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7. Heat capacity Basel, 2008 7. Heat capacity 1. Introduction 2. Heat capacity at constant volume 3. Internal energy 4. Classic approach – gases (Internal energy/heat capacity) 5. Quantum mechanics approach – gases (Internal energy/heat capacity) References: Supplementary material: German version for this chapter (Prof. Huber lecture from 2007, Kapitel 7). Web: Maple-Sheet „Cv-Rotat.“ •P. Atkins, J. de Paula, “Atkins‘ Physical Chemistry”, Oxford Univ. Press, Oxford, 8th ed., 2006, Chapter 2 and 20. • I. Tinoco, K. Sauer, J.C. Wang, J.D. Puglisi “Physical Chemistry, Principles and applications in biological sciences”, PrenticeHall, New Jersey, 4th ed., 2002, Chapter 2. 7.1 Introduction Heat capacity of the local surroundings of a system is essential in thermodynamics, because the heat absorbed or evolved by the system can be monitored as a temperature change. When heat is transferred to a system (proportional to the heat supplied): > change in temperature, T – temperature dT = Coefficient × dq (6.1) q - heat Heat capacity, C is a measure of the heat energy required to increase the temperature of a substance, by a certain temperature interval (specific heat), which depends on the size, composition and state of the system. dq = C × dT (6.2) Heat capacity Molar heat capacity: Cm = C/n (6.3) n – number of moles M – mass in kg Specific heat capacity: c = C/M (6.4) When C is large > a given amount of heating results in only a small temperature rise (the system has a large capacity for heat) The heat capacity depends on conditions: - system constrained to have constant volume > Cv (heat capacity at constant volume, or isochoric heat capacity) - system subject to constant pressure > Cp (heat capacity at constant pressure, or isobaric heat capacity) 7.2 Heat capacity at constant volume Heat capacity at constant volume represents the heat supplied to the system under specific conditions (constant volume and no other work). It can be related to the increase in internal energy that accompanies the heating: δ- partial derivatives with respect to the (6.5) δ U ⎞ ⎛ CV = ⎜ ⎟ ⎝ δT ⎠V changing variable (when one or more variables are held constant during the change of another variable) U – internal energy of the system T - temperature Molar heat capacity constant volume: CV ,m ⎛ δU m ⎞ =⎜ ⎟ ⎝ δT ⎠V at Specific heat capacity at constant volume: (6.6) Um – molar internal energy [CV,m ] = J/(mol K) 1 cV = M ⎛ δU ⎞ ⎟ ⎜ ⎝ δT ⎠V (6.7) [cV ] = J/(kg K) 7.3 Internal energy of the system Internal energy, U: total kinetic and potential energy of the particles in the system (summ of all contributions from all atoms, ions, and molecules in the system). U = Uo + Utrans + Urot + Uvib + Uel translation rotation (6.8) U0 – internal energy at T = 0 [U] = J = 1 kg m2s-2 electronic vibration 1 cal = 4.184 J 1 eV = 0.16 10-18J Characteristics of the internal energy: • U = f(T, p) • U is an extensive property of the system Internal energy – function of state U is a function of state (depends only on the present state of the system and is independent on the path by which the state was reached). A property which depends on two variables is function of state if it depends only on the current state of the system. ∆U is independent on the path between these states. ab le 2 P0 Va ri A change in this property is independent on the path between the initial state (P0), and the final state (Pf). (6.9) ∆U = Uf - Ui Pf Variable 1 Convention: ∆U > 0 – a flow of energy into the system ∆U < 0 – a flow of energy out of the system Internal energy of the system Total internal energy of a system – too complicate to measure it directly > ∆U can be measured (it is possible to monitor the energy supplied or lost by the system, as heat or work: ∆U = w + q (6.10) w – energy transfered to the sytem by dooing work q – energy transfered to the system, by heating Heat capacity is a summ of various terms according to the type of energy which is contained in the internal energy of the system. C = Ctrans + Crot + Cvib + Cel (6.11) 7.4 Classic approach - gases Degree of freedom of a molecule (3N): - Translation (3) - Rotation (linear molecules: nrot = 2, non-linear molecules nrot = 3) - Vibration (linear molecules nvib = 3N-5, non-linear molecules nvib = 3N-6) Internal energy of a molecule, due to translation (kinetic energy): U tr = ( ) 1 3 m v 2x + v 2y + v 2z = kT 2 2 (6.12) Equipartition theorem: for a collection of particles in thermal equilibrium (T), the average value of each quadratic contribution to the energy is the same = ½ kT (or ½ RT if we consider the molar contribution). Total energy of the gas of N particles (considering only the translation): U = U0 + 3 NkT 2 (6.13) 7.4.1 Total internal energy of a gas Total molar energy of a