Ideal Gas Law. September 2, 2014

Transcription

1 Ideal Gas Law Setember 2, 2014 Thermodynamics deals with internal transformations of the energy of a system and exchanges of energy between that system and its environment. A thermodynamic system refers to a seciied collection of matter. Such a system is said to be closed in no mass is exchanged with its surroundings and oen otherwise. The air arcel that will serve as our system is in rincile closed. In ractice, mass can be exchanged with its surroundings through entrainment and mixing across the system s country, which is the control surface. A thermodynamic state of a system is defined by the various roerties characterizing it. Two tyes of thermodynamic roerties characterize the state of a system. A roerty that does not deend on the mass of the system is said to be intensive (a secific roerty), and extensive otherwise. Intensive roerties are denoted with lower case, and extensive with uer case. An intensive roerty can be derived from an extensive roerty by dividing by mass z = Z/m. In general, describing the thermodynamic state of a system requires us to secify all of its roerties. However, that requirement can be relaxes for gases an other substances at normal temeratures and ressures. An ideal gas is a ure substance whose thermodynamic state is uniquely determined by any two intensive roerties, which are then referred to as state variables. That means the z 3 = g(z 1, z 2 ). The ideal gas law is the equation of state for an ideal gas. 1. Ideal Gas Law It is convenient to exress the amount of a gas as the number of moles n. One mole is the mass of a substance that contains molecules (N A, Avogadro s number).n = m/m where m is the mass of a substance and M is the molecular weight. Boyle s Law: For constant temerature. V = constant. Charle s First Law: For constant ressure. V/T = constant. Charle s Second Law: For constant volume. P/T = constant. 1

2 These all come together as the ideal gas law: V = nr T (1) = m M R T = mrt where R is the universal gas constant J K 1 mol 1, is ressure, T is temerature, M is molar weight of the gas, V is volume, m is mass and n = m/m is the molar abundance of a fixed collection of matter (an air arcel). The secific gas constant R is related to the universal gas constant R as R = R /M. The form of the gas law that doesn t deend on the dimensions of the system is = ρrt and ν = RT where ν = 1/ρ. A mixture of gases obeys similar relationshis as do its individual comonents. The artial ressure i of the ith comonent-the ressure the ith comonent would exert in isolation at the same volume and temerature as the mixture-satisfies the equation: i V = m i R i T (2) where R i is the secific gas constant of the ith comonent. The artial volume V i -the volume that the ith comonent would occuy in isolation at the same ressure and temerature as the mixture-satisfies the equation: V i = m i R i T (3) Dalton s law asserts that the ressure of a mixture of gases equals the sum of their artial ressures, and the volume of the mixture equals the sum of the artial volumes. = i i (4) V = i V i (5) The equation of state for the mixture can be obtained by summing over all of the comonents: V = T i m i R i (6) Then, defining the mean secific gas constant i R = m ir i m We can obtain the equation for the mixture (7) V = mrt (8) 2

3 We can also define the molar weight of the mixture as: i M = n im i i n = m i n and R = R M For dry air (air without water vaor) the ideal gas law becomes: (9) (10) d = ρ d R d T (11) where α d = 1/ρ d. and R d is the gas constant for 1 kg of dry air. For dry air M = M d = 28.97g (you will do this in HW 2). R d = R M d = = 287JK 1 kg 1 (12) For water vaor the ideal gas law is eα v = R v T. M v = g and R v = JK 1 kg 1. a. Virtual Temerature R d R v = M v M d = ɛ = (13) When there is a mixture of water vaor and dry air, it is convenient to retain the gas constant for dry air and use a fictitious temerature called virtual temerature in the ideal gas equation. The density of a mixture of dry air and water vaor is: where ρ = m d + m v V = ρ d + ρ v = e R d T + T v e R v T = R d T [1 e (1 ɛ) ] (14) = ρr d T v (15) T 1 e (1 ɛ) (16) Virtual temerature is the temerature that dry air would have to attain in order to have the same density as the moist air at the same ressure. Because moist air is less dense than dry air at the same temerature and ressure, the virtual temerature is always greater than the actual temerature by a few degrees. 2. Stratification of Mass If vertical accelerations are ignored, Newton s second law of motion alied to a column of air between ressure and + d reduces to a balance between the weight of that column and the net ressure force acting on it 3

