338
6 Molecular Systems
S(0) = 0. For water, S(0) has the value 3.39 J mol
−1 K −1 , while S(0) = 4.64, 4.77,
11.72 J mol
−1 K −1 , respectively, for CO, N 2 O, and CH 3 D.
The following analysis may be carried out for the entropy of water. Let us
examine the entropy as calculated from our formulae for water at a temperature
of 298.16 K and at a pressure of 1 bar. For the calculated entropy we obtain
S calc S trans
Sackur–Tetrode
144.80 J mol
−1 K −1
S rot
Classical
43.85 J mol
−1 K −1
S vib
ν i = 3652, 1592, 3756 cm −1
0.00 J mol
−1 K −1
S calc =
188.66 J mol
−1 K −1 ,
while Giauque and Stout [41] obtained the following calorimetric results:
S exptl
0–10 K
Debye
0.092 J mol
−1 K −1
10–273.16 K
C p ice
37.99 J mol
−1 K −1
273.16 K
melting
21.99 J mol
−1 K −1
273.16–298.16 K
C p water
6.61 J mol
−1 K −1
298.16 K
vaporization
149.36 J mol
−1 K −1
Compression to 1 bar
−6.79 J mol
−1 K −1
gas imperfections
0.01 J mol
−1 K −1
S expt =
185.13 J mol
−1 K −1 ,
with estimated experimental uncertainty ±0.08 J mol
−1 K −1 . The quantity
S calc (T ) − S expt (T ) ≥ 0 ,
is referred to as the ‘residual entropy’ of the substance. From the above results, we
see that water has a residual entropy S(0) = 3.39 J mol
−1 K −1 . In many cases the
residual entropy can be explained in terms of a partial randomness in the crystalline
form at very low temperatures. Another possibility is to consider the role of ‘frozen
in’ rotational states, especially for centrosymmetric molecules.
Let us first examine the case of the simplest centrosymmetric molecule, H 2 . For
H 2 there can be no partial randomness of the crystalline solid, and we are left with
the nuclear spin symmetry considerations associated with ortho and para hydrogen:
as you may recall, nH 2 is a 3:1 metastable (or nonequilibrium) mixture of oH 2
and pH 2 . At very low temperatures all pH 2 molecules lie in their ground j = 0
rotational states, while all oH 2 molecules lie in their ‘ground’ j = 1 rotational
states. We thus have an additional contribution to the spectroscopic entropy of
6 Molecular Systems
S(0) = 0. For water, S(0) has the value 3.39 J mol
−1 K −1 , while S(0) = 4.64, 4.77,
11.72 J mol
−1 K −1 , respectively, for CO, N 2 O, and CH 3 D.
The following analysis may be carried out for the entropy of water. Let us
examine the entropy as calculated from our formulae for water at a temperature
of 298.16 K and at a pressure of 1 bar. For the calculated entropy we obtain
S calc S trans
Sackur–Tetrode
144.80 J mol
−1 K −1
S rot
Classical
43.85 J mol
−1 K −1
S vib
ν i = 3652, 1592, 3756 cm −1
0.00 J mol
−1 K −1
S calc =
188.66 J mol
−1 K −1 ,
while Giauque and Stout [41] obtained the following calorimetric results:
S exptl
0–10 K
Debye
0.092 J mol
−1 K −1
10–273.16 K
C p ice
37.99 J mol
−1 K −1
273.16 K
melting
21.99 J mol
−1 K −1
273.16–298.16 K
C p water
6.61 J mol
−1 K −1
298.16 K
vaporization
149.36 J mol
−1 K −1
Compression to 1 bar
−6.79 J mol
−1 K −1
gas imperfections
0.01 J mol
−1 K −1
S expt =
185.13 J mol
−1 K −1 ,
with estimated experimental uncertainty ±0.08 J mol
−1 K −1 . The quantity
S calc (T ) − S expt (T ) ≥ 0 ,
is referred to as the ‘residual entropy’ of the substance. From the above results, we
see that water has a residual entropy S(0) = 3.39 J mol
−1 K −1 . In many cases the
residual entropy can be explained in terms of a partial randomness in the crystalline
form at very low temperatures. Another possibility is to consider the role of ‘frozen
in’ rotational states, especially for centrosymmetric molecules.
Let us first examine the case of the simplest centrosymmetric molecule, H 2 . For
H 2 there can be no partial randomness of the crystalline solid, and we are left with
the nuclear spin symmetry considerations associated with ortho and para hydrogen:
as you may recall, nH 2 is a 3:1 metastable (or nonequilibrium) mixture of oH 2
and pH 2 . At very low temperatures all pH 2 molecules lie in their ground j = 0
rotational states, while all oH 2 molecules lie in their ‘ground’ j = 1 rotational
states. We thus have an additional contribution to the spectroscopic entropy of