Part I: calculation of the evaporation heat for liquid gases and metals

General expressions for the transition heat of first-order phase transitions: The conventional expression for a transition heat λ=TΔS has two substantial drawbacks. First, in some phase transitions, for example, in evaporation, not only entropy changes but a system does the work, which can only be supplied by an external source of heat. Second, the transition heat is expressed in terms of the entropy variation ΔS, which cannot be measured experimentally. In the preset work, the transition heat is defined as

λ=TΔS+A

Where T is the transition temperature in K°, ΔS is the change of system entropy, A is the work that the system does. Entropy is calculated from the general definition [3]

Where K is the Boltzmann factor, V_ph is the volume of phase space occupied by the system, and S is the number of degrees of freedom. Thus, we obtain:

where and are the volumes of the old and new phases, respectively. The general expression for the heat for a phase transition has the form:

(1)

The volumes of phase spaces and the expressions for the work are specified for each particular phase transition. In the present work, all calculations are performed for one mole of substance; hence, all extensive values refer to one mole.

Approximate calculation of the phase space volumes for liquid and gaseous states

For liquid and gaseous states, the energy of system has the form:

(2)

Since we consider one mole of a single-component substance, all masses are equal and N=N_A is the Avogadro number. The volume of the phase space is:

Equation (2) can be rewritten in the form:

which is the equation of a 3N-dimensional sphere in -space; hence, the 3N-dimensional integral over pulses is equal to the volume of this sphere of the radius , and the expression for the volume of phase space takes the form:

In the result of integration the dimensionality of the integral has changed from 6 to 3; however, one cannot take the rest integral without further assumptions. For calculating the 3N-dimenstional integral, let us consider the behavior of the function of kinetic energy distribution near a point of a first-order phase transition. Not specifying exactly the distribution function one may assert that it has a bell shape with a maximum that shifts with temperature to right. Near the point of phase transition such an evolution of the distribution function is impossible because the temperature of the system does not change and energy incoming still continues. Hence, the only way for the distribution function to vary is its narrowing and, in the limit, it transfers to the δ -function. In the latter case, the most probable and the average values of the kinetic energy will coincide. It is not a rigorous proof of the distribution function narrowing; however, it suggests a principal assumption of the present work, which is confirmed by a satisfactory agreement between experimental data and calculation results.

Note: Near the first-order phase transition the most of atoms (molecules, ions) are in the state with the average kinetic energy.

Since (E-U(r₁,..r_n)) is the kinetic energy of the system, according to the theory about equal distribution of a kinetic energy over degrees of freedom [4] one can substitute it for the average value 3N_AKT/2=3RT/2, where R=KN_A is the universal gas constant:

Thus, the volume of a phase space for liquid and gas is expressed as

(3)

(4)

Where V_L and V_G are the volumes of liquid and gas, respectively.

Calculation of the evaporation heat for liquid gases

By using the general expression (1) for a transition heat one can find the expression for the evaporation heat on the saturation curve. Since the volumes of phase spaces for liquid and gas are known (3), (4), the change of entropy in evaporation has the form:

Where ΔV is the volume jump, V_L is the volume of liquid, V_G is the gas volume.

The work on volume expansion is A₁=PΔV. In the transit liquidgas, in addition to the work on volume expansion, also the work against surface tension forces is done A₂=σFN₁, where σ is the surface tension coefficient, F is the area of liquid surface, N₁=V_a/Fd is the number of mono-molecular layers, and V_a is the volume occupied by atoms (molecules, ions), d is the thickness of a mono-molecular layer (in the present work it is αr, where r is the radius of atom (molecule, ion), α=1.717 is the packing factor).

Thus, we have A₂=σFN₁=σFV/Fd=σV_α/d.

The expression for the evaporation heat on the saturation curve has the form

(5)

Where R is the universal gas constant; T, P are the temperature and pressure on the saturation curve, respectively; ΔV is the jump of volume in the process of evaporation; V_L is the volume of liquid; V_α is the volume occupied by atoms (molecules, ions); r_α is the effective atomic (molecular, ion) radius; and α=1.817 is the sphere packing factor. All extensive values refer to one mole of substance.

The expression for the evaporation heat λ comprises the volume occupied by atoms V_α and the volume occupied by liquid V_L. The question arises which volume one should employ – the geometrical (experimental) volume or the free volume V_Lf=V_Lf–N_AV₀, where V₀is the volume occupied by atom (molecule, ion). Here, the evaporation heat is calculated by using as the geometrical liquid volume V_L, so and the free volume of liquid V_Lf.

Experimental data on the saturation curve are taken from Ref. [10], the radii of atoms and ions are taken from Ref. [11,12]. The radii of two-atomic molecules are taken as half the distance between nuclei centers [13] plus the Van der Waals radii [12]. The results are given in Table 1 and Figure 1.

Figure 1: Experimental and calculated evaporation heat versus temperature.

TK° is the evaporation temperature, P*10^-5 [Pa] is the pressure, V_{L*10⁵ [m³/mole] is the molar volume of liquid, ΔV*10⁵ [m³/mole] is the jump of volume in evaporation, σ*10³ [N/m] is the surface tension coefficient, r*10¹⁰ [m] is the radius of atom (molecule, ion), λex [J/mole] is the experimental value of molar evaporation heat, λT1 [J/mole] is the molar evaporation heat calculated by using the geometrical volume, λT2 [J/mole] is the molar evaporation heat by using the free volume, δ1 and δ2 [%] are inaccuracies of λT1 and λT2, respectively [14].}

The calculated values of evaporation heat in Table 1 are only presented for the evaporation lines, for which the experimental values of surface tension coefficient have been found. For hydrogen, data on a surface tension coefficient along the entire evaporation curve are known; the corresponding experimental and calculated values of evaporation heat are plotted in Figure 1. A small difference between experimental and calculated values at low temperatures is related with the fact that the calculation of a phase volume should make allowance for quantum effects.

