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Home , Equipartition theorem, Molar heat capacity, Monatomic gas

Chapter 06 Equipartition of

P. J. Grandinetti

Chem. 4300

P. J. Grandinetti Chapter 06: Equipartition of Energy Energetically accessible degrees of freedom

In kinetic theory of we consider translational motion of monatomic and learned that is a measure of average translational , 2 3 pV = n N 휖 = nRT or 휖 = k T 3 A k k 2 B What about a polyatomic gas where there is translational, rotational, and vibrational motion? Definition Equipartition of energy theorem says that energy is distributed, on average, equally among all energetically accessible degrees of freedom, such as those associated with molecular translations, molecular , bond vibrations, and electronic motion.

More specifically, it says there will be an average of 1 k T associated with each 2 B coordinate that shows up squared in the total energy expression.

P. J. Grandinetti Chapter 06: Equipartition of Energy Monoatomic gas Simplest model of monoatomic gas only considers translational motion. Translational Energy For each particle () the energy is

2 2 2 p py p E = x + + z trans 2m 2m 2m

3 independent coordinates, px, py, and pz, appear squared. According to equipartition of energy theorem the average thermal energy of a monoatomic gas atom is ( ) 1 3 휖 = 3 k T = k T trans 2 B 2 B or internal molar energy of 3 U = RT monoatomic gas m 2

P. J. Grandinetti Chapter 06: Equipartition of Energy Diatomic gas

For diatomic gas total energy expression will include same contribution for translational motion as monoatomic gas except coordinates are associated with center of mass of molecule. In addition to translation, energy expression for diatomic molecules will include additional contributions from internal motions such as rotational and vibrational motion.

Translational Rotational Vibrational

P. J. Grandinetti Chapter 06: Equipartition of Energy Diatomic gas molecules Rotational Energy For a rotational energy is 1 1 E = I 휔2 + I 휔2 rot 2 b b 2 c c 휔 휔 Ib and Ic are principal moments of inertia and b and c are components of angular vector. Recall: diatomic molecule is linear so Ia = 0.

Vibrational Energy For a diatomic molecule vibrational energy (modeled as ) is

2 p 1 E = r + 휅 (r − r )2 vib 2m 2 f e ⏟⏟⏟r ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ Kinetic Potential

P. J. Grandinetti Chapter 06: Equipartition of Energy Diatomic gas molecules Applying the equipartition of energy theorem for a diatomic we count

2, 2, 2 ← 3 for px py pz translational 휔2, 휔2 ← +2 for b c rotational 2, 2 ← +2 for pr r vibrational 7 ← total

Average thermal energy of gas atom is ( ) 1 7 휖 = 7 k T = k T 2 B 2 B For diatomic gas this predicts internal molar energy of

7 U = RT diatomic gas m 2

P. J. Grandinetti Chapter 06: Equipartition of Energy Linear polyatomic gas molecules For a polyatomic gas molecule containing X we will consider two cases: linear and non-linear molecules. In the case of linear molecules we count 2, 2, 2 ← 3 for px py pz translational 휔2, 휔2 ← +2 for b c rotational +2(3X − 5) ← vibrational 6X − 5 ← total Average thermal energy of linear polyatomic gas molecule is ( ) 1 휖 = (6X − 5) k T 2 B For linear polyatomic gas molecule this predicts internal molar energy of ( ) 1 U = (6X − 5) RT , linear polyatomic gas m 2

P. J. Grandinetti Chapter 06: Equipartition of Energy Non-linear polyatomic gas molecules In the case of non-linear molecule containing X atoms we count

2, 2, 2 ← 3 for px py pz translational 휔2, 휔2, 휔2 ← +3 for a b c rotational +2(3X − 6) ← vibrational 6X − 6 ← total

Average thermal energy of non-linear polyatomic gas molecule is ( ) 1 휖 = (6X − 6) k T 2 B For non-linear polyatomic gas molecule this predicts internal molar energy of

, Um = 3(X − 1)RT non-linear polyatomic gas

P. J. Grandinetti Chapter 06: Equipartition of Energy capacity of a gas

P. J. Grandinetti Chapter 06: Equipartition of Energy of a gas

Molar heat capacity at constant is given by ( ) 휕 Um CV,m = 휕 T V

How does prediction do predicting heat capacities?

