Section 3: Thermal Properties:

Topic 3.1 Thermodynamics: states and phases Topic 3.2 Critical temperature and latent heat Topic 3.3 Thermal expansion in solids

1 3. Thermal Properties > 3.1 Thermodynamics: states and phases Topic 3.1 Thermodynamic aspects of stability

• Solid: - definite volume and shape • Gas: - volume and shape dependent on the container • Liquid:- definite volume, shape dependent on the container This behaviour relates to: COMPRESSIBILITY (C) : response to an attempt to change the volume VISCOSITY (V) and RIGIDITY (R): response* to an attempt to change the shape (*N.B. the measurement time-scale is important) • Solid: - low C, infinite V, high R • Gas: - high C, low V, zero R • Liquid:- low C, intermediate V, low R These properties relate to the PACKING and ORDER of the atoms:- • Solid: close packing, long range order • Gas: dilute packing, no order • Liquid: close packing, short range order 2 3. Thermal Properties > 3.1 Thermodynamics: states and phases States of Matter • The State of a substance depends on the values of P, V and T (pressure, volume and thermodynamic (i.e. Kelvin) temperature). • For a particular amount (e.g. 1 mole) of substance, in a particular state, these quantities are linked by an EQUATION of STATE (e.g. PV = RT for ideal gas) • Possible values of P,V and T form a surface in PVT space. Different regions correspond to different STATES • Boundaries between these regions correspond to transitions between STATES

• The links between PVT and STATE are usually displayed on P-T or P-V PHASE DIAGRAMS.

3 3. Thermal Properties > 3.1 Thermodynamics: states and phases Experimental determination

• Fixed amount (1 mole) of substance in a closed cylinder, P pressure p and T can be varied. constant • Each pair of P,T values leads to temperature enclosure value(s) of V and STATE T

• Only some [P,V,T, state] combinations possible. system under investigation • SYSTEM must be at EQUILIBRIUM

4 3. Thermal Properties > 3.1 Thermodynamics: states and phases P-T Phase Diagram • The range of variables where the solid, liquid and gas phases exist are shown as areas. P • Notice diagram doesn’t show regions where phases coexist. SOLID Reason: while a mix of phases exists P is constant at constant T. States of mixed phases are lines not regions CP in the PT diagram LIQUID TP • Lines are boundaries and represent conditions where phase transitions take GAS place T • TP: Triple Point : Solid, liquid, gas coexist Brown: sublimation curve Blue: melting/fusion curve • CP: Critical Point : highest temperature at Red: vaporization/condensation curve which liquid can exist (also vapour pressure)

5 3. Thermal Properties > 3.1 Thermodynamics: states and phases P-T Phase Diagram (2) α β γ P Changes at constant P: isobars

SOLID Changes at constant T: isotherms CP

LIQUID TP

GAS

Examples: 3 isotherm line α, β and γ T α GAS -> SOLID (below triple point temperature, sublimation) β GAS -> LIQUID -> SOLID γ GAS -> SOLID (above critical temperature, no liquid phase)

6 3. Thermal Properties > 3.1 Thermodynamics: states and phases P-V Phase Diagram Separates regions of P and V where substance is in different STATES

In PV diagram: regions of single phase and regions of mixed phases

S+G Critical point curve (fixed T) SOLID CP GAS S+L L

L+G TP-line

SOLID+GAS Triple point line (fixed T)

7 3. Thermal Properties > 3.1 Thermodynamics: states and phases P-V Phase Diagram (2) T information can be included by ISOTHERMS

ISOTHERMS: α: compression of gas (ideal gas: P ∝ 1/V) ⇒ solidification (g+s) ⇒ compression of solid β: compression of gas ⇒ liquefaction/condensation (g+l) α ⇒ compression of liquid ⇒ solidification/fusion (l+s) ⇒ compression of solid β γ: compression of gas γ ⇒ solidification/sublimation (s+g) ⇒ compression of solid

Notice in all (phase transition) regions of mixed phase, P constant along isotherm. 8 3. Thermal Properties > 3.1 Thermodynamics: states and phases The PVT surface

If the physically allowed sets of PVT values are plotted as points along 3 axes => PVT surface with regions for each PHASE

PT and PV PHASE DIAGRAMS are projections of the 3D PVT surface onto the 2D PT and PV planes.

The surface is described by the Equation(s) of State of the material and describes all equilibrium states of the material.

9 3. Thermal Properties > 3.1 Thermodynamics: states and phases What determines the state of a system? (Microscopically)

The balance between the interatomic (intermolecular) potential energy* and the kinetic energy+ (thermal energy) of the atoms (molecules)

*depends on separation (i.e. P or V) + depends on temperature (T)

• interatomic p.e. dominant (low T or high P) => SOLID

• thermal energy dominant (high T or low P) => GAS

• both important (intermediate T & P) => LIQUID

10 3. Thermal Properties > 3.1 Thermodynamics: states and phases Phase transitions

• The distinctions between the three states outlined above may not always be clear cut. The transitions (phase transitions) may not always be sharp or well-defined.

