The Third Law of Thermodynamics TL-1

Nernst suggested that the change in entropy for chemical reactions approached 0 as the temperature approached 0 K.

as Δr S → 0 T → 0

I think that the entropy of a pure substance approaches 0 at 0 K! Walther Nernst

Max Planck

The Third Law: Every substance has a finite positive entropy, but at 0 K the entropy may become 0, and does so in the case of a perfectly crystalline substance. Statistical Mechanics and the 3rd Law TL-2

The third law was formulated before the full development of quantum theory. However, statistical thermodynamics gives us molecular insight to the third law.

At 0 K, we expect that the system = lnWkS will be in its lowest energy state B and therefore W = 1, S = 0.

p = 1 and all other p ’s = 0. S = 0. −= ∑ jB ln ppkS j 0 j j 1st and 2nd Law vs 3rd Law TL-3

The 1st and 2nd Law of Thermodynamics introduced new state functions. The 3rd Law of Thermodynamics simply provides an absolute scale for entropy. Law 2 Law 1 δ rev = TdSq

= δ rev +δqwdU rev − PdV

= − PdVTdSdU 1st and 2nd Law

= + )( = + +VdPPdVdUPVUddH = +VdPTdSdH 1st and 2nd Law EX-TL1 Results of EX-TL1 TL-4

⎛ ∂S ⎞ CV ⎛ ∂S ⎞ C ⎜ ⎟ = ⎜ ⎟ = P ⎝ ∂T ⎠V T ⎝ ∂T ⎠P T

⎛ ∂S ⎞ 1 ⎡ ⎛ ∂U ⎞ ⎤ ⎛ ∂S ⎞ 1 ⎡⎛ ∂H ⎞ ⎤ ⎜ ⎟ ⎢P += ⎜ ⎟ ⎥ ⎜ ⎟ = ⎜ ⎟ −V ∂ TV ∂V ⎢ ⎥ ⎝ ⎠T ⎣ ⎝ ⎠T ⎦ ⎝ ∂ ⎠T TP ⎣⎝ ∂P ⎠T ⎦

We work at constant pressure most of the time… ⎛ ∂S ⎞ C ⎜ ⎟ = P Integrate with respect to T at constant P ⎝ ∂T ⎠P T T 2 P TC )( 2 TSTSS 1)()( =−=Δ dT ∫T1 T T 2 P TC )( If T1 = 0 K TSS 2 )( ==Δ dT ∫0 T Phase Transitions!! TL-5

T2 TC )( P If we know Cp(T) we can find ΔS TSS 2 )( ==Δ dT ∫0 T

What happens at phase transitions?

q rev Δtrs H trs S =Δ At constant P: S =Δ T trs trs T trs Practical Absolute Entropies TL-6

Table 21.1 Figure 21.1

N2

Values of entropies for gases given in the literature are standard entropies. These are by convention corrected for the non-ideality of gases (to be discussed in detail in Ch 22). Low T and Debye Theory TL-7

s 3 P )( → TTC as T → 0 Less than 15 K

3 12π 4 ⎛ T ⎞ Peter Debye ⎜ ⎟ 0 < ≤ TT CP = R⎜ ⎟ low 5 ⎝ ΘD ⎠

4 12π R T′ TC )( TS )( = 2dTT = P 3 ∫0 5ΘD 3 Partition Functions and 3rd Law TL-8

Remember: ⎛ ∂ lnQ ⎞ B ln += BTkQkS ⎜ ⎟ ⎝ ∂T ⎠ ,VN

− / Bj TkE ∑ jeE − / TkE 1 = ln ekS Bj + j B ∑ − / Bj TkE j T ∑e j

How does S behave as the temperature goes to 0?

Is this consistent with the 3rd Law of Thermodynamics?

See page 861-862 for proofs. 862 and 863 also discusses diatomic and polyatomics. Literature Values and Trends TL-9

Table 21.2 What are the trends for: Phase: Gas, liquid, solid Mass # of atoms

Tables are often a combination of statistical thermodynamics and

Table 21.3 calorimetric values. Entropy and Molecular Structure TL-10

Function of Mass

From what E term?

Acetone Trimethylene oxide O O

C3H 6 O

3CH CH3 298 J·K-1·mol-1 274 J·K-1·mol-1 Why? Sometimes there is not agreement TL-11 ≡ OC At 81.6 K: -1 -1 Scalc = 3.160 J·K ·mol calc > SS exp -1 -1 Sexp = 6.155 J·K ·mol

CO has a very small dipole moment residual calc −= SSS exp so the molecules do not have a strong tendency to line up in an N W = 2 = B lnWkS energetically favorable way. As a N result, in the crystal (i.e., low T form) residual kS B == NkB 2ln2ln gets “locked” into its own orientation and cannot find the state of lowest RS == 7.52ln J·K-1·mol-1 energy (i.e., where W = 1). -1 -1 calc res SSS exp =+= 3.161 J·K ·mol Entropy changes of chemical reactions TL-12

As used in Homework #3…

+ +→ zZyYbBaA

oo o o o r −−+=Δ BbSAaSZzSYySS ][][][][