to appeal to experimental measurements.

  20. We define two quantities which can be measured and are often tabulated: Volume expansivity: Isothermal compressibility almost always > 0 and is always > 0  and can often be taken as approximately constant over a limited range of the thermodynamic coordinates. Students: What are and for an ideal gas?

  21. The above definitions can be integrated to obtain a change in the thermodynamic coordinates. (1) can now be written as: When we introduce other quantities and develop relationships among various quantities, we will need to manipulate partial derivatives. Hence we will briefly consider partial derivatives.

  22. Theorem: If f(x,y,z) = 0 then Proof:

  23. Now x and z may be considered as independent variables and so the equation must be true when dx=0 and dz0. This implies that The equation must also be true for dz=0 and dx0 which implies that

  24. Equation (1) can now be rewritten as or Notice the cyclic permutation of the variables. These two theorems will be used very often in the course.

  25. Condition for exact differential. Suppose that a relationship exists among x,y and z. We can then consider z=z(x,y) and write dz is said to be an exact differential if where C is any closed path. We write The condition for an exact differential is (see Appendix A)

  26. Example: N M dF is an exact differential

  27. Example: Given thatis an exact differential and using the equation on slide20, i.e. The condition for an exact differential gives: A slightly different approach is given on the next slide. (Problem 2-9)

  28. dV is an exact differential. Using the condition for an exact differential:

  29. It is important in thermodynamics to show what the variables are! is not the same as and is ambiguous The notation indicating which variable is held fixed is always used. Using the above theorems for the PVT system: It is not necessary to obtain additional data for this partial relationship!

  30. Writing P=P(T,V)

  31. Example Suppose temperature is changed a finite amount from TiTf at constant volume. Using equation (1) Over a not too great temperature range

  32. EXAMPLE: Given the equation of state for superheated steam, determine the volume expansivity. a,b,r and m constants Multiply by

  33. To recapitulate: In Chapter 1 we introduced a number of definitions such as isolated system, intensive variables, isobaric process etc. Zeroth Law of thermodynamics and temperature. We discussed the problem of assigning a temperature to a system. In the present chapter we introduced the concept of an ideal gas which has the equation of state: PV = n RT For gases this approximation is usually sufficient, but a better approximation is given by the van der Waals Equation. We briefly discussed PVT surfaces and, in particular, phase diagrams which are projections of PVT surfaces onto the PT plane.

  34. In making calculations involving the macroscopic properties of systems, we must have recourse to experimental data. For example, we might want to know how the volume changes if the system undergoes a change of temperature. Two very useful quantities for which extensive tabulations exist are: Volume expansivity Isothermal compressibility We also discussed some mathematical theorems that are useful in the study of thermodynamics. In the next chapter we will discuss the work that must be done to change a system.

More Related