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

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

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

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

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

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

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

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)/

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)