Measurement Techniques
4.2. GASES 1. Introduction
-.:::-~ 40 °60 80 100 a..
20 solid
o
L -________ ~ ______ ~~ T __________ ~~o
50 100 150T (K)
Fig. 4.1. Phase diagram of argon. T is the triple point. C is the critical point. Inset shows region near T with pressure scale enlarged.
4.4, including the unique properties of the helium isotopes.
4.2. GASES 4.2.1. Introduction
Under typical working conditions, the thermodynamic properties of most gases approximate quite closely to those of a classical ideal gas, and the ideal equation of state [Eq. (4.1)] is then adequate for most engineering purposes. However, the heat
Table 4.1. Boiling points (T b), triple points (Tt ), and critical points (Tc ), for some liquids of cryogenic interest; density is at
boiling point under 100 kPa (1 bar) [Jac97, Qui90J
Liquid Tb Tc Pc T, P, p
(K) (K) (kPa) (K) (kPa) (g/cm3)
3He 3.19 3.32 115 0.059
4He 4.207 5.1953 227.5 2.1768 (A) 4.8565(A) 0.125 n-H2 20.345 33.19 1315 13.95 7.20 0.0707 p-H2 20.233 32.94 1284 13.8033 7.034 0.0708 Ne 27.061 44.492 2679 24.5561 43.4 1.208 N2 77.237 126.19 3398 63.1504 12.52 0.807 Ar 87.17 150.663 4860 83.8058 68.89 1.397 Ch 90.062 154.58 5043 54.3584 0.1463 1.141
CCh 194.6 304.14 7375 216.589 518 1.18
Fluids 131
capacity of an ideal gas depends on its molecular composition. In most cryogenic gases electronic and vibrational degrees of freedom are not excited, while rotational contributions to Cv have reached their full classical value. Under these conditions, the heat capacity depends only on whether the gas is composed of isolated atoms (e.g., Ar), linear molecules (e.g., N2, C(h), or non-linear molecules (e.g., Ca.).
The values of Cv for these three groups are then respectively ~R, ~R, and 3R, or numerically 12.41,20.79, and 24.94 J·mol-1K-l; the corresponding values of Cp are 20.79, 29.10, and 33.25 J·mol-1K-1. The data for Cv in Table 4.2 for some real gases at atmospheric pressure shows that these values are a good approximation over wide ranges of temperature, although there are large deviations for hydrogen below about 200 K (Section 4.2.3), and for methane at higher temperatures due to vibrational excitation. Small deviations also occur as the temperature is lowered towards the boiling point, due to intermolecular interaction.
We treat first the ideal monatomic gas; next the ideal molecular gas, and the rotational quantum effects seen in hydrogen; and then departures from ideal behavior as the pressure is increased.
Classical Ideal Monatomic Gases. The ideal gas limit has already been dis-cussed briefly in Sections 1.3.5 and 2.5.1. The equation of state is the same for all gases, giving
PV=nRT, BT=P, (3 = liT, Cp-Cv =nR (4.1) where R is the gas constant (8.314 J·mol-1·K- 1) and n is the number oflIloles of the gas. The ratio of heat capacities (traditionally called 'Y but here denoted by 'Y*) is simply related to the Griineisen function; by Eq. (2.8)
.. Cp Bs
'Y
== -
= - = 1+
(3'YT = 1+
'YCv BT (4.2)
For a monatomic gas the partition function depends only on translational degrees of freedom, and is given by Eq. (2.60) when the potential energy function q, is put to zero:
_ _ VN (2'1rmkT)3N/2
Z - Ztrans - N! h2 (4.3)
From this the thermodynamic properties follow. In particular
3 3
Cv =
2
Nk= 2
nR (4.4)and the entropy at pressure P can be evaluated from the Sackur-Tetrode formula:
5 2'1rm :z s 5
{ [ 3
1 }
S=Strans =nR 2InT-InP+ln
(y)
k:Z +2 (4.5)132 Chapter 4
For mixtures of gases the partition function becomes
(4.6) The equation of state and the heat capacity are unchanged, but the entropy changes both because of the distribution of masses and because there is an additional entropy of mixing
Smix
=
Nk( -Xa InxA -Xb InxB _ ... )=
R( -na lnxa - nb lnxb _ ... ) (4.7) where na ... and Xa ... are respectively the number of moles and the atomic fractions of the component gases.4.2.2. Ideal Molecular Gases
The partition function. In the ideal limit the translational kinetic energy of the center of mass of a molecule does not interact with its remaining degrees of freedom, and the partition function is given by
(4.8) where Ztrans is unaltered except that mA is now the mass of molecule A, and ZA,int is the internal partition function obtained by summing over all internal energy states of the molecule, which are independent of volume. The Helmholtz energy is therefore of the form
F
=
Ftrans(T, V)+
Fint (T) (4.9)The second term does not affect the equation of state, and so the bulk modulus, thermal expansion coefficient and Cp - Cv are the same as for a monatomic gas (Eq. 4.1); but it does contribute additive terms to the entropy and heat capacity.
