Thermodynamics and Chemistry

CHAPTER 6 THE THIRD LAW AND CRYOGENICS

6.2 MOLARENTROPIES 155

Table 6.1 Standard molar entropies of several substances (ideal gases

atTD298:15K andpD 1 bar) and molar residual entropies

Sm=J K^1 mol^1

Substance calorimetric spectroscopica Sm,0=J K^1 mol^1

HCl 186:30:4b 186:901 0:60:4

CO 193:40:4c 197:650:04 4:30:4

NO 208:00:4d 210:758 2:80:4

N 2 O (NNO) 215:30:4e 219:957 4:70:4

H 2 O 185:40:2f 188:8340:042 3:40:2

aRef. [ 28 ]. bRef. [ 60 ]. cRef. [ 34 ]. dRef. [ 83 ]. eRef. [ 15 ]. fRef. [ 59 ].

When the spectroscopic method is used to evaluateSmwithpset equal to the standard
pressurepD 1 bar, the value is thestandardmolar entropy,Sm, of the substance in the
gas phase. This value is useful for thermodynamic calculations even if the substance is not
an ideal gas at the standard pressure, as will be discussed in Sec.7.9.

6.2.3 Residual entropy

Ideally, the molar entropy values obtained by the calorimetric (third-law) method for a gas
should agree closely with the values calculated from spectroscopic data. Table6.1shows
that for some substances this agreement is not present. The table lists values ofSm for
ideal gases at298:15K evaluated by both the calorimetric and spectroscopic methods. The
quantitySm,0in the last column is the difference between the twoSmvalues, and is called
the molarresidual entropy.
In the case of HCl, the experimental value of the residual entropy is comparable to its
uncertainty, indicating good agreement between the calorimetric and spectroscopic meth-
ods. This agreement is typical of most substances, particularly those like HCl whose
molecules are polar and asymmetric with a large energetic advantage of forming perfectly-
ordered crystals.
The other substances listed in Table6.1have residual entropies that are greater than
zero within the uncertainty of the data. What is the meaning of this discrepancy between
the calorimetric and spectroscopic results? We can assume that the true values ofSm at
298:15K are thespectroscopicvalues, because their calculation assumes the solid has only
one microstate at 0 K, with an entropy of zero, and takes into account all of the possible
accessible microstates of the ideal gas. Thecalorimetricvalues, on the other hand, are
based on Eq.6.2.2which assumes the solid becomes a perfectly-ordered crystal as the
temperature approaches 0 K.^4
The conventional explanation of a nonzero residual entropy is the presence of random
rotational orientations of molecules in the solid at the lowest temperature at which the heat

(^4) The calorimetric values in Table6.1were calculated as follows. Measurements of heat capacities and heats
of transition were used in Eq.6.2.2to find the third-law value ofSmfor the vapor at the boiling point of the
substance atpD 1 atm. This calculated value for the gas was corrected to that for the ideal gas atpD 1 bar
and adjusted toTD298:15K with spectroscopic data.