Heat and Related Properties:

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1 Heat and Related roperties: Enthalpy & Heat apacities 1 rof. Zvi. Koren

2 Heat and Related roperties Enthalpy H + (a convenient definition) Greek: en + thalpein = to heat in Sometimes H is referred to as the heat content of the system I don t like it!!! Significance of H is not so much in H itself, but in H H = f(state functions) H is a state function: 2 dh H 2 H1 H and dh 1 Significance of dh and of H: [p ex = in ] dh = d + d( in ) = d + p ex d = đq + đw + p ex d = đq Note: đw = 0 dh = đq d = đq H = q = q he change in enthalpy (or internal energy) is the energy transferred as heat at constant (or ). -p ex d 0 (for reversible -work only) (for -work only) 2 rof. Zvi. Koren

3 p ex in q [] dh = đq H = q d = đq = q q = Heating at []:, const. atmosphere p ex in q solid, liquid (reversible -work) q vs. q (which is greater?) q = Heating at []:, const. q (work) Difference between q and q is the difference between H and : H + dh = d + d() H = + d() đq = đq + d() q = q + d() d() = () 2 () 1 = (). For an ideal gas: () = nr 3 For a solid, small, constant: () 0.[ rof. Zvi. Koren

4 )קיבול חום( Heat apacities dq d Amount of heat needed to raise a certain amount of substance by 1 0 For example: s specific molar heat capacity, heat capacity, For H 2 O(l): s 1 cal/g deg cal or J g deg cal or J mol deg Is a state function? 4 rof. Zvi. Koren

5 & dq d dq d dq = dh (rev -work) dq = d (-work) dh d d d H Which is larger for a particular substance, or? What about for solids? is directly dependent on the expansion ability of the substance: oefficient of thermal expansion hermal expansivity ubic expansion coefficient α 1 For an ideal gas, = 1/ (Do it!) H (rev -work only) (-work only) (Note each factor) 5 rof. Zvi. Koren

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8 For any ure Substance For an Ideal gas: d d H dh d For an Ideal gas: dh d( ) d d d d() d d R nr 8 rof. Zvi. Koren

9 as a function of for Some Gases - Graph Alberty R A, hysical hemistry, 7 th ed., Wiley, NY, 1987, p. 45 Notes: he more complex the molecule, 1. the greater its, and 2. the greater the increase with rising. 9 rof. Zvi. Koren

10 as a function of - able as a function of emperatur e from 300 to 1500 K : a b c 2 Molecule (g) a / JK -1 mol b / JK -2 mol c / JK -3 mol -1 H O l N Hl H 2 O O H H (many other tables for solids, liquids, gases) Note: In some tables, = a + b + c rof. Zvi. Koren

11 0 of SO 2 vs. /K (Alberty, p. 98) Log (/K) S 11 rof. Zvi. Koren

12 S 12 rof. Zvi. Koren

13 Molecular Interpretation of for an Ideal Gas Molecular Energies Associated With Atomic (Nuclear) Motions Each atom in a molecule can move independently in 3 basic directions, i.e., each movement can be described by a vector: # of degrees of freedom for the movement of each atom in a molecule = 3 For N atoms in a molecule: otal # of degrees of freedom = 3N But this seemingly chaotic atomic motion that is executed simultaneously by all of the atoms can be dissected into a set of ordered motions: ranslation, Rotation, and ibration. ib: E 2 = 1½h N A J/mol r Rot Heat (thermal energy) mainly influences translational, rotational, and vibrational motions and not the high-lying electronic energy levels (1 e = 96,500 J/mol). E thermal k (per atom: KM). R2500 J/mol,298 K E vib, J/mol (4000 cm -1 ) ib: E 1 = ½h N A J/mol, wavenumber cm 13 rof. Zvi. Koren

14 otal # of Degrees of Freedom for a molecule with N atoms = ranslation: Molecule moves as if frozen ; movement of the center of mass as a vector. # of independent translational motions (along 3 axes) = 3 Rotation: Whole molecule twists as a frozen block about axes of symmetry, without moving the center of mass. For non-linear molecules, # of independent rotations (about 3 axes) = 3 For linear molecules, # of independent rotations (about 2 axes only) = 2 (No energy is needed to spin a linear molecule about the internuclear axis, because the atoms (as point masses) do not really move; no movement, no moment of inertia, I.) ibrations (the remaining motions): Each atom can twist & shout : Bonds are like springs and bond angles are flexible; bonds can stretch and contract, bonds can bend, etc., without moving the center of mass, and without rotating the molecule. So, here s what s left of the total 3N motions: For non-linear molecules, # of independent motions = 3N 6 3N For linear molecules, # of independent motions = 3N 5 Examples for H 2 O & O 2 (continued) 14 rof. Zvi. Koren

15 Energies of Independent Motions for an Ideal Gas: rinciple of Equipartition of Energy Recall the results of the Kinetic Molecular heory for an Ideal Gas: K.E. of translation = 3/2 k or 3/2 R = 3(½)R Axiom 1: Energy of each independent translational direction = ½R. (he total translational energy has been equally partitioned into 3 parts.) Axiom 2: Energy of any independent motion (rotation or vibration) = ½R. Note: otential Energy of translation or rotation = 0 Intramolecular.E. associated with every vibrational mode 0 Recall: he bond is a spring and a restoring force acts on it as it is stretching: Hooke s Law: F = kx, k force (or spring) constant F = d/dr, =.E. Motion ranslation Rotation ibration Molecule (any) Non-linear Linear Non-linear Linear For an ideal gas: KE 3 (½R) 3 (½R) 2 (½R) (3N 6) (½R) (3N 5) (½R) d/d & E 0 0 (3N 6) (½R) (3N 5) (½R) R (total) 3 (½R) 3 (½R) 2 (½R) 2 (3N 6) (½R) 2 (3N 5) (½R) (cont d) 15 rof. Zvi. Koren

