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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure

Chapter 8. Entropy

Adopted Database

Author(s)	Key #	Method	Range (K)
Singsass Table 1	1	Fountain Pressure	1.59 T 2.1705
Van den Meijdenberg et al.	2	Fountain Pressure	1.15 T 2.05
Singsass Table 4	3	Fountain Pressure	2.15 T 2.168
Singsass Table 2	4	Fountain Pressure	1.6 T 2.159
Singsass Table 5	5	Fountain Pressure	(T-T)/T = 0.1007 to 0.00970

Comments and Key to Authors

1,3,4,5) Singsass and Ahlers; Tables from Singsass Thesis, Ref. 7.
2) Ref. 4.
6) The spline has been adjusted to agree approximately with Singsass and Ahlers at the lambda point. The spline returns =1.579 J/g.K and =3.025 J/g.K².
7) Uncertainties: Singsass and Ahlers quote 0.1% precision, 0.5% accuracy. Van den Meijdenberg et al. quote 1% precision, 3% accuracy.
8) Following the fountain pressure entropy spline are the knots and coefficients of the fit to entropy data integrated from the recommended values of heat capacity by

This spline fit provides entropy data over both helium I and helium II regions.
9) Most heat capacity measurements are reported in Joules/mol.K. Entropy, however is usually quoted in gram units. To convert data from J/g.K to J/mol.K multiply by 4.0026. To convert from J/g.K to J/kg.K multiply by 1000.

Table 8.1. Adopted database for the fountain pressure entropy of liquid ⁴He.

T90 (K)	Cs (J/mol·K)	Key	T90	u1 (m/s)	Key
1.15302927	3.950000E-2	2	1.90415059	7.320000E-1	2
1.20326685	5.150000E-2	2	1.90415059	7.356230E-1	4
1.25320267	6.650000E-2	2	1.95418895	8.420000E-1	2
1.30332399	8.550000E-2	2	1.95418895	8.472770E-1	4
1.35344288	1.075000E-1	2	2.00303241	9.663170E-1	1
1.40351505	1.325000E-1	2	2.00423358	9.630000E-1	2
1.45355661	1.620000E-1	2	2.00423358	9.711100E-1	4
1.50359791	1.960000E-1	2	2.05429341	1.105000E+0	2
1.55363712	2.370000E-1	2	2.05429341	1.113220E+0	4
1.59708138	2.812150E-1	1	2.10389749	1.271440E+0	1
1.60369037	2.840000E-1	2	2.10439892	1.275540E+0	4
1.60369037	2.900930E-1	4	2.15456275	1.468060E+0	4
1.65377936	3.370000E-1	2	2.15466305	1.469430E+0	1
1.65377936	3.425750E-1	4	2.15469110	1.469300E+0	3
1.69937308	3.977850E-1	1	2.16429053	1.512190E+0	1
1.70388228	3.980000E-1	2	2.16429053	1.513560E+0	4
1.70388228	4.034210E-1	4	2.16429267	1.511520E+0	3
1.75398060	4.660000E-1	2	2.16970354	1.540000E+0	3
1.75408079	4.725490E-1	4	2.16970398	1.540690E+0	1
1.80106080	5.449310E-1	1	2.17290000	1.557640E+0	1
1.80406509	5.450000E-1	2	2.17290343	1.557990E+0	3
1.80406509	5.502870E-1	4	2.17460000	1.570540E+0	1
1.85411884	6.330000E-1	2	2.17510000	1.571870E+0	1
1.85411884	6.381940E-1	4	2.17530000	1.573910E+0	1
1.90214931	7.301360E-1	1	2.17680000	1.578970E+0	5

Figure 8.1. The recommended values of the fountain pressure entropy of liquid 4He as a function of temperature at saturated vapor pressure.

Table 8.2. Knots and coefficients for the spline fit of the fountain pressure entropy.

