The consequences of a new model for calculating heat transfer phase transitions of the first kind.
A.A. Sobko.
A M Prokhorov Academy of Engineering Sciences, Moscow.
Keywords: The heat of fusion, heat of vaporization, the volume of atoms (molecules), the size of atoms (molecules).
The relationship between thermodynamic parameters at the critical point is derived on the basis of the results of the previous work. Additionally, the expression for the effective volume of atom (molecule) is obtained.
The expressions derived for the evaporation heat and melting heat can be substituted into the Clapeyron-Clausius equation [1,2]:
Satisfactory agreement between experimental and calculated values of the heat of first-order phase transitions allows us to hope that utilization of the obtained relationships for the analysis of the behaviour of melting and saturation curves will provide reliable results.
1. The behaviour of the melting curve at high temperatures and pressures.
As it was shown before the melting heat at high pressures is described by the following expression:
where is a nonzero constant. Inserting this relationship into the Clapeyron-Clausius equation with the passage to the limit which corresponds to high temperatures and pressures we obtain
Thus, if with increasing temperature and pressure then and, consequently, in the framework of the considered model the critical point is absent on the melting curve (contrary to the critical point on the evaporation curve).
2. The relationship between thermodynamic parameters at the critical point.
One can perform the operations analogous to the calculations done in the previous subsection and substitute the expression for the evaporation heat into the Clapeyron-Clausius equation: or , with the passage to the limit , i.e. the passage to the critical point. Consequently, we obtain: ;
The argumentation for the latest expression may be the following. It was shown in [2] that , (where in the classical critical point theory and according to the experimental data). And in [3] it was shown that the surface tension constant conducts itself as near the critical point. Consequently, ~ ~ , and this expression tends to zero at the approach to the critical point. The validity of the statement can also be understood by considering the curve depicted in Fig. 1.
Fig. 1. Dependence of on the temperature of hydrogen.
In Fig. 1 the dependence of on the temperature is given for hydrogen for which the values of the surface tension constant are known along the saturation curve. Hence, the following relation between the thermodynamic parameters at the critical point is valid:
(1)
the subscript “c” indicates that all the values of the thermodynamic parameters are taken at the critical point.
3. Calculation of the experimental values of the derivative at the critical point and comparison of these to the theoretical ones.
The relationship (1) is purely derived from the Clapeyron-Clausius equation and the expression for the evaporation heat, which was proved to be valid by a good agreement between experimental data and numerical results on evaporation heat. However, one can test this relationship with the use of experimental data.
Experimental value of the derivative is calculated in two ways.
а) Linear approximation: ,
where P1 and T1 are the experimental values of pressure and temperature that are the closest to the critical point.
б) Quadratic approximation:
Critical values Pc and Tc, as well as two closest experimental points are approximated using the polynomial: P = AT2 +BT + C, and then the coefficients A, B and are determined.
The results of calculation of the derivatives are given in Table I. The values of A and B are also presented there. All experimental data required for the calculations are taken from [4].
Table I. Calculation results for experimental values of derivatives.
Ma-terial / T1 / T2 / Tc / P1 / P2 / Pc / A / B / (dP/dT)1 / (dP/dT)2 / /Ne / 43 / 44.0 / 44.4 / 22.16 / 25.22 / 26.54 / 0.18 / -12.32 / 3.31 / 3.38 / 2.09
Ar / 149 / 150.0 / 150.86 / 45.58 / 47.39 / 48.98 / 0.02 / -4.43 / 1.85 / 1.87 / 0.96
Kr / 206 / 208.0 / 209.39 / 50.05 / 52.91 / 54.97 / 0.02 / -4.92 / 1.48 / 1.50 / 1.42
Xe / 286 / 288.0 / 289.74 / 54.10 / 56.31 / 58.28 / 0.01 / -3.07 / 1.13 / 1.14 / 1.10
H2 / 32.5 / 33.0 / 33.23 / 11.84 / 12.73 / 13.16 / 0.12 / -6.26 / 1.87 / 1.90 / 1.49
N2 / 125 / 126.0 / 126.25 / 32.05 / 33.57 / 33.96 / 0.03 / -6.51 / 1.56 / 1.57 / 0.51
O2 / 153 / 154.0 / 154.77 / 47.51 / 49.39 / 50.87 / 0.02 / -5.42 / 1.92 / 1.94 / 0.94
F2 / 140 / 142.5 / 144.00 / 44.69 / 49.87 / 53.25 / 0.05 / -10.73 / 2.25 / 2.32 / 2.93
Cl2 / 410.95 / 416.5 / 417.17 / 70.36 / 76.34 / 77.09 / 0.01 / -4.37 / 1.12 / 1.12 / 0.00
CH4 / 189 / 190.0 / 190.55 / 43.95 / 45.52 / 46.41 / 0.03 / -10.21 / 1.62 / 1.64 / 1.05
NH3 / 390 / 400.0 / 405.60 / 86.06 / 102.8 / 113.0 / 0.01 / -5.79 / 1.82 / 1.87 / 2.82
CO / 125.99 / 129.87 / 132.94 / 25.32 / 30.39 / 34.98 / 0.03 / -5.63 / 1.50 / 1.58 / 5.27
Tc, T2, T1 [K], Pc, P2, P1 *10-5[Pa] are the experimental values of the critical temperature and pressure and experimental values of temperature and pressure near the critical point, A and B are the parameters of the quadratic trinomial used for interpolation, P1′, P2′ *10-5[Pa/K] are the experimental values of the derivatives calculated with the use of linear and quadratic approximations, δ is fractional accuracy.