• Certain materials or materials under certain conditions can exhibit intermediate properties.

IN PHYSICS: • LIQUIDS are sometimes associated with solids as CONDENSED MATTER (emphasising the close packing) • LIQUIDS are sometimes associated with GASES as FLUIDS (emphasizing the low viscosity)

11 3. Thermal Properties > 3.1 Thermodynamics: states and phases 3 Thermal Properties Topic 3.2 Critical temperature and latent heat We will look at

(i) critical temperature (the Collision cross-section temperature above which no V(r) liquefaction occurs) density

a0 r

-ε thermal expansion, compressibility Critical temperature (ii) latent heat for the van der Waals Latent heat solid Surface energy

(iii) latent heat for the ionic solid 12 3 Thermal Properties > 3.2 Critical temperature and latent heat Critical temperature (For the L-J 6-12 van der Waals potential)

12 6  a   a   V (r) = ε  0  − 2 0    r   r   • -ε is the potential energy between two molecules at their equilibrium separation ∴ +ε is the BINDING ENERGY of the pair

≡ energy needed to separate them (to ∞)

–23 • THERMAL ENERGY ≡ kBT (Boltzmann’s constant: kB = 1.38×10 J/deg)

when THERMAL ENERGY > BINDING ENERGY molecules do not stay together

∴ substance does not liquefy irrespective of pressure applied

13 3 Thermal Properties > 3.2 Critical temperature and latent heat Critical temperature (For the L-J 6-12 van der Waals potential)

∴ at CRITICAL TEMPERATURE kBTC ≅ ε ε ∴TC ≅ kB

For example: argon TC=151 K => ε = 2.1x10-21 J (measured ~2.2x10-21 J)

= 0.013 eV (1 eV = 1.6×10–19 J)

14 3 Thermal Properties > 3.2 Critical temperature and latent heat Latent Heat (van der Waals solid/liquid)

• The MOLAR LATENT HEAT OF SUBLIMATION (or

VAPORISATION) is the energy required to change one mole (NA molecules) of a substance from solid (or liquid) to a gas. [Note: SPECIFIC LATENT HEAT is for 1 kg]

To change separation from:

r = a0 r ≈ 10 a0 (value for solid or liquid) (value for a gas)

requires approximately an energy ε per pair of molecules.

15 3 Thermal Properties > 3.2 Critical temperature and latent heat Latent Heat (van der Waals solid/liquid) cont.

• In 1 mole there are 1/2 n NA pairs. – factor of 1/2 ensures pairs are not counted twice, – n is coordination number (lecture 2.1). For sublimation use n for solid ; For vaporisation use n for liquid 1 L = nN ε MOLAR LATENT HEAT 0 2 A

∴ using assumption kinetic energy is small compared to potential energy (i.e. low temperatures)

Heats of sublimation Calculated (kJ/mol) Measured (kJ/mol) V(r) E.g. He 0.33 0.08 Ne 1.7 1.3 Ar 5.9 7.3 N 4.7 5.4 -ε 2

16 3 Thermal Properties > 3.2 Critical temperature and latent heat Latent Heat: temperature variation • The derivation above ignores the kinetic energy of the atom/molecules and is valid only at low V(r) temperatures. • Experimentally latent heats vary strongly with temperature (as do density, compressibility etc.) -ε • e.g. Argon (the best L-J 6-12 atomic system) sublimation 8 2 solid

6 evaporation ρ L0 4 1 liquid 103Jmol-1 103kgm-3 2 melting 0 0 0 50 100 150 T/K 70 100 150 T/K (At higher T part of the increase in separation happens in between phase transitions (decompression). i.e. it isn’t counted in the latent heats) 17 3 Thermal Properties > 3.2 Critical temperature and latent heat Latent heat of ionic crystal m α e2  1  a  a  Recall the interatomic potential is: V(r) =   0  − 0  4πε 0a0 m  r  r  ε  1  α e2 and at r=a0, V(a0)= -ε, therefore = 1−   m  4πε 0a0

This is the binding energy per pair of ions in a crystal. ε ∴ binding energy per mole (NA pairs of ions) ≡ molar latent heat  1  α e2 L0 ()= N A = N A 1−   m  4πε 0a0

18 3 Thermal Properties > 3.2 Critical temperature and latent heat Latent heat of ionic crystals

• Example: crystalline NaCl 5 -1 Experimentally L0 = 7.63x10 J mol

-18 ∴ Binding energy per pair of ions ε = 1.27x10 J = 7.92 eV L0 = N Aε

Does this agree with our potential energy model? Using: ε a = 2.8x10-10 m, m ~ 10, α for NaCl = 1.75, 2 0  1  α e = 1−  ∴ ε = 1.29x10-18 J = 8.09 eV  m  4πε 0a0

5 -1 ∴ L0=7.79x10 J mol .

