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Chapter 19 The Kinetic Theory of Gases. Chapter-19 The Kinetic Theory of Gases  Topics to be covered:  Avogadro number  Ideal gas law  Internal energy.

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Presentation on theme: "Chapter 19 The Kinetic Theory of Gases. Chapter-19 The Kinetic Theory of Gases  Topics to be covered:  Avogadro number  Ideal gas law  Internal energy."— Presentation transcript:

1 Chapter 19 The Kinetic Theory of Gases

2 Chapter-19 The Kinetic Theory of Gases  Topics to be covered:  Avogadro number  Ideal gas law  Internal energy of an ideal gas  Distribution of speeds among the atoms in a gas  Specific heat under constant volume  Specific heat under constant pressure.  Adiabatic expansion of an ideal gas

3 Kinetic theory of gases It relates the macroscopic property of gases (pressure - temperature - volume - internal energy) to the microscopic property - the motion of atoms or molecules (speed)

4 What is one mole? Avogadro’s Number N A = 6.02 X One mole of any element contains Avogadro’s number of atoms of that element. One mole of iron contains 6.02 X iron atoms. One mole of water contains 6.02 X water molecules. From experiments: 12 g of carbon contains 6.02 X carbon atoms. Thus, 1 mole carbon = 12 g of carbon. 4 g of helium contains 6.02 X helium atoms. Thus, 1 mole helium = 4 g of helium. Avogadro’s Number N A = 6.02 X per mole = 6.02 X mol -1

5 Avogadro's Number Formula - number of moles n = N /N A n = number of moles N = number of molecules N A = Avogadro number M = Molar mass of a substance M sample = mass of a sample n = M sample /M

6 Ideal Gas Law At low enough densities, all gases tend to obey the ideal gas law. Ideal gas law p V=n R T where R= 8.31 J/mol.K (ideal gas constant), and T temperature in Kelvin!!! p V= n R T = N k T; N is the number of molecules and K is Boltzman constant k = R/N A

7 Ideal Gas Law

8 Isothermal process Isothermal expansion (Reverse is isothermal Compression) isotherm Quasi-static equilibrium (p,V,T are well defined) p =n R T/V = constant/V

9 Checkpoint 1

10 Work done at constant temperature W = n R T Ln(V f /V i )

11 Work done at constant pressure isobaric process W = p (V f -V i )

12 Work done at constant volume isochoric process W = 0

13 Root Mean Square (RMS) speed v rms For 4 atoms having speeds v 1, v 2, v 3 and v 4 V rms is a kind of average speed

14 Pressure, Temperature, RMS Speed The pressure p of the gas is related to root-mean -square speed v rms, volume V and temperature T of the gas p=(nM v rms 2 )/3V Equation in the textbook V rms =  (3RT)/M but pV/n = RT

15 Continue… V rms =  (3RT)/M R is the ideal gas constant T is temperature in Kelvin M is the molar mass (mass of one mole of the gas) At room temperature (300K) Gas Molar Mass (g/mol) V rns (m/s) Hydrogen21920 Nitrogen28517 Oxygen32483

16 K avg =(3/2)(R/N A )T=(3/2)kT Translational Kinetic Energy K Average translational kinetic energy of one molecule K avg =(mv 2 /2) avg =m(v rms 2 )/2 K avg =m(v rms 2 )/2=(m/2)[3RT/M] =(3/2)(m/M)RT=(3/2)(R/N A )T

17 Continue At a given temperature, all ideal gas molecules – no matter what their masses – have the same average translational kinetic energy.

18 Checkpoint 2 A gas mixture consists of molecules of type 1, 2, and 3, with molecular masses m 1 >m 2 >m 3. Rank the three types according to average kinetic energy, and rms speed, greatest first.

19 The Molar Specific Heat of an Ideal Gas For a monatomic gas (which has individual atoms rather than molecules), the internal energy E int is the sum of the translational kinetic energies of the atoms. Internal energy of an ideal gas E int E int = N K avg = N (3/2) k T = 3/2 (N k T) = 3/2 (n R T) E int = 3/2 n R T The internal energy E int of a confined ideal gas is a function of the gas temperature only, it does not depend on any other variable.

20 Change in internal energy E int = 3/2 n R T,  E int = 3/2 n R  T

21 The Molar Specific Heat of an Ideal Gas Heat Q 1 Heat Q 2 Eventhough T i and T f is the same for both processes, but Q 1 and Q 2 are Different because heat depends on the path!

22 For an ideal gas process at constant volume p i,T i increases to p f,T f and heat absorbed Q = n c v  T and W=0. Then  E int = (3/2)n R  T = Q = n c v  T c v = 3R/2 Q = n c V  T Heat gained or lost at constant volume where c v is molar specific heat at constant volume

23 For an ideal gas process at constant pressure V i,T i increases to V f,T f and heat absorbed Q = n c p  T and W=P  V. Then  E int = (3/2)n R  T = Q  P  V = Q  n R  T Q = (3/2nR+nR)  T = 5/2 n R  T c p = 5/2 R Q = n c p  T Heat gained or lost at constant pressure where c p is molar specific heat at constant pressure

24 The Molar Specific Heats of a Monatomic Ideal Gas C p = C V + R; specific heat ration  = C p / C V For monatomic gas C p= 5R/2, C V = 3R/2 and  = C p / C V = 5/3 (specific heat ratio)

25 Checkpoint 3

26 The Molar Specific Heat of an Ideal Gas monatomic diatomic polyatomic

27 Internal energy of monatomic, diatomic, and polyatomic gases (theoretical values) (3/2) R = 12.5 (3/2) nRT5/2 R3 Diatomic gas (5/2) R = 20.8 (5/2) nRT7/2 R5 (6/2) R = 24.9 (6/2) nRT8/2 R6 E int =n C V T Monatomic gas Polyatomic gas CvCv E int =n C V T C p =C v +R Degrees of freedom (translational + rotational)

28 C v of common gases in joules/mole/deg.C (at 15 C and 1 atm.) GasSymbol C v   (experiment) (theory) HeliumHe ArgonAr NitrogenN2N OxygenO Carbon DioxideCO

29 Adiabatic Expansion for an Ideal Gas In adiabatic processes, no heat transferred to the system Q=0 Either system is well insulated, or process occurs so rapidly  E int = - W In this case

30 Adiabatic Process  P, V and T are related to the initial and final states with the following relations: P i V i  = P f V f  T i V i  -1 = T f V f  -1  Also T  /(  -1) V  =constant then p i T i (  -1)/  = p f T f (  -1)/ 

31  An ideal gas expands in an adiabatic process such that no work is done on or by the gas and no change in the internal energy of the system i.e. T i =T f  Also in this adiabatic process since ( pV=nRT), p i V i =p f V f ( not P i V i  = P f V f  ) Free Expansion of an Ideal Gas

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