Heat Capacity: Theory and measurement of heat capacity ratio,

Transcription

1 Heat Capacity: Theory and measurement of heat capacity ratio, Physical Chemistry Laboratory Chemistry 4396 Dr. Jonathan Smith

2 Statistical Mechanics Statistical Mechanics: Classical Equipartition theorem Definition: Equipartition of energy: each quadratic term in the energy is associated with ½ k b T ½ RT (molar) energy k b = x J/K = cm -1 /K R = J/K mole Degree of freedom: total = 3 N N= # of atoms Types: translation, rotational, vibrational, etc.

3 Monatomic gas Monatomic gas: <E trans > = ½ m<v x > 2 + ½ m<v y > 2 + ½ m<v z > 2 U m =3/2 RT # C V! "U & % ( $ "T ' V C v,monatomic = 3/2R

4 Polyatomic gas <E trans > = ½ m<v x > 2 + ½ m<v y > 2 + ½ m<v z > 2 Rotation Linear: 2 degrees of freedom Non-linear: 3 degrees of freedom Vibration: linear 3N-5 degrees of freedom Non-linear 3N-6 degrees of freedom Diatomic: 1 degree of freedom but 1/2 RT for kinetic and potential each = RT Polyatomic: RT for each degree of freedom at high temperature

5 Vibrations

6 Vibrational Contribution Vibrational temperature:! vib =! k k = cm -1 /K * Atkins and De Paula, Physical Chemistry, 9 th edition

7 Heat Capacity Comparison Computed Cp Cv Translational rotational T+R diff vibrations He(g) Ne(g) H2(g) O2(g) H2O(g) CH4(g) (2) 3019(3) 1306(3) 0.006

8 C p Ideal monatomic gas C p =C v +R = C p /C v monatomic = 5/2R / 3/2R = 5/3

9 Equipartition Questions: Given heat capacity data can for a given gas can we determine some structural/geometric details? Do we expect exact agreement between e

10 Adiabatic expansion de=dq+dw = 0 pdv C v,m ln T 2 T 1 =!Rln V m,2 V m,1 2 steps: Expand gas adiabatically and reversibly p 1,V 1, T 1 p 2,V 2, T 2 At constant volume restore temperature to T 1 with p 2 p 3

11 Adiabatic expansion (2) T 2 T 1 = p 2 V m,2 p 1 V m,1 C v,m ln T 2 T 1 =!Rln V m,2 V m,1 ( ) ln p 2 =! C v,m + R " ln V % 2,m p 1 C $ ' v,m # & =! C p,m " ln V % 2,m C $ ' v,m # & V 1,m V 1,m V 2,m V 1,m = p 1 p 2! " C p,m C v,m = ( ) ( ) ln p 1 p2 ln p 1 p3 p 1 = initial high p, p 2 = atm. p, p 3 = final p

12 Adiabatic expansion

13 Abiabatic expansion Apparatus Garland, Nibler, and Shoemaker, 8th editio, Exp. 3

14

15 Analysis! " C p, m Cv, m = ln ln ( ) ( ) p1 p1 p2 p3 Uncertainty analysis: Propogate experimental errors Compare results with theory, within error? Use pressure-time values, calibrate versus accurate pressure gauge.

