Chapter 14. The Ideal Gas Law and Kinetic Theory

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1 Chapter 14 The Ideal Gas Law and Kinetic Theory

2 14.1 Molecular Mass, the Mole, and Avogadro s Number To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass scale has been established. The unit is called the atomic mass unit (symbol u). The reference element is chosen to be the most abundant isotope of carbon, which is called carbon-1. Carbon-1 has exactly 1 atomic mass units. 1u kg The atomic mass is given in atomic mass units. For example, a Li atom has a mass of 6.941u.

3 14.1 Molecular Mass, the Mole, and Avogadro s Number One gram-mole of a substance contains as many particles as there are atoms in 1 grams of the isotope carbon-1. The number of atoms per mole is known as Avogadro s number, N A. N A mol 1 n N N A number of moles number of atoms

14.1 Molecular Mass, the Mole, and Avogadro s Number n N N A n m n m m Mass per mole particle N particlen A The mass per mole (in g/mol) of a substance has the same numerical value

4 14.1 Molecular Mass, the Mole, and Avogadro s Number n N N A n m n m m Mass per mole particle N particlen A The mass per mole (in g/mol) of a substance has the same numerical value as the atomic or molecular mass of the substance (in atomic mass units). For example Hydrogen has an atomic mass of g/mol, while the mass of a single hydrogen atom is u.

5 14.1 Molecular Mass, the Mole, and Avogadro s Number Example: The Hope Diamond and the Rosser Reeves Ruby The Hope diamond (44.5 carats) is almost pure carbon. The Rosser Reeves ruby (138 carats) is primarily aluminum oxide (Al O 3 ). One carat is equivalent to a mass of 0.00 g. Determine (a) the number of carbon atoms in the Hope diamond and (b) the number of Al O 3 molecules in the ruby. (a) n Mass m per mole ( 44.5 carats) [( 0.00 g) ( 1carat) ] 1g mol N nn A N 1 ( )( mol 6.0 mol ) 0.74 moles N atoms

6 14.1 Molecular Mass, the Mole, and Avogadro s Number (b) the number of Al O 3 molecules in the ruby. m n Mass per mole ( 138 carats) [( 0.00 g) ( 1carat) ] ( 6.98) + 3( 15.99) g mol N nn A n 0.71moles N nn A 1 ( )( mol 6.0 mol ) atoms

7 14. The Ideal Gas Law An ideal gas is an idealized model for real gases that have sufficiently low densities. The condition of low density means that the molecules are so far apart that they do not interact except during collisions, which are effectively elastic. No interactions All collisions are elastic At constant volume the pressure is proportional to the temperature. P T

8 14. The Ideal Gas Law At constant temperature, the pressure is inversely proportional to the volume. P 1 V The pressure is also proportional to the amount of gas. P n P nt V

9 14. The Ideal Gas Law The absolute pressure of an ideal gas is directly proportional to the Kelvin temperature and the number of moles of the gas and is inversely proportional to the volume of the gas. P nt V THE IDEAL GAS LAW P nrt V PV nrt ( mol K) R 8.31J

10 14. The Ideal Gas Law n N N A PV nrt k PV R N A.31J NkT PV ( mol K) mol N R N A T J K Boltzmann Constant PV nrt OR PV NkT No. of moles No. of particles

11 14. The Ideal Gas Law Example: Oxygen in the Lungs In the lungs, the respiratory membrane separates tiny sacs of air (pressure 1.00x10 5 Pa) from the blood in the capillaries. These sacs are called alveoli. The average radius of the alveoli is 0.15 mm, and the air inside contains 14% oxygen. Assuming that the air behaves as an ideal gas at 310 K (body temperature), find the number of oxygen molecules in one of these sacs. N PV NkT N PV kt [ ] 3 ( 5 ) ( Pa m) 4 π 3 ( J K)( 310 K) molecules of Number of molecules of oxygen in one sac ( ) ( 0.14)? air molecules

12 14. The Ideal Gas Law Consider a sample of an ideal gas that is taken from an initial to a final state, with the amount of the gas remaining constant. PV nrt PV T nr constant P V f T f f PV i T i i Constant T, constant n: Constant P, constant n: P V PV Boyle s law f f i i i V f V i Charles law T f T

13 14.3 Kinetic Theory of Gases The particles are in constant, random motion, colliding with each other and with the walls of the container. Each collision changes the particle s speed. As a result, the atoms and molecules have different speeds.

