Absolute entropies and the Third Law of thermodynamics

Entropy

2.6 Absolute entropies and the Third Law of thermodynamics

To calculate entropy changes associated with biological processes, we need to see how to compile tables that list values of the entropies of substances.

The graphical procedure summarized by Fig. 2.5 and eqn 2.3 for the determination of the difference in entropy of a substance at two temperatures has a very impor- tant application. If T_i0, then the area under the graph between T0 and some temperatureTgives us the value of SS(T)S(0). However, at T0, all the motion of the atoms has been eliminated, and there is no thermal disorder. More- over, if the substance is perfectly crystalline, with every atom in a well-defined lo- cation, then there is no spatial disorder either. We can therefore suspect that at T0, the entropy is zero.

The thermodynamic evidence for this conclusion is as follows. Sulfur under- goes a phase transition from its rhombic form to its monoclinic polymorph at 96°C (369 K) and the enthalpy of transition is 402 J mol¹. The entropy of transition is therefore 1.09 J K¹mol¹at this temperature. We can also measure the mo- lar entropy of each phase relative to its value at T0 by determining the heat ca- pacity from T0 up to the transition temperature (Fig. 2.7). At this stage, we do not know the values of the entropies at T0. However, as we see from the illus- tration, to match the observed entropy of transition at 369 K, the molar entropies of the two crystalline forms must be the same at T0. We cannot say that the entropies are zero at T0, but from the experimental data we do know that they are the same. This observation is generalized into the Third Law of thermodynamics:

The entropies of all perfectly crystalline substances are the same at T0.

For convenience (and in accord with our understanding of entropy as a measure of dispersal of energy), we take this common value to be zero. Then, with this con- vention, according to the Third Law,

S(0)0for all perfectly ordered crystalline materials.

TheThird-Law entropyat any temperature, S(T), is equal to the area under the graph of C/TbetweenT0 and the temperature T(Fig. 2.8). If there are any phase transitions (for example, melting) in the temperature range of interest, then the entropy of each transition at the transition temperature is calculated like that in eqn 2.4 and its contribution added to the contributions from each of the phases, as shown in Fig. 2.9. The Third-Law entropy, which is commonly called simply

“the entropy,” of a substance depends on the pressure; we therefore select a stan- dard pressure (1 bar) and report the standard molar entropy,S_m^両, the molar en- tropy of a substance in its standard state at the temperature of interest. Some val- ues at 298.15 K (the conventional temperature for reporting data) are given in Table 2.2.

It is worth spending a moment to look at the values in Table 2.2 to see that they are consistent with our understanding of entropy. All standard molar entropies

Temperature, T/ K Temperature, T/ K Molar entropy, Sm/(J K−1 mol−1)

369 0

Monoclinic

Rhombic

?

?

?

(a)

Molar entropy, Sm/(J K−1 mol−1)

369 0

(b)

+1.09 J K⁻¹ mol⁻¹

Fig. 2.7 (a) The molar entropies of monoclinic and rhombic sulfur vary with temperature as shown here. At this stage we do not know their values at T0. (b) When we slide the two curves together by matching their separation to the measured entropy of transition at the transition temperature, we find that the entropies of the two forms are the same at T0.

Temperature, T Temperature,T

Entropy, S

C/CCT

Area

Fig. 2.8 The absolute entropy (or Third-Law entropy) of a substance is calculated by extending the measurement of heat capacities down to T0 (or as close to that value as possible) and then determining the area of the graph of C/TagainstT up to the temperature of interest. The area is equal to the absolute entropy at the temperatureT.

(a)

(b) S(0)0

Temperature, T Entropy, SCp/T

T_f T_b T

T_f T_b T

Solid Gas

Liquid

Melt Boil

ΔvapS ΔfusS

Fig. 2.9 The determination of entropy from heat capacity data. (a) Variation of Cp/T with the temperature of the sample. (b) The entropy, which is equal to the area beneath the upper curve up to the temperature of interest plus the entropy of each phase transition between T0 and the temperature of interest.

are positive, because raising the temperature of a sample above T0 invariably increases its entropy above the value S(0)0. Another feature is that the stan- dard molar entropy of diamond (2.4 J K¹mol¹) is lower than that of graphite (5.7 J K¹mol¹). This difference is consistent with the atoms being linked less rigidly in graphite than in diamond and their thermal motion being correspond- ingly greater. The standard molar entropies of ice, water, and water vapor at 25°C are, respectively, 45, 70, and 189 J K¹mol¹, and the increase in values corre- sponds to the increasing dispersal of matter and energy on going from a solid to a liquid and then to a gas.

Heat capacities can be measured only with great difficulty at very low tem- peratures, particularly close to T0. However, it has been found that many non- metallic substances have a heat capacity that obeys the DebyeT³-law:

At temperatures close to T0,C_V,maT³ (2.9a)

whereais a constant that depends on the substance and is found by fitting this equation to a series of measurements of the heat capacity close to T0. With a determined, it is easy to deduce the molar entropy at low temperatures, because

At temperatures close to T0,S_m(T)¹⁄3C_V,m(T) (2.9b) That is, the molar entropy at the low temperature T is equal to one-third of the constant-volume heat capacity at that temperature.

Table 2.2

Standard molar entropies of some substances at 298.15 K*

Substance Sm^両/(J K¹mol¹) Gases

Ammonia, NH3 192.5

Carbon dioxide, CO2 213.7

Hydrogen, H₂ 130.7

Nitrogen, N2 191.6

Oxygen, O2 205.1

Water vapor, H2O 188.8

Liquids

Acetic acid, CH3COOH 159.8 Ethanol, CH3CH2OH 160.7

Water, H2O 69.9

Solids

Calcium carbonate, CaCO3 92.9

Diamond, C 2.4

Glycine, CH2(NH2)COOH 103.5

Graphite, C 5.7

Sodium chloride, NaCl 72.1 Sucrose, C12H22O11 360.2

Urea, CO(NH2)2 104.60

*See the Data section for more values.

COMMENT 2.1 The text’s web site contains links to online databases of thermochemical data, including tabulations of standard molar entropies. ■

DERIVATION 2.2 Entropies close to T0

Once again, we use the general expression for the entropy change accompany- ing a change of temperature deduced in Section 2.3, with S interpreted as S(T_f)S(T_i), taking molar values, and supposing that the heating takes place at constant volume:

S_m(T_f)S_m(T_i)

冕

_T^T_i^{f dT}

If we set T_i0 and T_fsome general temperature T, we can rearrange this ex- pression into

S_m(T)S_m(0)

冕

₀^{T dT}

According to the Third Law, S(0)0, and according to the Debye T³-law, C_V,maT³, so

S_m(T)

冕

₀^{T dTa}

冕

₀^TT2dT

At this point we can use the standard integral

冕

^{x2dx1⁄3x3constant}

to write

冕

₀^TT2dT

_冢

^{1⁄3T3constant}

_冣兩

^T0

冢

^{1⁄3T3constant}

冣

^constant

¹⁄3T³ We can conclude that

S_m(T)¹⁄3aT³¹⁄3C_V,m(T) as in eqn 2.9b.