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a monoatomic gas 3 For monoatomic gas equipartition theorem predicts Um = RT. ( ) 2 휕U 3 m . J/(mol⋅K) CV,m = 휕 = R = 12 47 T V 2

and experimental heat capacities for a few monoatomic gases are ...

⋅ Gas CV,m (measured) / J/(mol K) He 12.47 Ne 12.47 Ar 12.47

Impressive agreement!!

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a diatomic gas 7 For diatomic gas the equipartition theorem predicts U = RT m 2 ( ) 휕U 7 m . J/(mol⋅K) CV,m = 휕 = R = 29 10 T V 2

Experimental heat capacities for a few diatomic gases are ...

⋅ Gas CV,m (measured) / J/(mol K) H2 20.50 N2 20.50 O2 21.50

Measured heat capacity is consistently lower than prediction by value close to R ≈ 8.31446 J/(mol⋅K). What went wrong?

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a gas Equipartition theorem assumes thermal energy is distributed equally among all types of motion.

Thermal energy E distributed equally

translation vibration It did not escape attention of Boltzmann, Maxwell, and others that equipartition theorem could give right answer if vibrational or rotation contributions to heat capacity were dropped. Boltzmann thought somehow energy couldn’t transfer into molecular vibrations, but could not find suitable explanation.

3 >0 5 U = RT + RT +(6 − 5)RT = RT = 20.79 J/(mol⋅K) m 2 ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ 2 ⏟⏟⏟  rot vib trans  ⋅ Gas CV,m (measured) / J/(mol K) H2 20.50 N2 20.50 O2 21.50

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a gas Quantum Effects

In molecules can only be found in discrete energy states. < < Spacing between energy levels depends on type of motion: ΔEtrans ΔErot ΔEvib

Spacings between vibrational energy levels, ΔEvib, exceeds kBT at room temperature.

Thermal energy E distributed equally

translation rotation vibration

Thermal energy at room temperature is not high enough to excite molecular vibrations. With no ability to excite diatomic or polyatomic molecule vibrations, gas has lower heat capacity than predicted—greater temperature increase for fixed amount of heat.

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a gas Quantum Effects At even lower we can lose rotational motion contribution to heat capacity. Quantum effects to step-like heat capacity for gases as a function of temperature.

heat capacity of gas, H2

4 7/2 3 Vibrational 5/2 2 Rotational 3/2 1 Translational 1/2 0 20 50 100 200 500 1000 2000 5000 10000

(note the logarithmic temperature scale)

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a solid We can use equipartition of energy theorem to predict heat capacity of . Unlike situation in gas phase, in solid phase we expect no translational motion. Let’s also assume no rotational motion—may not be good assumption for molecular solids. Leaves only vibrational motion as the primary sink for thermal energy.

Each atom in solid has vibrational energy motion in 3 dimensions giving 3 kinetic energy contributions and 3 contributions—each of which involve a squared coordinate. Equipartition of energy predicts average thermal energy per atom: ( ) 1 휖 = 6 k T 2 B

For 1 of atoms in solid this predicts :

Um = 3RT P. J. Grandinetti Chapter 06: Equipartition of Energy Dulong and Petit rule for the( heat) capacity of a solid 휕U Given we obtain m J/(mol⋅K), Dulong and Petit rule. Um = 3RT CV,m = 휕 = 3R ≈ 25 T V

Quantum effects cause the heat capacities of solids to decrease with decreasing temperature.

Why do different materials reach Dulong-Petit limit at different temperatures?

P. J. Grandinetti Chapter 06: Equipartition of Energy Heat capacity of a solid ⋅ ⋅ Name Cp/J/(mol K) Name Cp/J/(mol K) Aluminum 24.20 25.86 25.23 29.600 24.64 27.20 28.07 25.48 16.443 24.98 25.52 31.060 26.020 24.06 25.929 29.54 (graphite) 8.517 25.52 26.94 25.363 Cesium 32.210 19.99 23.35 25.350 ( ) 24.81 28.230 휕 24.440 26.79 Um 26.03 (rhombic) 22.70 J/(mol⋅K) 23.222 25.36 CV,m = 휕 = 3R ≈ 25 T V 25.418 25.73 25.10 28.91 25.10 26.32 Lead 26.84 27.32 24.860 27.03 24.869 (white) 26.99 26.32 25.060 27.983 24.27 24.06 27.665 27.45 24.89 26.07 26.74 24.60 26.53 25.98 25.390 25.36

P. J. Grandinetti Chapter 06: Equipartition of Energy