Heat Capacities:Heat Capacities:q ∝∝∝∝∆∆∆∆Tq =C ∆∆∆∆TThe value of C depends on the process, T ↑↑↑↑Closed systemConstant pressureClosed systemConstant volumeExperimentally: ∆∆∆∆UT ↑↑↑↑1 atm∆∆∆∆Hqv=Cv∆∆∆∆Tqv= Cv∆∆∆∆Tqp=Cp∆∆∆∆Tqp= Cp∆∆∆∆THeat capacity of an “ideal” gas:Heat capacity of an “ideal” gas:VIBRATIONROTATIONTRANSLATIONClassically heat energy can be stored in a gas as internal energy with:Classically heat energy can be stored in a gas as internal energy with:½ kT per degree of freedom½ kT per degree of freedomkT per degree of freedomWhen quantization of energy levels is taken into account, contributions to the heat capacity can be considered classically only if En<< kT. Energy levels with En≥≥≥≥kT contribute little, if at all, to the heat capacity.When quantization of energy levels is taken into account, contributions to the heat capacity can be considered classically only if En<< kT. Energy levels with En≥≥≥≥kT contribute little, if at all, to the heat capacity.Since vibrational energies are generally comparable to or greater than kT, only rotations and translations store much energy under normal conditions.Classical internal energy of an “ideal” gas:Classical internal energy of an “ideal” gas:ATOMS:ATOMS:3 Translations per atom32U = n RTU = N(3 ××××½ kT)# of atoms (or molecules)# of molesLINEAR MOLECULES:LINEAR MOLECULES:3 Translations per molecule2 Rotations per molecule52U = n RT32U = N kT + N(2 ××××½kT)NON-LINEAR MOLECULES:NON-LINEAR MOLECULES:3 Translations per molecule3 Rotations per moleculeU = n 3RT32U = N kT + N(3 ××××½kT)qp= Cp∆∆∆∆TConstant Pressureqv= Cv∆∆∆∆TConstant VolumeMolar heat capacity of an “ideal” gas:Molar heat capacity of an “ideal” gas:ATOMS:1.1.∆∆∆∆U + P∆∆∆∆V = Cp∆∆∆∆T∆∆∆∆U = Cv∆∆∆∆T∆∆∆∆H = Cp∆∆∆∆T∆∆∆∆U + R∆∆∆∆T = Cp∆∆∆∆TPV = nRTP∆∆∆∆V = nR∆∆∆∆T1 moleCv= ∆∆∆∆U/∆∆∆∆T32U = n RTCv= ∆∆∆∆U/∆∆∆∆T = R 3252Cp= ∆∆∆∆U/∆∆∆∆T + R = R LINEAR MOLECULES:2.2.NON-LINEAR MOLECULES:3.3.52U = n RTCv= ∆∆∆∆U/∆∆∆∆T = R 5272Cp= ∆∆∆∆U/∆∆∆∆T + R = R U = n 3RTCv= ∆∆∆∆U/∆∆∆∆T = 3R Cp= ∆∆∆∆U/∆∆∆∆T + R = 4R 20.8 J K-1/mol29.1 J K-1/mol33.3 J K-1/mol1 moleCp= ∆∆∆∆U/∆∆∆∆T + RSample Problem:Sample Problem:Sample Problem:When 229 J of energy is supplied as heat to 3.00 mol Ar (g) at constant pressure the temperature of the sample increases by 2.55 K. Calculate the molar heat capacity Cpfor Ar (g). How well does this compare to the ideal gas limit? Estimate Cvfor Ar (g).Sample Problem:Sample Problem:Sample Problem:10.4 kcal of heat are evolved at constant pressure when 1.00g of water combines with Al3+in forming Al(H2O)63+. Calculate ∆H for the reaction:Al3+(aq) + 6 H2O (l) →Al(H2O)63+(aq)∅Molar heat capacity of a monatomic solid:Molar heat capacity of a monatomic solid:For vibrational motion U = kT per degree of freedom.For a solid:d.o.f = 3N - 6 ≈≈≈≈3NU = 3 nRTU = 3 NkTCv= ∆∆∆∆U/∆∆∆∆T = 3R 24.9CvJ K-1/molTAt low T Cv→→→→0At low T Cv→→→→0At high T Cv→→→→3RAt high T Cv→→→→3REnergyQuantized Energy LevelsPn∝∝∝∝ e−−−−∆∆∆∆E/kTClassical Behavior:∆∆∆∆E << kTNon-Classical Behavior:∆∆∆∆E