Kinetic Theory of Gases I Ideal Gas The number of molecules is large The average separation between molecules is large Molecules moves randomly Molecules obeys Newtons Law Molecules collide elastically with each other and with the wall Consists of identical molecules The Ideal Gas Law PV nRT in K n: the number of moles in the ideal gas N n NA total number of molecules Avogadros number: the number of atoms, molecules, etc, in a mole of a substance: NA=6.02 x 1023/mol.

R: the Gas Constant: R = 8.31 J/mol K Pressure and Temperature Pressure: Results from collisions of molecules on the surface Pressure: F P A Force: dp F dt Force Area Rate of momentum given to the surface Momentum: momentum given by each collision times the number of collisions in time dt Only molecules moving toward the surface hit the surface. Assuming the surface is normal to

the x axis, half the molecules of speed vx move toward the surface. Only those close enough to the surface hit it in time dt, those within the distance vxdt The number of collisions hitting an area A in time dt is 1 N A vx dt 2 V Average density The momentum given by each collision to the surface 2mvx Momentum in time dt: 1 N dp 2mv x A v x dt 2 V Force: dp 1 N F 2mvx Av x

dt 2 V Pressure: F N 2 P mv x A V Not all molecules have the same v x average v2x N P mv 2x V 2 vx 1 2 1 2 2 2 = v v x v y vz 3 3 2 vx 1 2 1 2 = v vrms

3 3 vrms is the root-mean-square speed vrms v 2 2 vx 2 + vy 2 + vz 3 1N 2 2 N 1 2 Pressure: P mv mv

3V 3 V 2 Average Translational Kinetic Energy: 1 2 1 2 K mv mvrms 2 2 2 N P K 3 V Pressure: From 2 PV N K and 3 Temperature: PV nRT

3 nRT 3 K k BT 2 N 2 R 23 Boltzmann constant: k B 1.38 10 J/K NA 1 2 From PV N mvrms 3 and N PV nRT RT NA Avogadros number vrms

3RT M N nN A Molar mass M mN A Pressure Density x Kinetic Energy Temperature Kinetic Energy Internal Energy For monatomic gas: the internal energy = sum of the kinetic energy of all molecules: Eint 3 3 N K nN A k BT nRT 2 2 Eint

3 nRT T 2 HRW 16P (5th ed.). Consider a given mass of an ideal gas. Compare curves representing constant-pressure, constant volume, and isothermal processes on (a) a p-V diagram, (b) a p-T diagram, and (c) a V-T diagram. (d) How do these curves depend on the mass of gas? p p constant pressure isothermal V isothermal constant pressure constant volume isothermal constant volume constant volume V (d)

constant pressure T T mass of gasn pV nRT n p nR n Constant volume T V V Constant temperature PV nRT Constant pressure nR n T p HRW 18P (5th ed.). A sample of an ideal gas is taken through the

cyclic process abca shown in the figure; at point a, T = 200 K. (a) How many moles of gas are in the sample? What are (b) the temperature of the gas at point b, (c) the temperature of the gas at point c, and (d) the net heat added to the gas during the cycle? PV nRT (b) pBVB TB 1.8 103 K nR (c) pC VC TC 6.0 102 K nR Eint Q W Pressure (kN/m2) (a) p AVA n 1.5 mol.

RTA b 7.5 2.5 a c 1.0 3.0 (d) Cyclic process EEint = Volume (m3) 0 Q = W = Enclosed Area= 0.5 x 2m2 x 5x103Pa = 5.0 x 103 J HRW 30E (5th ed.).(a) Compute the root-mean-square speed of a nitrogen molecule at 20.0 C. At what temperatures will the rootmean-square speed be (b) half that value and (c) twice that value? vrms (a) vrms 3RT

M 3RT 38.31 J/mol K 293 K 511 m/s -3 M 28.0 10 kg/mol (b) Since 2 vrms 2 vrms vrms T for 0.5 vrms for 2 vrms T T