As one can see from Table I, the values of (dP/dT)ex derivatives calculated by means of two different methods match with a good agreement. The derivative calculated with the use of linear approximation will be used hereafter due to its calculation simplicity. Theoretical value of is calculated with the use of expression (1):
.
For the calculation of the “free volume” the values equal to the half of internuclear distances plus van der Waals radii of the atoms of these molecules were taken as the radii of binary molecules like it was done before for the calculation of evaporation heat [5,6]. Results of calculations of are given in Table II, along with the experimental values of derivative.
Table II. Calculation results for theoretical values of derivatives.
Ma-terial / Tc / Pc / Vc / R / Vcf / (dP/dT)ex / (dP/dT)T / /Ne / 44.4 / 27.60 / 4.18 / 1.60 / 3.15 / 3.31 / 3.26 / 1.43
Ar / 150.86 / 48.98 / 7.49 / 1.92 / 5.71 / 1.85 / 1.78 / 3.73
Kr / 209.39 / 54.97 / 9.20 / 1.98 / 7.24 / 1.48 / 1.41 / 4.73
Xe / 289.74 / 58.28 / 11.94 / 2.18 / 9.33 / 1.13 / 1.09 / 3.37
H2 / 33.23 / 13.16 / 6.39 / 1.44 / 5.63 / 1.87 / 1.87 / 0.06
N2 / 126,25 / 33.96 / 8.95 / 2.09 / 6.65 / 1.56 / 1.52 / 2.64
O2 / 154.77 / 50.87 / 7.34 / 2.00 / 5.32 / 1.92 / 1.89 / 1.58
F2 / 144.00 / 52.20 / 4.38 / 2.07 / 4.91 / 2.25 / 2.26 / 0.35
Cl2 / 417.17 / 77.09 / 12.38 / 2.47 / 8.58 / 1.12 / 1.15 / 3.00
CH4 / 190.55 / 46.41 / 9.88 / 2.30 / 6.81 / 1.62 / 1.46 / 9.68
NH3 / 405.60 / 113.0 / 7.25 / 1.88 / 5.57 / 1.82 / 1.77 / 2.76
CO / 132.94 / 34.98 / 9.36 / 2.16 / 6.81 / 1.50 / 1.48 / 1.06
Tc [K], Pc*10-5[Pa], Vc*105[m3/mol] are the critical values of temperature, pressure and volume, respectively, ra*1010 [m] is the radius of atom (molecule), Vcf *105[m3/mol] is the “free critical volume”, Pex′*10-5, PT′*10-5[Pa/K] are the experimental and theoretical values of the derivatives, δ is fractional accuracy.
A good agreement of the experimental values and the results of calculations by formula (1) allows one to assert that the following relationship is valid at the critical point:
.
4. Calculation of the volumes of atoms (molecules, ions)
The expression (1) which was proved to be valid by the analysis of the results given in Table II involves free critical volume as one of its parameters. Therefore, the expression (1) can be used to determine effective volumes of atoms (molecules, ions). Indeed, , which gives ,
and then the volume of atom (molecule) is: (2)
where is the critical geometric (experimental) volume.
The expression (2) allows one to determine the effective (intrinsic) volume using the experimental data for the critical point.
Results of calculation of the volumes of atoms and simple symmetrical molecules as well as their parameters are given in Tables VI-VI. The volumes of complex molecules are given in Table III.
Table III. Volumes of complex molecules.