Excellent agreement! 19 3 Thermal Properties > 3.2 Critical temperature and latent heat 3 Thermal Properties Topic 3.3 Thermal expansion in solids

Collision cross-section We have already discussed: V(r) density (i) Density (ii) Critical temperature a 0 (iii) Latent heat r Next we discuss: Thermal expansion -ε thermal expansion, compressibility Critical temperature Latent heat Surface energy

20 3. Thermal properties > 3.3 Thermal expansion Thermal expansion in solids

MACROSCOPICALLY (HRW) ∆L Coefficient of linear expansion = α ∆T L β = 3α ∆V Coefficient of volume expansion = β ∆T V

MICROSCOPICALLY (QUALITATIVE)

Thermal expansion arises from increasing atomic vibration through increasing temperature in combination with the fact that the V(r) curve is not symmetric

(i.e. two atoms are more easily pushed apart than pushed together).

21 3. Thermal properties > 3.3 Thermal expansion Thermal expansion Consider two atoms only: ∴ increasing temperature ⇒ +ε increasing mean separation

V(r)

a0 If V(r) were symmetric 0 r (parabolic ≡ SHM) there would be no thermal Increasing expansion. T -ε T=0 ∴ anharmonicity ⇒ thermal expansion

22 3. Thermal properties > 3.3 Thermal expansion Thermal expansion cont. • MICROSCOPICALLY (QUANTITATIVE) How does V(r) change around the equilibrium position (r= a )? 0 ∆V

Putting ∆V = V(r)-V(a0) parabola and x = r- a0, (=SHM) ∆V can be approximated by

∆V = Bx2 + Cx3 0 x harmonic anharmonic

C(<0) and B are constants relating to the shape of the V(r) curve

23 3. Thermal properties > 3.3 Thermal expansion Thermal expansion cont. • If the total mean energy (k.e. + p.e.) of the vibrating system is E , then this equals the p.e. at the end-points of the vibration x1 and –x2.

2 3 2 3 ∴E = Bx1 + Cx1 = Bx2 − Cx2 2 2 3 3 ∴ B(x1 − x2 ) = −C(x1 + x2 ); 3 3 ∴ B(x1 − x2 )(x1 + x2 ) = −C(x1 + x2 ); x − x − C(x 3 + x 3 ) Mean position ≡ 1 2 = 1 2 2 2B()x1 + x2

Then for x1 ≈ x2 (i.e. small anharmonicity), 3 3 3 x1 + x2 ≈ 2x; x1 + x2 ≈ 2x , where x = amplitude of vibration x − x − 2Cx 3 C ∴ Mean position : 1 2 = = − x 2 2 4Bx 2B

This corresponds to the increased separation between two atoms 2 originally a0 apart (i.e. a0 → a0 – (C/2B)x , where C<0) 24 3. Thermal properties > 3.3 Thermal expansion Thermal expansion cont. • The coefficient of linear expansion relates to how x − x (i.e. ∆L/L) varies with T (at constant pressure) 1 2 a 2 0 C ∂ 2 ∴ α = − (x )P 2Ba0 ∂T Around the equilibrium separation the thermal energy of vibration k.e. = E (p.e. at minimum): k.e. =Bx2 +Cx3 ≈Bx2 (neglecting the small 2nd term)

∴ the molar heat capacity for NA atoms vibrating in 3D is given by ∂ C = 3N B (x2 ) P A ∂T P Thus α is directly related to the heat capacity C . C C P ∴ α = − P 2 This is known as the 6B a0 N A Gruneisen relation 25 3. Thermal properties > 3.3 Thermal expansion Thermal expansion (L-J solid) 12 6  a   a   • For L-J 6-12, V (r) = ε  0  − 2 0    r   r  

Using Taylor’s theorem, 36ε 252ε ⇒B = 2 & C = − 3 (see Flowers and Mendoza) a0 a0

(This is a good approximation up to x = 0.1 a0, ~melting point.) α 7 CP Putting in values of C and B = 216ε N A

26 3. Thermal properties > 3.3 Thermal expansion Example: Argon

For Ar, in the high temperature regime:

Molarα heat capacity: Cp ~ 3NAkB (= 3R)

7 ε CP 1 ∴ = ≈ k B 216 N A 10ε

For Ar: Triple point 3 ε =1.7x10-21J

∴ε/kB ~ 120 K

so α ~10-3 K-1: about right! Density, g/cm Lattice constant, 0.1nm

27 3. Thermal properties > 3.3 Thermal expansion Temperature, K Next Lecture

Problem session

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