16 heoretical (lassical) vs. Experimental Heat apacities translationa heoretical (assuming ideal gas) Experimental Molecule rotational vibrational total l /R /R /R d/d R /R 298 K 3000 K He H O I O H 2 O NH H Heat mainly influences translational, rotational, and vibrational motions and not the high-lying electronic energy levels. 2. he classically calculated assumes that all molecular motions contribute equally at all temperatures. 3. Experimental s: At low, vibrational motions are negligible or small: exp < class. At high, vibrations are important: exp class. 4. As, real gas ideal gas 16 rof. Zvi. Koren / R /R /R

17 /3R Heat apacities of Atomic Solids At Room and Higher emps: Law of Dulong et etit (1819) For metallic elements, 3R = 6 cal/mol deg (actually 6 0.3) onsider an atom bound in an atomic crystal: Each atom oscillates (with ) about equilibrium position in 3 independent directions (no translations, no rotations): vib = KE + E = 3(½R) + 3(½R) = 3R (for a mole) = d/d = 3R At Low emps: Debye heory (1912): 3 Alberty R A, hysical hemistry, 7 th ed., Wiley, NY, 1987, p. 49 / D Heat capacities of all atomic solids should lie on the same curve when is plotted versus / D : D Debye temperature h/k B, frequency of oscillation f(m,k), k force constant At low ( 15 K): = a(/ D ) rof. Zvi. Koren

18 If alculations: =q from & H=q from d dq d dq In general : n d constant f() a b c q q n n 2 d or m (-work only) (or something similar), [, ] dh dq d dq H nd H q n d (rev -work only) If In general : constant f() H a' b' q n c' 2 m, 18 rof. Zvi. Koren or (or something similar) [, ]

19 Degrees of Freedom or Number of Independent ariables Every substance has an Equation of State, which relates the State ariables (,,,n) to each other: otal # of degrees of freedom = 3 But, for a given quantity n, # of degrees of freedom = 2 hermodynamic roperties are State Functions that are expressed in terms of the independent state variables. For example: = (,) or = (,) or = (,) H = H(,) or H = H(,) or H = H(,) (continued) 19 rof. Zvi. Koren

20 = (,): H = H(,): d dh H d d internal pressure coefficient π d H d π d d π d H For an ideal gas : = () only! (Recall the Kinetic Molecular heory) d dh d H = H() only! roof: H + = () + nr = H() only! H For an isothermal process involving an ideal gas: H = = 0 n n d d 0 0 For any pure [] For any pure [] First Law roblems: heat capacity: 9, 11, 12, 66. & d dh (note the units) n n d d For an ideal gas 20 rof. Zvi. Koren

21 internal pressurecoefficient π 21 (continued) rof. Zvi. Koren

22 internal pressurecoefficient π is a measure of the intermolecular forces < 0 ( slope) > 0 (+ slope) r (for stable systems) 22 rof. Zvi. Koren

23 Mathematical Intermezzo 1: Relating Similar artial Derivatives We often encounter the following types of D s: How do we relate one to the other? For example, how do we relate the first in terms of the second? he rick is to start with the 2 nd and create the 1 st : 1. Begin with the total differential equation that includes the second D: = (,) d d d and 2. Next, REAE the first D from this eqn by dividing by d, and impose the condition of constant on each new D: 23 rof. Zvi. Koren

24 H similar partial derivatives Relationship Between and H=+ From before, for similar D s: work produced per unit increase in emp. energy per unit emp required to separate the molecules against intermolecular attraction (continued) 24 rof. Zvi. Koren

25 + + + α( 0 coefficient of thermal expansion For an ideal gas: (for stable sys.; see before) internal pressure coefficient ) = nr Work done by a mole of ideal gas in pushing back a piston when []: w = = = R = R(1 0 ) = R (for any pure substance) 1 α R extra energy required to heat a mole of ideal gas by 1 0 at const over that required to heat it at const For a solid : (continued) 25 rof. Zvi. Koren

π α( ) (for any pure substance) (proved later, from S) (proved later, from α/κ Euler s ycle Rule) Isothermal compressibility coefficient of thermal expansion alculate for van der Waals gas and for

26 π α( ) (for any pure substance) (proved later, from S) (proved later, from α/κ Euler s ycle Rule) Isothermal compressibility coefficient of thermal expansion alculate for van der Waals gas and for another real gas. α 2 /κ or ( kappa ) engineering 1 α 1 alculate and for some real gases and compare with an ideal gas. hus, especially for liquids, it s not enough to look just at, but we need to look at 2 /. Examples appear on next slide. 26 (continued) rof. Zvi. Koren Nike Nike

27 Substance Water Benzene Lead Diamond coefficient of thermal expansion isothermal compressibility 10 4 / K α /κ α / atm Examples: For water For at 25 0 : benzene at J/K mol. alculate its : 134 J/K mol. alculate its (Note: d = g/ml).. Answer : 75.3 Answer :89 J/K mol J/K mol 27 First Law roblems: heat capacity: rof. Zvi. Koren

Thermodynamics & Statistical Mechanics

Thermodynamics & Statistical Mechanics hysics GRE: hermodynamics & Statistical Mechanics G. J. Loges University of Rochester Dept. of hysics & Astronomy xkcd.com/66/ c Gregory Loges, 206 Contents Ensembles 2 Laws of hermodynamics 3 hermodynamic