Material / Tc / Pc / Vc / / V0 /H2O / 647.29 / 221.15 / 5.67 / 2.54 / 31.43
CO2 / 34.19 / 73.82 / 9.40 / 1.64 / 57.75
SO2 / 430.7 / 77.81 / 12.20 / 1.23 / 71.41
CH4 / 190.55 / 46.41 / 9.9 / 1.62 / 64.07
C2H6 / 305.5 / 49.13 / 14.18 / 1.00 / 70.39
C3H8 / 370.0 / 42.65 / 19.6 / 0.76 / 109.87
n-C4 H10 / 425.18 / 37.96 / 25.49 / 0.61 / 159.33
i-C4 H10 / 408.15 / 36.47 / 26.30 / 0.61 / 170.92
n-C5 H12 / 469.79 / 33.74 / 31.10 / 0.56 / 233.17
i-C5 H12 / 460.97 / 33.3 / 30.83 / 0.46 / 156.4
n-C6 H14 / 507.87 / 30.31 / 36.81 / 0.41 / 220.77
n-C7 H16 / 540.18 / 27.36 / 42.64 / 0.39 / 297.36
n-C8 H18 / 569.37 / 24.96 / 48.61 / 0.33 / 317.85
i-C8H18 / 544.27 / 25.83 / 48.20 / 0.35 / 337.36
c-C5 H10 / 511.77 / 45.13 / 25.97 / 0.55 / 133.31
c-C6 H12 / 532.78 / 37.84 / 31.88 / 0.44 / 153.93
c-C7 H14 / 569.47 / 33.96 / 37.47 / 0.40 / 218.81
c-C6 H12 / 553.07 / 4030 / 31.17 / 0.47 / 173.41
C2 H4 / 282.67 / 50.6 / 12.75 / 1.2 / 76.61
C4 H8 / 419.57 / 40.20 / 24.08 / 0.63 / 142.93
i-C4 H6 / 417.9 / 39.69 / 23.98 / 0.63 / 141.29
C4 H6 / 425.17 / 43.3 / 22.08 / 0.67 / 124.38
C2 H2 / 308.7 / 62.45 / 11.32 / 1.86 / 104.54
c-C6 H6 / 562.6 / 49.24 / 25.69 / 0.58 / 148.4
o-C8 H10 / 631.61 / 38.08 / 36.99 / 0.45 / 264.32
p-C8 H10 / 618.17 / 36.17 / 37.78 / 0.46 / 283.74
m-C8H10 / 619.17 / 36.5 / 39.32 / 0.44 / 290.87
Tc [K], Pc*10-5[Pa], Vc*105[m3/mol] are the critical values of temperature, pressure and geometric volume, respectively, Pex′*10-5[Pa/K ] is the experimental value of the derivative at the critical point, V0*1030[m3] is the volume of molecule.
It is necessary to consider geometric models of atoms (molecules) to obtain their linear size. Corresponding models are regarded below.
5. Geometric models of atoms and molecules
a) Spherical model
Spherical model is relevant to noble gasses as well as CH4 and NH3 molecules due to small size of hydrogen atoms. Using the volumes of atoms (molecules) calculated according to formula (2) one can find their radii from the following relationships: or . These results are presented in Table IV.
Table IV. Volumes and radii of atoms and molecules in spherical model.
Ma-terial / Tc / Pc / Vc / (dP/dT)ex / ra / V0 / raT / 1 / rw / δ2 /Ne / 44.4 / 26.54 / 4.18 / 3.31 / 1.60 / 18.47 / 1.64 / -2.51 / 1.77 / -10.51
Ar / 150.86 / 48.98 / 7.49 / 1.85 / 1.92 / 33.22 / 1.99 / -3.89 / 2.14 / -11.66
Kr / 209.39 / 54.97 / 9.20 / 1.48 / 1.98 / 39.61 / 2.11 / -6.82 / 2.30 / -16.16
Xe / 289.74 / 58.28 / 11.83 / 1.13 / 2.18 / 48.23 / 2.26 / -3.60 / 2.50 / -14.73
CH4 / 190.55 / 46.41 / 9.90 / 1.34 / 2.00 / 38.87 / 2.10 / 1.82 / 2.36 / -17.95
NH3 / 405.60 / 113.0 / 7.25 / 1.82 / 1.82 / 30.92 / 1.95 / -6.92 / 2.12 / -41.63
Tc [K], Pс*10-5[Pa], Vc*105 [m3/mol] are the critical values of temperature, pressure and volume, respectively, Pex′*10-5[Pa/K] is the experimental value of the derivative at the critical point, ra*1010[m] is the tabulated radius of atom, V0*1030[m3] is the volume of atom (molecule), raT is the theoretical radius of atom (molecule), δ1 is fractional accuracy, rw *1010[m] are the radii of atoms (molecules) calculated under the assumption that the parameter “b” in the van der Waals equation is the volume occupied with atoms (molecules), δ2 is fractional accuracy.
As one can see from the results given in Table IV, the radii of atoms of noble gasses are in a good agreement with the literature data [5,6]. In Table IV the radii of atoms rw obtained under assumption that the parameter “b” in the van der Waals equation is the volume occupied with atoms are given as well. Considering that b = Vс/3, in the van der Waals equation gives and as can be seen from Table IV these radii are substantially different from the conventional ones as substituting the “free volume” into the van der Waals equation is not fully consistent.