Knots	Coefficients
K(1) = 1.153029	C(1) = 3.949990E-2
K(2) = 1.153029	C(2) = 5.660103E-2
K(3) = 1.153029	C(3) = 9.933250E-2
K(4) = 1.153029	C(4) = 2.321544E-1
K(5) = 1.402666	C(5) = 5.183673E-1
K(6) = 1.501863	C(6) = 9.994261E-1
K(7) = 1.801948	C(7) = 1.506163E+0
K(8) = 2.154592	C(8) = 1.549628E+0
K(9) = 2.164268	C(9) = 1.575018E+0
K(10) = 2.172874	C(10) = 1.578977E+0
K(11) = 2.176800
K(12) = 2.176800
K(13) = 2.176800
K(14) = 2.176800

Figure 8.2. The fractional deviation of values of the adopted database from the recommended values for the fountain pressure entropy of liquid 4He expressed in percent.

Table 8.3. Recommended values of the fountain pressure entropy of liquid ⁴He.

T90 (K)	S (J/g K)
1.20	5.059E-2
1.25	6.560E-2
1.30	8.397E-2
1.35	1.057E-1
1.40	1.310E-1
1.45	1.600E-1
1.50	1.940E-1
1.55	2.343E-1
1.60	2.815E-1
1.65	3.357E-1
1.70	3.972E-1
1.75	4.662E-1
1.80	5.429E-1
1.85	6.279E-1
1.90	7.233E-1
1.95	8.319E-1
2.00	9.561E-1
2.05	1.099E+0
2.10	1.262E+0
2.15	1.450E+0
2.1761	1.577E+0
2.1762	1.577E+0
2.1763	1.577E+0
2.1764	1.578E+0
2.1765	1.578E+0
2.1766	1.578E+0
2.1767	1.579E+0
2.1768	1.579E+0

Figure 8.3. The recommended values of the entropy of liquid 4He obtained by integration of the heat capacity spline.

Table 8.4. Knots and coefficients for the spline fit of entropy integrated from the recommended values of heat capacity.

Knots	Coefficients
K(1) = 0.10000	C(1) = 6.60E-6
K(2) = 0.10000	C(2) = 7.60E-6
K(3) = 0.10000	C(3) = 1.60E-5
K(4) = 0.10000	C(4) = 4.61413E-5
K(5) = 0.11123	C(5) = 1.8400E-4
K(6) = 0.21625	C(6) = 4.68675E-4
K(7) = 0.27672	C(7) = 0.00121
K(8) = 0.44817	C(8) = 0.00236
K(9) = 0.56369	C(9) = 0.00561
K(10) = 0.71969	C(10) = 0.01978
K(11) = 0.85360	C(11) = 0.08337
K(12) = 1.00186	C(12) = 0.35824
K(13) = 1.40184	C(13) = 0.76882
K(14) = 1.72912	C(14) = 1.17803
K(15) = 1.99887	C(15) = 1.38629
K(16) = 2.07775	C(16) = 1.53613
K(17) = 2.15384	C(17) = 1.58271
K(18) = 2.17680	C(18) = 1.62419
K(19) = 2.17680	C(19) = 1.72261
K(20) = 2.17680	C(20) = 1.91862
K(21) = 2.21160	C(21) = 2.53529
K(22) = 2.34820	C(22) = 3.16601
K(23) = 2.69075	C(23) = 4.10379
K(24) = 4.38277	C(24) = 4.44077
K(25) = 4.76361	C(25) = 4.61570
K(26) - K(29)
= 5.000000

Figure 8.4. The fractional deviation of values of the adopted database from the recommended values for the entropy integrated from the heat capacity spline.

Table 8.5. Recommended values of entropy of liquid ⁴He obtained by integration of the heat capacity spline.