14 14.3 Kinetic Theory of Gases KINETIC THEORY F ma aa vv tt Average force F v m t ( mv) t pp tt Final momentum - Initial momentum Time between successive collisions Average force ( mv) ( + mv) L v mv L

15 14.3 Kinetic Theory of Gases For a single molecule, the magnitude of average force is: F mv L For N molecules, the average force is: F N mv L 3 root-mean-square speed Since PP FF AA and A L P F F N mv 3 A L 3 L volume

16 14.3 Kinetic Theory of Gases P N mv 3 L 3 P N 3 mv V NkT KE PV N 1 3 ( ) ( ) 1 mv N rms 3 mv rms NNNNNN 3 NN(KKKK) KE 1 3 mv rms KKKK 3 kkkk kt

17 14.3 Kinetic Theory of Gases THE INTERNAL ENERGY OF A MONATOMIC IDEAL GAS 1 mv 3 rms KE kt 3 U N kt 3 nrt

18 14.3 Kinetic Theory of Gases Example: The Speed of Molecules in Air Air is primarily a mixture of nitrogen N molecules (molecular mass 8.0 u) and oxygen O molecules (molecular mass 3.0 u). Assume that each behaves as an ideal gas and determine the rms speeds of the nitrogen and oxygen molecules when the temperature of the air is 93 K. 3kT 1 3 mv rms kt v rms m For Nitrogen m 8.0g mol mol g kg ( J K)( 93K) 3kT v rms m kg v rms 511m s For Oxygen v rms 478m s

19 . Problem: 3 A mass of 135 g of an element is known to contain 30.1 x 10 3 atoms. What is the unknown element? nn NN NN AA moles Since the sample has a mass of 135 g, the mass per mole is 135 g 5.00 mol 7.0 g/mol Atomic mass of the substance is 7.0 u The periodic table of the elements Aluminum

20 Problem: 16 A Goodyear blimp typically contains 5400 m 3 of helium (He) at an absolute pressure of 1.10 x 10 5 Pa. The temperature of the helium is 80 K. What is the mass (in kg) of the helium in the blimp? nn PPPP RRRR PV nrt ( PPPP)(5400 mm 3 ) 8.31 JJ/(mmmmmm. KK)(80 KK) mmmmmmmmmm From Periodic table 1 mole of Helium is g Therefore, the mass m of helium in the blimp is, mm ( mmmmmm)(4.006 gg mmmmmm ) gg kkkk

21 Problem: mv rms An oxygen molecule is moving near the earth's surface. Another oxygen molecule is moving in the ionosphere (the uppermost part of the earth's atmosphere) where the Kelvin temperature is 3.00 times greater. Determine the ratio of the translational rms speed in the ionosphere to that near the earth's surface. kt and Dividing the first equation by the second and using the fact that ( v ) rms ionosphere ( v ) rms T earth ionosphere 3kT 3kT m earth m 3T ionosphere earth's surface ( v ) ( v ) rms ionosphere rms earth's surface 3kTionosphere Tionosphere m kT T earth's surface m earth's surface

22 HWAssignment#4 on WileyPlus is due by Thursday, June 8 th For Practice Ch. 14 FOC: 1, 3, 6, 7 & 8. Problems 13, 16, & 58. Reading For Next Class Ch. 16 & Ch 17

Chapter 14. The Ideal Gas Law and Kinetic Theory

Chapter 14. The Ideal Gas Law and Kinetic Theory Chapter 14 The Ideal Gas Law and Kinetic Theory 14.1 Molecular Mass, the Mole, and Avogadro s Number To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass