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PHY 231 1 PHYSICS 231 Lecture 26: Ideal gases Remco Zegers Walk-in hour: Thursday 11:30-13:30 am Helproom

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PHY 231 2 Problems in the book for extra practice 8: 39, 41, 47, 51 9: 5, 13, 19, 26, 27, 40, 43, 55 10: 9, 13, 29, 31, 37, 42 11: 1,4,13,21,33

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PHY 231 3 Ideal Gas: properties Collection of atoms/molecules that Exert no force upon each other The energy of a system of two atoms/molecules cannot be reduced by bringing them close to each other Take no volume The volume taken by the atoms/molecules is negligible compared to the volume they are sitting in

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PHY 231 4 R Potential Energy 0 -E min R min Ideal gas: we are neglecting the potential energy between The atoms/molecules R Potential Energy 0 Kinetic energy

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PHY 231 5 Number of particles: mol 1 mol of particles: 6.02 x 10 23 particles Avogadro’s number N A =6.02x10 23 particles per mol It doesn’t matter what kind of particles: 1 mol is always N A particles

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PHY 231 6 What is the weight of 1 mol of atoms? X Z A Number of protons molar mass Name

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PHY 231 7 Weight of 1 mol of atoms 1 mol of atoms: A gram (A: mass number) Example: 1 mol of Carbon = 12 g 1 mol of Zinc = 65.4 g What about molecules? H 2 O 1 mol of water molecules: 2x 1 g (due to Hydrogen) 1x 16 g (due to Oxygen) Total: 18 g

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PHY 231 8 Example A cube of Silicon (molar mass 28.1 g) is 250 g. A) How much Silicon atoms are in the cube? B) What would be the mass for the same number of gold atoms (molar mass 197 g) A) Total number of mol: 250/28.1 = 8.90 mol 8.9 mol x 6.02x10 23 particles = 5.4x10 24 atoms B) 8.90 mol x 197 g = 1.75x10 3 g

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PHY 231 9 Boyle’s Law P 0 V 0 T 0 2P 0 ½V 0 T 0 ½P 0 2V 0 T 0 At constant temperature: P ~ 1/V

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PHY 231 10 Charles’ law P 0 V 0 T 0 P 0 2V 0 2T 0 If you want to maintain a constant pressure, the temperature must be increased linearly with the volume V~T

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PHY 231 11 Gay-Lussac’s law P 0 V 0 T 0 2P 0 V 0 2T 0 If, at constant volume, the temperature is increased, the pressure will increase by the same factor P ~ T

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PHY 231 12 Boyle & Charles & Gay-Lussac IDEAL GAS LAW PV/T = nR n: number of particles in the gas (mol) R: universal gas constant 8.31 J/mol·K If no molecules are extracted from or added to a system:

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PHY 231 13 Example An ideal gas occupies a volume of 1.0cm 3 at 20 0 C at 1 atm. A) How many molecules are in the volume? B) If the pressure is reduced to 1.0x10 -11 Pa, while the temperature drops to 0 0 C, how many molecules remained in the volume? A) PV/T=nR, so n=PV/(TR) R=8.31 J/molK T=20 0 C=293K P=1atm=1.013x10 5 Pa V=1.0cm 3 =1x10 -6 m 3 n=4.2x10 -5 mol n=4.2x10 -5 *N A =2.5x10 19 molecules B) T=0 0 C=273K P=1.0x10 -11 Pa V=1x10 -6 m 3 n=4.4x10 -21 mol n=2.6x10 3 particles (almost vacuum)

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PHY 231 14 And another! An airbubble has a volume of 1.50 cm 3 at 950 m depth (T=7 o C). What is its volume when it reaches the surface ( water =1.0x10 3 kg/m 3 )? P 950m =P 0 + water gh =1.013x10 5 +1.0x10 3 x950x9.81 =9.42x10 6 Pa V surface =1.46x10 -4 m 3 =146 cm 3

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PHY 231 15 Correlations A volume with dimensions LxWxH is kept under pressure P at temperature T. A) If the temperature is Raised by a factor of 2, and the height of the volume made 5 times smaller, by what factor does the pressure change? Use the fact PV/T is constant if no gas is added/leaked P 1 V 1 /T 1 = P 2 V 2 /T 2 P 1 V 1 /T 1 = P 2 (V 1 /5)/(2T 1 ) P 2 =5*2*P 1 =10P 1 A factor of 10.

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PHY 231 16 Diving Bell A cylindrical diving bell (diameter 3m and 4m tall, with an open bottom is submerged to a depth of 220m in the sea. The surface temperature is 25 0 C and at 220m, T=5 0 C. The density of sea water is 1025 kg/m 3. How high does the sea water rise in the bell when it is submerged? Consider the air in the bell. P surf =1.0x10 5 Pa V surf = r 2 h=28.3m 3 T surf =25+273=298K P sub =P 0 + w g*depth=2.3x10 6 Pa V sub =? T sub =5+273=278K Next, use PV/T=constant P surf V surf /T surf =P sub V sub /T sub plug in the numbers and find: V sub =1.15m 3 (this is the amount of volume taken by the air left) V taken by water =28.3-1.15=27.15m 3 = r 2 h h=27.15/ r 2 =3.8m rise of water level in bell.

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PHY 231 17 A small matter of definition Ideal gas law: PV/T=nR PV/T=(N/N A )R n (number of mols)= N (number of molecules) N A (number of molecules in 1 mol) Rewrite ideal gas law: PV/T = Nk B where k B =R/N A =1.38x10 -23 J/K Boltzmann’s constant

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PHY 231 18 From macroscopic to microscopic descriptions: kinetic theory of gases For derivations of the next equation, see the textbook 1) The number of molecules is large (statistical model) 2) Their average separation is large (take no volume) 3) Molecules follow Newton’s laws 4) Any particular molecule can move in any direction with a large distribution of velocities 5) Molecules undergo elastic collision with each other 6) Molecules make elastic collisions with the walls 7) All molecules are of the same type

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PHY 231 19 Pressure Number of Molecules Volume Mass of 1 molecule Averaged squared velocity Average translation kinetic energy Number of molecules per unit volume

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PHY 231 20 Microscopic Macroscopic Temperature ~ average molecular kinetic energy Average molecular kinetic energy Total kinetic energy rms speed of a molecule M=Molar mass (kg/mol)

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PHY 231 21 example What is the rms speed of air at 1atm and room temperature? Assume it consist of molecular Nitrogen only (N 2 )? R=8.31 J/molK T=293 K M=2*14x10 -3 kg/mol v rms =511 m/s !!!!!

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PHY 231 22 And another... What is the total kinetic energy of the air molecules in the lecture room (assume only molecular nitrogen is present N 2 )? 1) find the total number of molecules in the room PV/T= Nk b P=1.015x10 5 Pa V=10*4*25=1000 m 3 k b =1.38x10 -23 J/K T=293 K N=2.5x10 28 molecules (4.2x10 4 mol) 2) E kin =(3/2)Nk B T=1.5x10 8 J (same as driving a 1000kg car at 547.7 m/s)

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