Helium entropy

Extracted from TsederLerg, Popov, et al., Theimodynamic and Theimophysical Propeities of Helium, Atomizdat, Moscow, 1969, and NBS-NSF TT 50096, 1971. Copyriglit material. Reproduced hy permission. This source contains entries for many more temperatures and pressures than can he reproduced here, v = volume, mVkg h = enthalpy, kj/kg s = entropy, kJ/(kg-K). 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

Helium is an interesting example of the application of the Third Law. At low temperatures, normal liquid helium converts to a superfluid with zero viscosity. This superfluid persists to 0 Kelvin without solidifying. Figure 4.12 shows how the entropy of He changes with temperature. The conversion from normal to superfluid occurs at what is known as the A transition temperature. Figure 4.12 indicates that at 0 Kelvin, superfluid He with zero viscosity has zero entropy, a condition that is hard to imagine.v... 

Figure 4.12 Entropy of liquid helium near absolute zero.

The enthalpy of copper at nitrogen temperature is H77K = 6 J/g, so the total entropy of the sphere will be about 6 x 106 J. The time needed to cool from 77 K down to 4K is of the order of 4h. The total helium consumption from room temperature to 4.2 K would be about 6001. The temperatures reached in a test run are reported in Table 16.2. The expected final sphere temperature is about 20 mK. A comparison of MiniGRAIL and Nautilus cool down is made in Table 16.2. The high power leak on the sphere has been attributed to a time-dependent heat leak caused by the ortho-para conversion (see Section 2.2) of molecular hydrogen present in the copper of the sphere (see Fig. 16.5) (the Nautilus bar instead is made by Al). A similar problem has been found in the cool down of the CUORICINO Frame (see Section 16.6). 

Helium (the simplest possible elemental species) remains a liquid at T = 0 (unless the pressure is increased above about 20 atm). Bizarre quantum fluid effects and nonzero entropies are exhibited by both 3He and 4He in the T —> 0 limit. 

The molecule He, is unknown since the number of antibonding electrons (2) is equal to the number of bonding electrons (2) and the net bond order is zero. With no bond energy14 to overcome the dispersive tendencies of entropy, two helium atoms in a "molecule will not remain together but fly apart. If it existed, molecular helium would have the electron configuration ... 

The star that exploded, SK-202-69, was, as theory required, a massive star. When it lived on the main sequence, it had a mass of 19 3 M . At the time it exploded it had a helium core mass of 6 1 M , a radius 3 1 xlO12 cm, a luminosity 3 to 6 xlO38 erg s 1, and an effective temperature 15,000 to 18,000 K. Further consideration of the stellar models (Woosley 1987 Nomoto, this volume) suggests that the iron core mass at the time of collapse was 1.45 0.15 M0. Adding 0.15 M0 for matter between the iron core and the entropy jump... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The stable phase of all substances, except helium, at sufficiently low temperatures is the solid phase. We therefore consider the solid phase as the condensed state whose entropy is zero at 0 K, and exclude helium from the discussion for the present. The absolute entropy of a pure substance in some state at a given temperature and pressure is the value of the entropy function for the given state taking the value of the entropy of the solid phase at 0 K... 

Liquid helium presents an interesting case leading to further understanding of the third law. When liquid 4He, the abundant isotope of helium, is cooled at pressures of < 25 bar, a second-order transition takes place at approximately 2 K to form liquid Hell. On further cooling Hell remains liquid to the lowest observed temperature at 10 5 K. Hell does become solid at pressures greater than about 25 bar. The slope of the equilibrium line between liquid and solid helium apparently becomes zero at temperatures below approximately 1 K. Thus, dP/dT becomes zero for these temperatures and therefore AS, the difference between the molar entropies of liquid Hell and solid helium, is zero because AV remains finite. We may assume that liquid Hell remains liquid as 0 K is approached at pressures below 25 bar. Then, if the value of the entropy function for sol 4 helium becomes zero at 0 K, so must the value for liquid Hell. Liquid 3He apparently does not have the second-order transition, but like 4He it appears to remain liquid as the temperature is lowered at pressures of less than approximately 30 bar. The slope of the equilibrium line between solid and liquid 3He appears to become zero as the temperature approaches 0 K. If, then, the slope is zero at 0 K, the value of the entropy function of liquid 3He is zero at 0 K if we assume that the entropy of solid 3He is zero at 0 K. Helium is the only known substance that apparently remains liquid as absolute zero is approached under appropriate pressures. Here we have evidence that the third law is applicable to liquid helium and is not restricted to crystalline phases. 

A design for purifying helium consists of an adiabatic process that splits a helium stream containing 30-mole-percent methane into two product streams, one containing 97-mole-percent helium and the other 90-mole-percent methane. The feed enters at 10 bar and 117 C the methane-rich product leaves at 1 bar and 27 C the helium-rich product leaves at 50°C and IS bar. Moreover, wort is produced by the process. Assuming helium an ideal gas with CP = (5/2)1 and methane an ideal gas with CP = (9/2)/ , calculate the total entropy change of the process on the basis of 1 mol of feed to confirm that the process does not violate the second law. 

Entropy evaluations from published cryothermal data on the lanthanide (III) oxides are summarized in Table II with an indication of the lowest temperature of the measurements and the estimated magnetic entropy increments below this temperature. Their original assignment of crystalline field levels from thermal data still appears to be in good accord with recent findings e.g., 17). Unfortunately, measurements on these substances were made only down to about 8°K. because the finely divided oxide samples tend to absorb the helium gas utilized to enhance thermal equilibration between sample and calorimeter. 

Thus the entropy of most crystalline substances may be regarded as zero at the absolute zero of temperature. There are a number of exceptions to this generalization, for example in the case of CO (c/. p. 196). In the CO crystal the molecules are still able to take up several different orientations even at the lowest experimental temperatures. In the same way, glasses and amorphous solids do not have zero entropy at 0 °K. On the other hand, the entropy of helium at 0 °K is zero even though helium is liquid at this temperature and at 1 atm. pressure. 

When the mass of the helium-layer accreted into the low-entropy core is sufficiently high, helium ignition will occur at the boundary between the non-degenerate and degenerate material. The star will expand on the short thermal timescale of the envelope ( 103 y) to become a yellow supergiant -a stage which will be described in more detail in Sect. 15.7. 

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