Calculation oj absolute entropiesJrom spectroscopic data. The additive terms in
F int are sums over all the non-translational energy states of an isolated molecule. For many simple gases the energies of such states have been the subject of precise study by spectroscopists, thus enabling the internal contribution to the entropy and other thermodynamic properties to be calculated. When this is added to the translational entropy given by Eq. (4.5), the absolute entropy of a dilute gas at a given temperature is obtained to a good accuracy purely from theory and spectroscopic data. Calorimetric data (latent heats and heat capacities) can then be used to find the difference in entropy between this dilute gas phase and other phases over the experimental temperature range. In this way absolute entropies of condensed phases are obtained, including the residual entropies of disordered material as T --t 0 (e.g., [Mo062, pp. 623-624];
see also Section 2.2.5).
Fluids 133
Rotational contributions. Usually the internal degrees of freedom are themselves separable to a good approximation. Most molecules are in their electronic ground state at cryogenic temperatures, and sufficiently rigid for the interaction between the rotational and vibrational motions to be small and for the vibrational frequencies to be too high for appreciable excitation at cryogenic temperatures. A non-linear rigid molecule has three rotational degrees of freedom, since two parameters are needed to give the direction of some axis within the molecule, and a third for rotation about that axis; and a linear molecule clearly has only two. Like the three translational degrees of freedom, the kinetic energy for each rotational degree of freedom contributes
!k
to the classical expression for Cv, thus giving the values of ~Nk (linear) and 3Nk quoted above. Apart from some molecules with low-lying vibrational (e.g., Ch) or electronic (e.g., NO) states, which start to become excited below room temperature, the only exceptions are H2 and its isotopic modifications D2 and HD. These are important cryogenic fluids whose rotational states we shall now discuss.
4.2.3. Ortho- and Para-Hydrogen
The rotational energy levels of a diatomic molecule are labelled by quantum number J, and given by
EJ = J(J
+
1)(1i2/21) = J(J+
I)kerot , (J=0,1,2,···) (4.10) where I is the moment of inertia about an axis through the center of mass perpen-dicular to the molecular axis. The spacing between the lower levels is thus inversely proportional to I, and of all molecules only hydrogen has a smaIl enough moment of inertia for the discrete nature of the levels to be reflected in the heat capacity when T>
Tt • For other molecular gases typical values of the characteristic temperaturee
rot lie between I and 15 K, much lower than the boiling points. But for H2, HD, and D2 the values ofe
rot are respectively 85.4, 65.7, and 43.0 K, well above the boiling points.The degeneracy (the number of quantum states in level J) depends on the con-stituent atoms, owing to the Pauli principle that the total wave function of the molecule must be anti-symmetric with respect to interchange of the coordinates of two identical fermions, and symmetric with respect to interchange of two identical bosons. The translational and vibrational factors of the wave-function are always symmetric, but rotational and nuclear spin factors can each be either symmetric or anti-symmetric.
Furthermore, the H nucleus (proton) is a fermion with spin
!,
and the D nucleus (deuteron) is a boson with spin 1. The degeneracies are therefore different for each of the homonuclear molecules H2 and D2, and different again for the heteronuclear molecule HD.Consider first HD. The nuclei are distinct, there is no symmetry requirement, and any rotational state can be combined with any of the six spin states. The rotational degeneracy (number of rotational states in level J) is 2J
+
1, and the spin-rotational134 Chapter 4
1.0 00:::
--
..-......
0...
> 0.5 U
OL-~----J---~---~---~
o 0.5 2.0
T/8rot
Fig. 4.2. Rotational heat capacity for a heteronuclear diatomic molecule. For HD,
e
rot=
65.7 K. From [Gop66, Fig. 6.1].partition function is
Zspin-rot = 6
L
(21+
1)e-J(1+1)(0ror/kT) ( 4.11) J=O,I,2···The only thermodynamic effect of the nuclear spin is thus to contribute an additional entropy of Rln6. The rotational heat capacity derived from Eq. (4.11), which is the same for all heteronucIear diatomic molecules, is shown in Fig. 4.2. The maximum in Crot is due to the low-lying triply degenerate first excited state.
For H2 the total state must be anti symmetric with respect to exchange of the nuclei. There are three symmetric spin states (l
=
1) and one anti-symmetric (I=
0).Rotational states of even 1 are symmetric, and to give total anti symmetry can be combined only with the single antisymmetric spin state, giving degeneracy 21
+
1;whereas those of odd 1 are anti symmetric and can be combined with any of the three symmetric spin states, giving degeneracy 3(21
+
1). The partition function is thereforeZ · Spin-rot -- '" LJ (21
+
1)e-J (J+I)(0rot/kT) J=O,2···+3
L
(21+1)e-J(1+I)(0rot /kT) (4.12)J=I,3···
From this is derived the rotational heat capacity of e-H2 ("equilibrium hydrogen") in Fig. 4.3. The large initial Schottky-like bump is due to the now nine-fold degenerate first excited level at 2k8rot ; the next level is not until6k8rot , and is then only five-fold degenerate.
However, the heat capacity of e-H2 is not that usually observed. The nuclear spins are so weakly coupled to other degrees of freedom that transitions between different
Fluids 135
400
Fig. 4.3. Rotational heat capacities of e-H2, P-H2 O-H2 and n-H2. 8ro, = 85.4 K. From [Gop66, Fig. 6.2].
spin states seldom occur; consequently, when the temperature is altered a new thermal equilibrium is established only among the even J states and among the odd J states, and not between them. Effectively therefore we have a mixture of two different species - 'para-hydrogen' (p-H2), whose molecules have the anti-symmetric spin state combined with one of the even J rotational states, and 'ortho-hydrogen' (0-H2), whose molecules have one of the symmetJic spin states combined with one of the odd J states. The total heat capacity is then the sum of contributions from each species.
The spacing between levels for each species is greater than for e-H2, and so their heat capacities do not approach the classical value until higher temperatures (Fig. 4.3).
The equilibrium ortho/para composition of hydrogen varies with temperature.
Below about 30 K it is virtually pure para; at the boiling point of nitrogen it is about half ortho; and at room temperature and above it is three quarters ortho. The relaxation time for equilibration between the species depends upon conditions and the catalytic action of the walls of the container. Typically in the gas it is of the order of years, in the liquid of the order of days, and in the solid of the order of hours.
Hydrogen is usually manufactured at room temperature or above, and then has an ortho to para ratio of 3: 1. This mixture is called 'normal hydrogen' (n-H2)' Its heat capacity is shown in Fig. 4.3, and tabulated in [Jac97], as is also that ofp-H2. Other compositions varying in proportions between pure p-H2 and n-H2 can be obtained by passing the gas over a catalyst that facilitates ortho-para conversion at the appropriate temperature.
The conversion of O-H2 to p-H2 at low temperatures is exothermic, and so if nor-mal hydrogen is rapidly liquefied and then immediately stored considerable heating and consequent evaporation occurs over a period of days. To avoid this, conversion to p-H2 should be carried out before storage.
136 Chapter 4
The same principles apply to the heat capacity of deuterium, where 8rol=43.0 K and the total wave function is symmetric with respect to deuteron exchange. Again there are two species: P-D2 with one of three antisymmetric spin states (I = 1), which for bosons must be combined with rotational states of odd J; and O-D2 with one of six symmetric spin states (I
=
0 or 1=
2) combined with rotational states of even J. At low temperatures the equilibrium composition is pure ortho, and for n-D2 the ortho-para ratio is 2: 1.Further details are given in many texts, including [Rus49, Gop66, Ric67]. Similar theory is needed to account for the spin-rotation states of other molecules containing identical nuclei, although it is only for the hydrogens that the heat capacity of the dilute gas is appreciably affected. But related effects sometimes occur when molecules are able to rotate in a condensed phase, for example in some cryocrystals (Ch. 8).
4.2.4. Non-Ideal Gases: Virial Expansion
The range of validity of the ideal gas expression for the virial P V can be extended considerably by taking it only as the first term in an expansion of the inverse molar volume V'; 1:
(4.13) where 8(T), C(T),··· are called respectively the second, third, ... virial coefficients.
Tabulations of experimental virial coefficients for a large number of gases are given in the compilation [Dym80].
Taking the expansion up to the third term usually gives a close approximation for gases for densities less than about half the critical density. Integrating the expression for P at constant T gives the expansion for the deviation in F from ideal behavior, and subsequent differentiation with respect to T gives expressions for S and C v. In partic-ular
C - C v - V,ideal - n RT
[d
2[TB(T)] dT2 V 1+
2 1d
2[TC(T)] dT2 V2 1 - ... ]m m (4.14)
The virial coefficients depend upon the intermolecular potential function: B de-pends only upon the interaction between two molecules, C also on the three-molecule potential, and so on. Since the interaction between molecules depends upon their rel-ative orientation, the integrals for the virial coefficients involve rotational degrees of freedom (e.g., [Gra84]); they also involve other coordinates if the molecules are floppy.
The simplest application however is to a classical monatomic gas. If the pair po-tential is c/>(r),
(4.15)
Fluids 137
At low temperatures the integral is dominated by the outer region where cf>( r) is nega-tive, and consequently B is negative. As T increases the repulsive range of the potential becomes more important, and B becomes positive at sufficiently high temperatures.
The temperature at which this occurs, the Boyle point TH, varies strongly from gas to gas. For helium it is about 22 K, for hydrogen about 110 K, for nitrogen about 330 K and for carbon dioxide over 700 K. The information contained in precise measure-ment of B as a function of temperature has been used in determining the shape of cf> (r ),
particularly in the attractive range of the potential. Information about three body po-tentials can in principle be derived from third virial coefficients, and so on [Bar76a].
4.2.5. Numerical Data
Whereas the small departures from perfection of the equation of state of gases such as helium and hydrogen have impact on their application as thermometric standards, these departures are trivial as far as cryogenic engineering applications are concerned.
Values of
f3
are sufficiently well established for most cryogenic purposes by assuming thatPV ~ nRT so thatf3
= d In V /dT ~ lIT.The heat capacity needs more discussion. The variation in Cv for a number of real gases may be judged from the data in Table 4.2. These few representative values are taken from the extensive compilation entitled Thermodynamic Properties o/Cryo-genic Fluids [J ac97]. This monograph gives both tables and computer fits for values of density, U, H, S, cv, cp, and velocity of sound at pressures from 0.1 MPa up to 20 MPa (or higher for some fluids) of air, Ar, CO, n-D2, C2~, F2, n-H2, Kr, CRt, Ne, N2, 02, p-H2, and Xe. Another useful source is the IUPAC series of International Ther-modynamic Tables o/the Fluid State, beginning in 1971 with Ar [Ang71]; subsequent volumes have been edited by S. Angus, B. Armstrong or K. M. de Reuck under the auspices of various publishers.
As discussed above, values of Cv at temperatures significantly above the boiling point are close to 3R12= 12.S J·mol-1·K-1 for the rare gases, andSRl2 =20.8J·mol-1.
K- 1 for oxygen and nitrogen. For n-H2, Cv falls well below SRl2 as T drops below about 300 K; forp-H2, Cv has a maximum near ISO K arising from rotational energy, as shown also in Fig. (4.3). When the temperature of a dilute gas held at constant pressure is lowered towards the boiling point, significant increases can occur in Cp and Cv (see Fig. 4.4 and Table 4.2), which are accompanied by a greater increase in density than predicted for an ideal gas, giving evidence of pre-condensation clustering.
4.3. LIQUIDS AND DENSE GASES