T90 (K)	S (J/g K)	T90 (K)	S (J/g K)
0.10	6.600E-6	2.25	1.720E+0
0.15	2.298E-5	2.30	1.783E+0
0.20	5.506E-5	2.35	1.838E+0
0.25	1.074E-4	2.40	1.890E+0
0.30	1.841E-4	2.45	1.939E+0
0.35	2.898E-4	2.50	1.985E+0
0.40	4.298E-4	2.55	2.030E+0
0.45	6.093E-4	2.60	2.074E+0
0.50	8.343E-4	2.65	2.116E+0
0.55	1.113E-3	2.70	2.158E+0
0.60	1.456E-3	2.75	2.200E+0
0.65	1.904E-3	2.80	2.242E+0
0.70	2.512E-3	2.85	2.284E+0
0.75	3.339E-3	2.90	2.326E+0
0.80	4.512E-3	2.95	2.367E+0
0.85	6.206E-3	3.00	2.409E+0
0.90	8.605E-3	3.05	2.451E+0
0.95	1.194E-2	3.10	2.492E+0
1.00	1.644E-2	3.15	2.534E+0
1.05	2.236E-2	3.20	2.576E+0
1.10	3.004E-2	3.25	2.619E+0
1.15	3.982E-2	3.30	2.661E+0
1.20	5.204E-2	3.35	2.704E+0
1.25	6.706E-2	3.40	2.747E+0
1.30	8.521E-2	3.45	2.790E+0
1.35	1.068E-1	3.50	2.834E+0
1.40	1.323E-1	3.55	2.878E+0
1.45	1.620E-1	3.60	2.923E+0
1.50	1.965E-1	3.70	3.014E+0
1.55	2.364E-1	3.75	3.060E+0
1.60	2.824E-1	3.80	3.107E+0
1.65	3.350E-1	3.85	3.154E+0
1.70	3.950E-1	3.90	3.203E+0
1.75	4.630E-1	3.95	3.251E+0
1.80	5.399E-1	4.00	3.301E+0
1.85	6.270E-1	4.05	3.351E+0
1.90	7.255E-1	4.10	3.402E+0
1.95	8.367E-1	4.15	3.454E+0
2.00	9.621E-1	4.20	3.507E+0
2.05	1.103E+0	4.25	3.561E+0
2.10	1.263E+0	4.30	3.616E+0
2.1760	1.578E+0	4.35	3.672E+0
2.1762	1.579E+0	4.40	3.728E+0
2.1763	1.580E+0	4.45	3.786E+0
2.1764	1.580E+0	4.50	3.846E+0
2.1765	1.581E+0	4.75	4.175E+0
2.1766	1.581E+0	4.80	4.250E+0
2.1767	1.582E+0	4.85	4.330E+0
2.1768	1.583E+0	4.90	4.416E+0
2.1769	1.583E+0	4.95	4.511E+0
2.1770	1.583E+0	5.00	4.616E+0
2.20	1.643E+0

Chronological Bibliography for Entropy

1	H. C. Kramers, J. D. Wasscher, and C. J. Gorter, "The Specific Heat of Liquid Helium Between 0.25 and 1.9 Degrees K," Physica 18, 329-337 (1952).
2	G. R. Hercus and J. Wilks, “The Specific Heat of Liquid Helium II as a Function of Pressure,” Philos. Mag. 45, 1163-1172 (1954).
3	O. V. Lounasmaa and E. Kojo, “The Specific Heat Cv of Liquid Helium Near the Lambda Curve at Various Densities,” Ann. Acad. Scien. Fennicae A VI, Physica 36, 1 (1959).
4	C. J. N. Van den Meijdenberg, K. W. Taconis, and R. De Bruyn Ouboter, “The Entropy of Helium II Under Pressure from Measurements on the Fountain Effect,” Physica 27, 197-218 (1961).
5	J. Wiebes, “Caloric Measurements on Liquid and Melting Helium Below 1.5 Kelvin,” Ph.D. Thesis, Kammerlingh Onnes Laboratory, 1969 (unpublished).
6	A. L. Singsaas and G. Ahlers, “Entropy of Helium II from 1.6 K to the Lambda Line,” Phys. Rev. B 29, 4951-4960 (1983).
7	A. L. Singsaas, “Entropy of Helium II and Universality Near the Superfluid Transition in 4He,” Ph.D. Thesis, University of California, 1984 (unpublished)