Table 1: Calculation results of evaporation heat.

From Table 1 and Figure 1 one can see that the calculations of the evaporation heat performed by the obtained formula well agree (within several percent) with experimental results. Hence, the assumption, that the most of atoms (molecules, ions) near a point of a first-order phase transition are in the state with the average kinetic energy, is valid. Also valid is the assumption that in a calculation of the evaporation heat one should take into account the work done by the system. The employment of the free volume in calculations also gives a better agreement between experimental and calculated results. Thus, one can assert that the molar evaporation heat λ on the saturation curve is described by the expression:

(6)

where all the values have been defined above.

Specific features of calculating the evaporation heat for liquid metals

As one can see from the results presented in Table 1, calculations of evaporation heat should be performed with the "free volume". For determining the "free volume" of liquid metals one should know the radii of ions. Handbooks [11,12] comprise two metal ion radii: radii for ions M ⁺¹, M ⁺² and so on, and for metal. Since it is not clear which radius should be used in the calculations of the free volume, the latter was calculated by using as the metal radius so and the ion radius. The results with the employment of the ion radius are given in Table 2, and the results based on metal radius are presented in Table 3.

Table 2: Calculation results for the evaporation heat on the saturation curve for metals by using the ion radii.

Table 3: Calculation results for the evaporation heat on the saturation curve for metals by using the metal radii.

T[K] is the evaporation temperature, P*10^-5 [Pa] is the pressure, V_L *10⁵ [m³/mole] is the molar volume of liquid, ΔV *10⁵ [m³/mole] is the volume jump in evaporation, σ*10³ [J/m²] is the surface tension coefficient, rm*10¹⁰ [m] is the ion radius in Table 2 and the metal radius in Table 3, V_m*10⁵ [m³/mole] is the volume occupied by ions, V_Lf*10⁵ [m³/mole] is the "free volume" of liquid, λ_ex [J/mole] is experimental evaporation heat, λ_T [J/mole] is calculated value of evaporation heat, and δ % is the calculation error λ_T.

From Table 2 one can see that the employment of ion radii in calculations of the free volumes of liquid gives a good agreement with experimental data; however, the presence of free electrons in liquid and screening of ions make one to assume that more realistic are metal radii. Results of calculations with metal radii are given in Table 3. One can see that the calculated values of the evaporation heat in this case are systematically greater than the experimental values. This can be explained by the fact that in evaporation of metals, in addition to endothermic processes, there are also exothermic processes.

Recombination of ions and electrons with the origin of neutral atoms occurs for all metals at the interface liquid-gas (vapor). In this case, the energy is released and completely or partially participates in the evaporation process. In addition, alkali metal atoms in the gaseous state form two-atomic molecules [10] with an energy release. Thus, Δλ=λ_T- λ_ex is the evaporation heat received by a system due to the exothermic processes considered above. The energy released due to generation of two-atomic molecules is proportional to the fraction of two-atomic molecules in gas, and the energy released in the ion recombination is constant and independent of thermodynamic parameters.

Hence,

Δλ=βC+Q₀, (7)

Where C is the part of two-atomic molecules in gas, β is the energy, released in the process of producing 0.5 mole of two-atomic molecules, Q₀ is the heat released in recombination of one mole of metal ions.

Calculation results for the evaporation heat, experimental values of evaporation heat in a wide temperature range on the evaporation curve for all alkali metals and mercury, and data on the part of two-atomic molecules in gas for alkali metals are given in Table 4.

Table 4: Calculation of Δλ.

Basing on these data, dependences Δλ=f (C) for alkali metals were plotted in Figure 2. One can see that the assumption about linear dependence of Δλ on C is confirmed with a high accuracy. Moreover, one can assert that the energies released in generating two-atomic molecules Na₂, K₂, Rb₂, Cs₂ are similar, and the energy released in generating Li₂ is substantially higher. The values of Q₀ are, respectively, Q₀^Li=4510 J/mole, Q₀ ^Na=3510 J/mole, Q₀ ^K=2510 J/mole, Q₀ ^Rb=2030 J/ mole, and Q₀ ^CS=1220 J/mole and well correlate with the ionization energy for these metals. This does not mean that the linear dependence will still exist at very high temperatures close to the critical temperature, because λ_ex and λ_T tend to zero as the temperature approaches the critical value; hence, Δλ also tends to zero. The mechanism of λ_ex reduction at high temperatures is most probably related with the fact that an exothermic process of dissociation of two-atomic molecules starts. Unfortunately, there are no experimental data for calculating λ_T and no information about the behavior of λ_ex and the part of two-atomic molecules in gas at high temperatures close to the critical temperature.

Figure 2: Dependence of Δλ=λ_ex-λ_T on the concentration of two-atomic molecules in gas for alkali metals.

In Figure 3 one can see a dependence of Δλ on temperature for mercury. A linear extrapolation of Δλ=f (T) turns Δλ to zero at a point close to the value of T_c.

Figure 3: Dependence of Δλ=λ_ex-λ_T on temperature for mercury.

We may assert that suggested expression (6), which only comprises measurable parameters, allows one to calculate the evaporation heat of liquid gases with a high accuracy taking into account considerations of Sect. 4, and the evaporation heat of liquid metals.