2 T 0.5 T 73.3K = -200 C 2 3 T 2 T 1.17 10 K = 899 C HRW 34E (5th ed.). What is the average translational kinetic energy of nitrogen molecules at 1600K, (a) in joules and (b) in electronvolts? (a) 3 3 23 K kBT 1.38 10 J/K 1600K 2

2 3.3110 20 J (b) 1 eV = 1.60 x 10-19 J 3.3110 20 J K 0.21 eV 19 1.60 10 J/eV 3 K kB T 2 Kinetic Theory of Gases II Mean Free Path Molecules collide elastically with other molecules Mean Free Path : average distance between two consecutive collisions 1

2 2d N / V the bigger the molecules the more collisions the more molecules the more collisions Q cm T Molar Specific Heat Eint Q W 3 Eint nRT 2 Definition: For constant volume: Q nCV T For constant pressure: Q nC p T The 1st Law of Thermodynamics:

Eint 3 nRT Q W 2 (Monatomic) Constant Volume 3 nRT Q W 2 (Monatomic) W PdV 0 3 nRT nCV T 2 3 CV R 2 Eint nCV T Q nCV T Eint

3 nRT 2 Constant Pressure 3 nRT Q W 2 W PV nRT 3 nRT nC p T nRT 2 CV Cp R Cp CV 5 Cp R 2 5

3 (Monatomic) Q nC p T 1st Law Adiabatic Process dEint dQ dW (Q=0) dEint dW pdV nCV dT Ideal Gas Law pV nRT Eint nCV T C p CV R pdV pdV Vdp nRdT nR nCV Divide by pV:

C p CV dV dV dp dV (1 ) V p V CV V Cp CV dV dp dV (1 ) V p V dp dV 0

p V ln p ln V ln( pV ) = const. pV = const. nRT ( )V = const. V TV 1 = const. Ideal Gas Law pV nRT Equipartition of Energy The internal energy of non-monatomic

molecules includes also vibrational and rotational energies besides the translational energy. Each degree of freedom has associated with 1 it an energy of k BT per molecules. 2 Monatomic Gases 3 translational degrees of freedom: Eint 3 3 kBT nN A nRT 2 2 1 dEint 3 CV R n dT 2 Eint nCV T

Eint nCV T Diatomic Gases 3 translational degrees of freedom 2 rotational degrees of freedom 2 vibrational degrees of freedom HOWEVER, different DOFs require different temperatures to excite. At room temperature, only the first two kinds are excited: Eint 5 nRT 2 5 CV R 2 HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while the pressure remains constant at 1.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas present is 2.00x10-3 mol, find the molar specific heat at (b) constant pressure and (c) constant volume. (a) Constant pressure:

W = pEV Eint Q W Q pV 20.9 1.0 105 Pa 100 cm3 50cm 3 10 6 m 3 /cm3 15.9 J Eint Q W HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while the pressure remains constant at 1.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas present is 2.00x10-3 mol, find the molar specific heat at (b) constant pressure and (c) constant volume. Q nC p T Q Q R Q

(b) C p nT npV / nR p V (c) pV nRT 8.31 J/mol K 20.9 J 1.0 10 5 Pa 50 10 6 cm 3 34.4 J/mol K

CV Cp R 34.4 J/mol K 8.31 J/mol K = 26.1 J/mol K HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression? (a) Adiabatic piVi p f Vf pi / p f Vf / Vi ln pi / p f 5

ln V / V 3 f Monatomic Cp CV i pV = const. HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression?

(b) piVi p f Vf Ti Tf Tf pf Vf piVi 1.0 10 5 atm 1.0 103 L 273 K T i 2.7 10 4 K 1.0 atm1.0 10 6 L pV nRT HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 C to p =1.0 x

105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression? (c) piVi 4 n 4.5 10 mol. RTi (d) For N/n = 1 3 3 Ki RTi 3.4 10 J 2 3 5 K f RTf 3.4 10 J 2 (Pay attention to the units)

3 nRT K 2 N pV nRT HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression? (e) 2 vrms,i Ti 0.01 2 vrms, f Tf 1 2 3 K mvrms kBT 2

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