4 which reduces to: da ( + d)da = ρgdv (17) d = ρg (18) dz Integrating between height z and the to of the atmoshere (TOA) ( ) (z) d = (z) = z z gρdz (19) gρdz (20) This imlies that the ressure at height z is equal to the weight of the air in the vertical columnof unit cross sectional area lying above that level. which is known as hydrostatic balance. This is a good aroximation even if the atmoshere is in motion because vertical dislacements of air and their time derivatives are small comared to the forces in Geootential Work that must be done against the earth s gravitational field to raise a mass of 1kg from sea level to that oint (gravitational otential er unit mass). The work (in Joules) in raising 1kg from z to z + dz is gdz, therefore, Integrating dφ gdz = αd (21) Φ(z) = z 0 gdz (22) The geootential doesn t deend on the ath through which the unit mass is taken in reaching that oint. We define the geootential height Z as Z Φ(z) g 0 = 1 g 0 z 0 gdz (23) g 0 is the globally averaged acceleration due to gravity (9.81 m s 2 ). z and Z are almost the same in the lower atmoshere where g 0 g. It is not convenient to deal with the density of a gas...we usually use: d dz = g RT = g (24) R d T v dφ = gdz = R d T v d (25) 4

5 where Φ2 2 d dφ = R d T v Φ d Φ 2 Φ 1 = R d T v Z 2 Z 1 = R d g T v d This is called the hysometric equation. This equation can also be exressed as: ( ) 1 Z 2 Z 1 = Hln = R ( ) dt v 1 ln g 2 2 (26) (27) T v 1 d 2 T v ln ( 1 2 ) (28) The difference Z 2 Z 1 is referred to as the geootential thickness of the layer between these two ressure levels. For an isothermal atmoshere, neglecting the virtual temerature correction, or where Z 2 Z 1 = Hln( 1 / 2 ) (29) [ 2 = 1 ex Z ] 2 Z 1 H (30) H RT v g 0 (31) H is the scale height which reresents the characteristic vertical dimension of the mass distribution, or the e-folding deth. H ranges from 7 to 8km in the lowest 100km of the atmoshere. Global mean ressure and density decrease with altitude aroximately exonentially, from about 1000mb at the surface to only about 100mb at 15km. 90% of the atmoshere s mass lies beneath this level. Turbulent mixing below 100km makes the densityies of assive constituents decrease with altitude at the same exonential rate, which gives air a homogeneous comosition with constant mixing ratios. r N2.78, r O2.21, M d = 28.96gmol 1, R d = Jkg 1 K 1. Above 100km, the mean free ath quickly becomes larger than turbulent dislacements of air. Turbulent air motions are strongly damed by diffusion of momentum and heat and diffusive transort becomes the dominant mechanism for transorting roerties vertically. Pressure decreases monotonically with height, so ressure surfaces never intersect. Because of hydrostatic equilibrium, the elevation of an isobaric surface may be interreted analogously to the 5

6 ressure on a surface of constant altitude. Therefore, centers of low elevation in the height of a 500mb surface imly centers of low ressure on a constant altitude surface. In mountainous terrain, differences in ressure are largely due to differences in elevation. These differences must be corrected to isolate the changes in ressure due to the assage of weather systems. For the layer between the Earth s surface (g) and sea level (0) the hysometric equation becomes Z g = Hln 0 g (32) and ( ) g0 Z g 0 = g ex (33) R d T V Radiosondes measure T(). Measurements are made twice er day at all of the world s meteorological stations at 0Z and 12Z. The heights calculated for a given ressure for all stations are then entered onto a ma. 6

7 List of Figures 1 Ahrens 8 2 Ahrens 9 3 Ahrens 10 7

8 FIG. 1. Ahrens 8

9 FIG. 2. Ahrens 9

10 FIG. 3. Ahrens 10