topic 1 CHM 2229 basic thermodynamics work

Thermodynamics, Gas Laws

1. Topic 1 will introduce you to laws governing properties exhibited by gas particles in systems with specified state variables/conditions Subtopics States of Matter Gas laws (Boyle’s law, Charles’ law, Avogadro’s hypothesis) kinetic theory of gases ideal and real gases, ideal gas equation, deviations from ideality  van der Waal gas equations at normal and critical points. Intended learning outcome State gas laws and use the state functions to derive gas equations that aid in explaining gas behaviour TOPIC 1: GAS LAWS AND PROPERTIES

2. STATES OF MATTER 2 Matter exists as solid, liquid or gas. These states differ in the degree of the forces of attraction between neighbouring molecules called cohesion. The gaseous state is the simplest of all the three states of matter because, the intermolecular separations are large relative to the size of the molecules 2 WEEK 1

3. 3 Thus, molecules in a gaseous state are essentially independent of each other and it is possible to describe the behaviour of the gas without referring to the nature of the individual molecules. Gases, which conform perfectly to this pattern, are called ideal gases . ( intermolecular forces and molecular sizes are totally ignored for ideal gas behaniour ). 3

4. 4 In real gas behaviour however, neither the molecular size nor intermolecular forces can be ignored. Real gases tend to become ideal as intermolecular effects are minimised under conditions of high temperatures and low pressures . Note: The physical state of a substance is defined by its physical properties, and two samples of a substance that have the same physical properties are in the same state. 4

5. 5 The state of a pure gas, is specified by giving the values of its; volume ( V), pressure (P) temperature (T) and moles (n). It has been established experimentally that it is sufficient to specify only three of these variables (the fourth is fixed). Each substance is described by an equation of state, an equation that interrelates these four variables. P = f (T, V, N) 5 Physical properties of gas

6. 6 Each substance can be described by its own equation of state. For an ideal gas the equation of state has the form. 6

7. GAS LAWS 7 The equation of state of a low-pressure gas was established by combining a series of empirical laws . The gas laws combined into the simple equation of state PV = nRT 7

8. 8 Robert Boyle , acting on the suggestion of a correspondent John Townley, showed in 1661 that, to a good approximation the pressure and volume of a fixed amount of gas at constant temperature are related by PV = constant This relation is known as Boyle’s law . That is, at constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume, (or the volume it occupies is inversely proportional to its pressure) . 8

9. 9 The molecular explanation of Boyle’s law is that, if a sample of gas is compressed to half its volume, then twice as many molecules strike the walls in a given period of time than before it was compressed. As a result, the average force exerted on the walls is doubled. Hence when the volume is halved, the pressure of the gas is doubled. 9

10. 10 2. Another important property of gases was established by the French scientist Jacques Charles commonly know as Charles’ law. He investigated the effect of T on the V of a sample of a gas that was subjected to constant P. His study revealed that the V increased linearly with the T irrespective of the identity of the gas, provided that it was at low P. 10

11. 11 The molecular explanation of Charles’ Law lies in the fact that raising the temperature of a gas increases the average speed of its molecules. The molecules collide with the walls more frequently and with greater impact. Therefore they exert a greater pressure on the walls of the container. 11

12. 12 3. Finally, at a given pressure and temperature, the molar volume, V m = V m is approximately the same regardless of the identity of the gas. This observation implies that the V of a sample of gas is proportional to the n present, and that the constant of proportionality is independent of the identity of the gas. V = Constant x n (at constant P and T) 12

13. 13 This expression is a modern form of Avogadro’s principle , stating that equal volumes of gases at the same pressure and temperature contain the same number of molecules. The three gas Laws that we have just described are the basis of the equation of a gas PV = Constant x nT The constant of proportionality, which experimentally is found to be the same for all gases, is denoted R (8.314 J K -1 mol -1 ) and is called the gas constant. 13

14. 14 NOTE A gas that obeys this equation exactly under all conditions is called a perfect gas (or Ideal gas ). A real gas , an actual gas, behaves more like a perfect gas and be described exactly by same equation as pressure tend to zero 14

15. KINETIC THEORY OF IDEAL GASES 15 The kinetic model of gases, assumes that the only contribution to the energy of the gas is from the kinetic energies of the molecules . Therefore, the potential energy of the interactions between molecules makes a negligible contribution to the total energy of the gas. 15 WEEK 2

16. 16 The Kinetic model of gases is based on three assumptions: The gas consists of molecules of mass m in ceaseless random motion. The size of the molecules is negligible , in the sense that their diameters are much smaller than the average distance travelled between collisions. The molecules do not interact , except that they make perfectly elastic collisions when they are in contact. 16

17. 17 An elastic collision defined as one in which no internal modes of motion are excited; that is to say the transitional energy remains constant in a collision. From the assumptions, it follows that the pressure and volume of the gas are related by the expression: 17

18. Justification : 18 Consider a particle of mass m is travelling with a component of velocity v x parallel to the x -axis.   18

19. 19 When this particle collides with the wall on the right of the box, it is reflected and its linear momentum changes from mv x before collision to -mv x after the collision The momentum therefore changes by 2mv x . This happens on each collision while the y- and z- components remain unchanged. 19

20. 20 20

21. 21 Many molecules collide with the wall in an interval  t, and the total change of momentum is the product of the change in momentum of each molecule multiplied by the number of molecules that reach the wall during the interval . Because a molecule with velocity component v x can travel a distance v x  t along the x-axis in an interval  t, all the molecules within a distance v x  t of the wall will strike it if they are travelling towards it . 21

22. 22 It follows that, if the wall has an area A, then all the particles in a volume A x v x  t will reach the wall (if they are travelling towards it). The number density of particles is , where n is the total amount of molecules in the container of volume V and N A is the Avogadro constant. 22

23. 23 So the number in the volume Av x  t is On the average, at any instant, half the particles are moving to the right and half are moving to the left. Therefore the average number of collisions with the wall during the interval  t is 23

24. 24 The total momentum change in that interval is the product of this number and the change 2mv x ; Momentum change = = =   24 Where M = mN A .