gas (considering only the translation movement): U m ,tr = U m (0) + 3 RT 2 (6.13) Um N = nNA R = Nak k – Boltzmann constant k = 1.38 10-23 JK-1 T Um, tr – linear in temperature Consider the rotation moverment of molecules: for each degree of freedom of rotation a contribution of 1/2kT (or ½ RT for molar internal energy) Total molar internal energy of the system, considering the translation and rotation: U m ,tr 3 1 = U m (0) + RT + nrot RT 2 2 (6.14) 7.4.2 Molar heat capacity at constant volume Total internal molar energy of the system, considering in addition the vibration movement: U m ,tr 3 1 = U m (0) + RT + nrot RT + nvib RT 2 2 (6.15) In vibration degree of freedom should be considered a potential energy term, of 1/2RT Molar heat capacity at constant volume: Cv , m ⎛ δU m ⎞ ⎛ 3 1 ⎞ =⎜ ⎟ = ⎜ + nrot + nvib ⎟ R ⎠ ⎝ δT ⎠ ⎝ 2 2 Cv, m (6.16) Do not depend on temperature T Molar heat capacity at constant volume Example: each HCl molecule of gas changes its movement due to the collisions with other molecules in the system (coloured points) Rotation Translation H Cv , m 7 ⎛3 1 ⎞ = ⎜ + nrot + nvib ⎟ R = R 2 ⎝2 2 ⎠ Cl Vibration Mass center nrot = 2, nvib=1 Cv,m = 29.10 J mol-1 K-1 > 20.6 J mol-1 K-1 (experimental value, at 300K) Quantum mechanics approach 7.5 Quantum mechanics approach A gas system formed by diatomic molecules at temperature T has a vibrational energy which is the summ of all vibrational contributions of the molecules. 1. Each molecule has a vibrational energy (harmonic oscillator approx.): 1⎞ ⎛ Evib (v ) = ⎜ v + ⎟hν vib 2⎠ ⎝ (6.17) 2. The v-th energy level is occupied by n(v) molecules, those number corresponds to the Boltzmann distribution: n(v ) = n0 e − Evib ( v ) (6.18) kT 7.5.1 Vibrational energy of a gas Total vibrational energy of the gas = summ of (all energies multiplied with the Boltzmann occupancy of every energy level) ET = ∑ E (v i ) = i 5 + hν ⋅ n0 e 2 5 − hν 2 1 hν ⋅ n0 e 2 kT 1 − hν 2 kT 3 + hν ⋅ n0 e 2 ⎛ 1⎞ + K + ⎜ i + ⎟hν ⋅ n0 e ⎝ 2⎠ 3 − hν 2 ⎛ 1⎞ − ⎜ i + ⎟ hν ⎝ 2⎠ kT kT + (6.19) +K Total number of molecules in the system = summ of molecules occupying each vibrational energy level: N T = ∑ n(v i ) = n0 e 1 − hν 2 kT + n0 e 3 − hν 2 kT i + n0 e 5 − hν 2 kT + K + n0 e ⎛ 1⎞ − ⎜ i + ⎟ hν ⎝ 2⎠ kT +K + (6.20) Mean vibrational energy of a molecule The mean vibrational energy of a gas molecule = total vibrational energy of the gas/total number of gas molecules (for complete calculus see: K7-4, Vorlesungsskript, 2007, H. Huber). ∑ E (v ) ⎡ 1 e ν = = hν ⎢ + ν ( ) n v 2 ⎢⎣ ∑ 1− e −h i Evib ,molecule kT i −h i i ⎤ ⎥ kT ⎥ ⎦ (6.21) Molar internal energy due to the vibration is: U vib (T ) = N A h ν ⋅ [ + (e 1 2 h ν / kT − 1) −1 ]= R ⋅ Θ ( + (e vib 1 2 Θ vib /T − 1) Θvib = hν/k −1 ) (6.22) 7.5.2 Heat capacity term - vibration The term of heat capacity corresponding to a mole of gas, due to the vibration movement is: C vib (T ) = R ⋅ (hν / kT ) e 2 hν /kT C vib (T ) = R ⋅ (Θ vib / T ) e 2 - Characteristic temperature: - Low temperatures (T = 0 K) - High temperatures (T >> Θ) (e hν /kT −Θ vib /T − 1) −2 (1 − e T=Θ (6.23) −Θ vib /T ) 2 (6.24) Cvib(T) = R(e -1)2/e3 Uvib = ½ NA h ν Uvib RT Cvib = 0 Cvib R 7.5.3 Mean rotational energy of a molecule The mean rotational energy of a gas molecule = total rotational energy of the gas/total number of gas molecules: ∑N E = ∑N i i Erot ,molecule i i i − hν ⎡1 e kT ⎤ ⎥ = hν ⎢ + − hν ⎢⎣ 2 1 − e kT ⎥⎦ (6.25) Internal energy due to the rotation movement: Urot = NA B/q Σ (2J+1)J(J+1) eBJ(J+1)/(kT) q= Σ(2J+1)e-BJ(J+1)/kT (6.27) (6.26) 7.5.4 Heat capacity term - rotation Molar heat capacity term at constant volume, due to the rotation of the molecules in the gas: Cv ,m ⎛ δU = ⎜⎜ m ,rot ⎝ δT ⎞ ⎟⎟ ⎠ Cv, rot (6.28) T To understand and learn - Wie ist die Molwärme bei konstantem Volumen definiert? - Was geben die Molwärmen an (in Worten)? - Was besagt der klassische Gleichverteilungssatz? - Wie gross ist klassisch die innere Energie des idealen Gases (ausgedrückt durch die Freiheitsgrade)? - Wie gross ist klassisch Cm,V des idealen Gases (ausgedrückt durch die Freiheitsgrade)? - -Wie gross ist Cm,V von Helium bei 300 K? - Wie gross ist Cm,V von HCl bei 300 K etwa? - Was bedeutet die Aussage, ein Vibrations-Freiheitsgrad sei aktiv bzw. inaktiv? - Welche Freiheitsgrade sind bei Zimmertemperatur typisch aktiv, inaktiv bzw. teilweise aktiv? - Wie lässt sich damit aussagen, wieweit ein Freiheitsgrad aktiv oder inaktiv ist? - Zeichnen Sie den Verlauf von Cm,V als Funktion der Temperatur für ein zweiatomiges Gas! - Skizzieren Sie das Vorgehen zur Berechnung der vibratorischen inneren Energie bei Quantelung! - Zeichnen Sie U, bzw. CV für eine Vibration klassisch bzw. bei Quantelung als Funktion von T! - Skizzieren Sie das Vorgehen zur Berechnung der rotatorischen inneren Energie bei Quantelung!