1. The Kinetic Theory of Gases
Chapter-19
The Kinetic Theory of Gases

2. Chapter-19 The Kinetic Theory of Gases
Topics to be covered:
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 volume.
Adiabatic expansion of an ideal gas

3. Ch 19-2, 3 Avogadro Number Ideal gas law: m = molecular mass M= m NA ;
N= n NA
Msamp=n M=m n NA
Ideal gas law:
An ideal gas obey the law
pV=nRT
where R=8.31 J/mol.K
Boltzman constant k
k = R/NA
pV=nRT=NkT
Kinetic Theory of gases: Relation of motion of atoms to the volume, pressure and temperature.
Mole: one mole is number of atoms in a 12 g sample of carbon-12
Avogadro Number: one mole contains Avogadro number NA of atoms
NA = 6.02 x 1023 atoms/mol
n= number of moles
N= number of molecules
M= Molar mass of a substance
Msamp= mass of a sample

4. Ch 19-3 Ideal Gas and work done by the ideal gas
Work done by an ideal gas at constant Temperature (Isothermal expansion ):
An ideal gas is allowed to expand from initial state pi,Vi to pf,Vf at constant T, the work W is:
W=VfVipdV=VfVi (nRT/V)dV= nRTln(Vf/Vi) Eint=0, Q=W= nRTln(Vf/Vi)
Work done by an ideal gas at constant volume:
W=VfVipdV=0 and Q=Eint=nCvT
Work done by an ideal gas at constant pressure
W=p(Vf-Vi)=pV=nRT

5. Kavg=(3/2)(R/NA)T=(3/2)kT
Ch 19-4,5 Pressure, Temperature, RMS Speed and Translational Kinetic Energy
Pressure P relation to root-mean -square speed vrms and temperature T
Pressure=Force/Area=(px/t)/L2=mvx2/L3
vx2 = vy2 = vz2 = vrms2 / 3
Pressure P=(nM vrms 2)/3V but pV=nRT then
vrms =  [(3RT)/M]
Translational Kinetic Energy K: Average translational kinetic energy of a molecule Kavg
Kavg=(mv2/2)avg=m (vrms2)/2=(3/2)(m/M)RT
Kavg=(3/2)(R/NA)T=(3/2)kT

6. Ch 19-8 The Molar Specific Heats of an Ideal Gas
Internal Energy Eint: Ideal gas is monatomic and its Eint is sum of translational kinetic energies of its atom. For a sample containing n moles, its internal energy Eint:
Eint=nNAKavg=nNA(3/2)kT= (3/2)n(NAk)T
Eint= (3/2)nRT
Molar Specific Heat at Constant Volume
For an ideal gas process at constant volume pi,Ti increases to pf,Tf and heat absorbed QV=nCVT and W=0. Then
Eint= (3/2)nRT= Q=nCVT
CV= 3R/2
Where CV is molar specific heat at constant volume

7. Ch 19-8 The Molar Specific Heats of an Ideal Gas
Cp :Molar Specific Heat at Constant Pressure:
For an ideal gas process at constant pressure Vi,Ti increases to Vf,Tf and heat absorbed QP=nCpT and W=pV= nR T. Then
Eint= (3/2)nRT= Q-W =nCpT- nRT.
Cp= 3R/2+R=5R/2
Where Cp is molar specific heat at constant pressure
Cp= CV + R; specic heat ration = Cp/ CV
For monatomic gas Cp= 5R/2 ; CV= 3R/2
= Cp/ CV = 5/3
For diatomic gas Cp= 7R/2 ; CV= 5R/2
= Cp/ CV = 7/5

8. Ch 19-11 The Adiabatic Expansion of an Ideal Gas
Adiabatic process: In an adiabatic expansion of an ideal gas no heat enters or leaves the system i.e. Q=0
P, V and T are related to the initial and final states with the following relations:
PiVi= PfVf
TiVi-1 = TfVf-1
Also T/( -1) V =constant then
piTi(-1)/ = pfTf(-1)/

9. Ch 19-11 The Adiabatic Expansion of an Ideal Gas-Free Expansion
Free Expansion of an ideal gas-
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. Ti=Tf
Also in this adiabatic process since ( pV=nRT), piVi=pfVf ( not PiVi= PfVf)