25. 25 Thus the rate of change of momentum = = = Force = PA This rate of change of momentum is equal to the force (by Newton’s second Law of motion) It follows that the pressure, which is the force divided by the area, is = = P 25

26. 26 Not all molecules travel with the same velocity, so the detected pressure P is just an average of the quantity calculated above. Thus, Since the molecules are moving randomly and there is no net flow in a particular direction, the average speed along x is the same as that in the y- and z- directions. It follows that C 2 = (v x 2 ) + (v y 2 ) + (v z 2 ) = 3 (v x 2 ) 26

27. 27 Implying that (v x 2 ) = and therefore And hence, 27

28. IDEAL AND REAL GASES, IDEAL GAS BEHAVIOUR AND DEVIATIONS FROM IDEALITY 28 In describing the kinetic model, one of the assumptions we made is that molecules in a gaseous state are essentially independent of each other. More precise measurements have however shown that no real gas conforms to such assumptions and therefore do not obey the equation of state exactly, except when the pressure is reduced to very low values. 28

29. 29 From a graphical view, we know that for 1 mole of an ideal gas PV = RT and the ratio is known as the compressibility factor z . For an ideal gas Z = 1 . If z is plotted against P for several gases, it will be observed that at around one atmosphere (101.325 KN-m -2 ) most gases behave ideally 29

30. 30 At higher pressures however, the kinetic theory assumption of point mass and no intermolecular forces become less realistic and Z becomes greater than unit. It is also noticeable that in the low-pressure region Z is less than unity (except for hydrogen). 30

31. The Van der Waals forces 31 Effects due to finite size and intermolecular forces are taken into account as follows: Since molecules are of finite size, no two molecules can simultaneously occupy the same point in space. 31 WEEK 3

32. 32 One can consider half the molecules as “excluding” a certain volume of the container from the other half. If b is the “excluded volume” then the actual volume available to the molecules is (V – b), and incorporating this into ideal gas law gives   P (V – b) = RT 32

33. 33 For N molecules of a gas, as regarded as hard spheres of radius then the presence of one molecule will exclude volume from the centre of the other. 33

34. 34 Since half the molecules exclude this volume from the other half, the total excluded volume is The factor b may be determined experimentally, and from the above equation, d i.e. the size of a molecule can be calculated 34

35. 35 Effects due to Intermolecular Interactions: Attractive forces exist between all types of molecule. There is in fact more than one type of forces, depending on whether the molecules are polar or non-polar, (the general name Van der Waals forces is used to describe them). Intermolecular interactions are taken into account by considering their effect on the speeds with which molecules strike the walls. 35

36. 36 A molecule in the centre of the gas is attracted symmetrically in all directions and so experiences no net force. However on approaching the wall, a net attraction is experienced back towards the centre of the vessel, since there are more molecules on the side remote from the wall. The resulting decrease in speed means that a real gas will exert less pressure than an ideal gas under the same conditions. 36

37. 37 The effect can be allowed for by a term of the form , since both the number of molecules exerting the retarding force and the number upon which the force is exerted depend upon concentration (i.e. both are proportional to a is a constant for a particular gas, independent of the temperature and pressure. 37

38. 38 Thus the Van der Waals equation may be expressed as For any amount of gas with n moles, 38

39. The critical point The term critical point is used to denote the vapor–liquid point of a material, above which distinct liquid and gas phases do not exist. 39

40. The Critical Constants 40 At the critical point Where P c , V c and T c are called the critical constants. Thus for a particular gas, the Van der Waals equation can be used to evaluate the critical constants. 40

41. 41 Multiplying out the parentheses in Van der Waals equation by ,rearranges to the equation below (n=1) at critical temperature , P c , V c and T c are critical P, V and T, respectively 41

42. 42 Therefore, at any temperature below T c an isothermal of a real gas is represented by a cubic equation and must therefore have three roots as shown in the figure below. 42

43. 43 The three roots in isothermal T 1 are V 1 , V 2 and V 3 . The equation of such a curve can also be written in terms of its roots i.e.   (V – V 1 ) (V – V 2 ) (V – V 3 ) = 0 As the temperature is increased, the roots V 1 , V 2 and V 3 approach each other until the critical temperature such that V 1 = V 2 = V 3 = V c 43

44. 44 Therefore equation (V – V c ) (V – V c ) (V- V c ) = 0 becomes V 3 – 3V c V 2 + 3V c 2 V – V c 3 = 0 The above equations are alternative descriptions of the critical Isotherm and therefore are the same i.e., 44

45. 45 Values of Z (compressibility factor) for the most gases lie in the range 0.27 – 0.30 Van der Waals equation provides a useful insight into molecular interactions, its description of the behaviour of real gas is only qualitative . Note: Limitation Reading assignment: Significance of compressibility factor (Z=1, Z > 1 and Z < 1 45

46. The Reduced equation of state 46 46

47. 47 Note Reduced Equation of State is an effective expression of the Van der Waals’ equation. Reason: Does not involve any constants, which are known to be specific to the gas under consideration. The equation therefore applies equally well to all gases. 47

48. SUMMARY Equations of state 48

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