16 Edited by RALPHK. BIRDWHISTELL textbook forum University of West Florida Pensacola. FL Gary L. Bertrand and H. O. McDonald J. Chem. Educ., 1986, 63 (3), p 252 Heat Capacity Ratio of a Gas by Adiabatic Expansion A Physical Chemistry Experiment with an Erroneous Assumption Gary L. Bertrand and H. 0. McDonald University of Missouri-Rolla, Rolla, MO Thlsrxprrinirnt apparently dates hack to 1819 ( I I as cited ~ vravleich (B). by Parr~ngtun(21. It is rited and discussed t. and appears in at least three laboratory manuals'i4-6). Problems based on variations of this experiment appear in two chemical thermodynamics texts (7,8). The experiment is performed by pressurizing a gas (PI) in a large vessel at ambient temperature (TI), then releasing the gas a t atmospheric pressure (P2) and quickly reclosing before an appreciable amount of heat transfer between the system and the surroundings has occurred. The vessel and remaining gas are then allowed to return to ambient temperature (TI) at constant volume, and the final pressure (Pa) is measured. The heat capacity ratio (C,/C,) for the gas is then calculated from the three pressures. An idealized version of this experiment would have the pressurized gas (P1,TJ insulated and confined by a weightless, frictionless piston expanding against the resistance of the atmosphere (Pd. After the gas has reached a uniform temperature (fi) and pressure (equal to the atmospheric pressure, P2, following this adiabatic expansion to some volume (V2). a volume of the gas equal to the initial volume (VJ is trapped and allowed to return a t constant volume to the initial temperature (TI) and a new pressure (Pa). In every discussion we have found, this process is treated as an adiabatic reversible process, in some cases explicitly and in other cases by referring to an isentropic process. The process is not truly reversible, however, due to a finite difference between the pressure (PI) within the system and the pressure (Pz) operating in the surroundings (see Moore (9)).Modifications of the experiment, such as slowing the escape of the gas (7) or actually measuring the temperature of the gas immediately after expansion ( R, I O, l l ), do not alter the irreversible nature of this process. The idealized version of this experiment is an irreversible, adiabatic expansion at constant opposing pressure (atmospheric). The actual process differs from the idealized version in that the insulation is not perfect, and the escapinggas mixes with the surrounding atmosphere. The actual experiment will also suffer from the fact that the gas cannot be trapped at the precise instant a t which the remaining gas would adiabatically equilibrate to atmospheric pressure. If precisely performed, however, the mixing of the escaping gas with the atmosphere will have little effect on the remaining gas. Failure to meet the truly adiabatic condition is a common flaw whether the process is considered to be reversible or irreversible but, because of the quickness of the process, should contribute little error. Based on these considerations, the experiment is more realistically considered as an irreversible adiabatic expansion at constant opposing pressure than as an adiabatic 252 Journal of Chemical Education reversible process. Application of the first law of thermodynamics to an adiabatic process involving only P - V work gives: In the special case of an ideal gas (which will he assumed throughout this derivation), the change in energy (du) depends only on the change in temperature, the number of moles of gas (n), and the molar constant-volume heat capacity (CJ: nc,dt = -P,,dV (2) At constant opposing pressure (Pz), this integrates to Substituting for initial and final volumes using the equation of state of an ideal gas: The number of moles (n)refers to the original amount in the idealized experiment. However, it is common to both sides of eq 4, and removal leaves only intensive properties of the gas in the initial and final states. The difference between the constant-pressure and Constant-volume heat capacities of an ideal gas (C, - C, = R ) is substituted for the gas constant: CJT, - T,) = -(C, - C&T, - T,P,IP,) (5) which rearranges to The temperature after the adiabatic expansion is unknown but may be calculated from the final pressure after the trapped gas returns to the initial temperature (TI): T,IP, = TIP3 and suhstituted in eq 6 to give The final expression derived for the adiabatic reversible process is (4) While eq 9 is based on the incorrect assumption of a reversible process, calculations of the heat capacity ratio for nearly ideal gases using this equation are normally found to agree quite well with literature data. If this were not the case, the Worthy of discussion

17 J. Chem. Eng. Data 2010, 55, Gaseous Phase Heat Capacity of Benzoic Acid Luís M. N. B. F. Santos* and Marisa A. A. Rocha Centro de Investigação em Química, Departamento de Química e Bioquímica, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, P Porto, Portugal Lígia R. Gomes CIAGEB, Faculdade de Ciências de Saúde Escola Superior de Saúde da UFP, Universidade Fernando Pessoa, Rua Carlos da Maia, 296, P Porto, Portugal, and REQUIMTE, Departamento de Química e Bioquímica, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, P Porto, Portugal Bernd Schröder and João A. P. Coutinho CICECO, Departamento de Química, Universidade de Aveiro, Campus Santiago, P Aveiro, Portugal The gaseous phase heat capacity of benzoic acid (BA) was proven using the experimental technique called the in vacuum sublimation/vaporization Calvet microcalorimetry drop method. To overcome known experimental shortfalls, the gaseous phase heat capacity of BA monomer was estimated by ab initio calculations and compared with experimental results. Gaseous phase heat capacities of BA were directly derived via calculated harmonic frequencies obtained by density functional theory (DFT) (B3LYP, BLYP, BP86, with G(d,p), TZVP, cc-pvtz basis sets) and the second-order Møller-Plesset theory, MP2/ G(d,p). To increase the accuracy of estimation of the thermal properties, a procedure based on the calculation of the heat capacity from quantum chemical calculations in combination with a heat capacity balance of isodesmic reactions is described and applied to calculate the gaseous phase heat capacity, C p,m,ofthemonomericspeciesoverthetemperaturerangeof( to 600) K. The gaseous phase thermodynamic properties of the monomeric form of the BA were also derived from the assignment of the fundamental vibrational frequencies using experimental IR spectra. An excellent agreement among the experimental gaseous phase heat capacities, the results obtained using the proposed ab initio procedure, and the results derived from the assignment of fundamental vibrational frequencies was found. The results for the monomeric form of the BA, directly or indirectly obtained, and conclusions of this work strongly support the thesis that the gaseous phase heat capacity data as currently found in the literature are underestimated to the order of 20 %.