Physical Model for Vaporization

Journal reference: Fluid Phase Equilibria, 183, 89-92 (2009)

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Abstract

Atomic model is proposed for vaporization. The two fundamental assumption of the model is that the surface layer is flexible, and that the internal energy of the latent heat of vaporization is completely utilized by the atoms for overcoming on the surface resistance of the liquid. Calculating the maximum surface area which covers the atom when breaking through the surface and multiplying this surface with the number of atoms and the surface tension gives the internal energy of the latent heat of vaporization. The model was tested to experiments of 45 elements with positive result.

CONTENTS

1 Text

2 Figure 1

3 Figure 2

4 Figure 3

4 Table 1

Text

The latent heat of vaporization is an extremely important physical process with many applications to physics, chemistry, and biology. Thermodynamic defines the latent heat of vaporization (L_V) as the energy that has to be supplied to the system in order to complete the liquid-vapor phase transformation. At constant pressure the energy is absorbed at constant temperature. The absorbed energy not only increases the internal energy of the system (U) but also used for the external work of the expansion (w). The latent heat of vaporization is then

L_V = Δ_VU + Δ_Vw       (1)

The work of the expansion at vaporization is

Δ_Vw = p (V_V - V_L)       (2)

where p is the pressure, V_V is the volume of the vapor, and V_L is the volume of the liquid.

Several empirical and semi-empirical relationships are known for calculating the latent heat of vaporization ^{1 and refs. therein}. Even though there is no consensus on the exact physics, there is a general consensus that the surface energy must be an important part of the latent heat of vaporization. The vaporization diminishes the surface energy of the liquid; thus this energy must be supplied to the system.

Laplace in his theory suggested that the ratio of the molar total surface energy and the molar internal latent heat is constant with respect to the temperature. Weisskopf ² determined the total surface area by slicing the substance at the linear dimension of one molecule and summing these surfaces. The latent heat of vaporization was calculated by multiplying the total surface area with the surface tension. The model was evaluated by comparing the theoretical atomic diameters to the actual ones

       (3)

where γ_LV is the surface tension at the boiling temperature, and ε_B is the binding energy of the substance contained in one cubic centimeter volume. Subscript L and V refers to liquid and vaporization respectively. The theoretical values of the investigated ten substances agreed reasonable well with experiments.

Agrawal and Menon ³ proposed that the atomic layers are removed one by one from the liquid. They calculated the area of one gram liquid as:

       (4)

where d is the average distance between the molecules and ρ_L is the density of the liquid. The internal energy required to remove all of the layers is then

       (5)

The latent heat was calculated by summing the internal energy [Eq.(5)] and the work required for the volume expansion at the boiling temperature [Eq.(2)]. The calculated latent heat values were compared to experiments of Li, Na, K, Rb, Cs, and water. In order to reproduce the experiments the introduction of a multiplier [f] was necessary. The value of the multiplier is between 4.3 and 7.1.

Analyzing the physical process of vaporization the following model is proposed. Sublimation like process, atoms escaping from the liquid surface, is minor in liquid gas interface because the surface layer of the liquid is flexible. If an inside atom with high energy hits the surface then the first reaction is that the surface absorbs the energy through deformation. The maximum resistance of the deformed surface is achieved when the center of the outgoing atom reaches the plane of the surface as shown in Fig. 1 . Beyond that maximum resistance the area of the surface starts to decrease and eventually leads to the detachment of the atom from the liquid. The extra energy required for an atom to escape from the liquid, is equal with the maximum surface area resistance. Assuming that the surface of the liquid contains one atomic layer and that the liquid is monoatomic as in Fig. 1 then the maximum surface area [A] around an atom is approximated as:

A = 2π(2r)² = 8πr²       (6)

where, r is the radius of the atoms. Multiplying the maximum surface area with the number of moles [n], Avogadro’s number [ N_A], and the surface tension at the boiling temperature gives the internal energy required for the vaporization of the bulk liquid atoms.

Δ_VU = 8n π N_A γ_LV r²       (7)

Equation (7) can be written in a more general form as:

Δ_VU = c γ_LV       (8)

where c is a constant, characteristic of the substance. The latent heat of vaporization is the sum of the internal energy Eq. (7) and the external work Eq. (2). Assuming that the ideal gas law is valid and the volume of the liquid is negligible then the extension work can be calculated as:

Δ_Vw = p V_V(T_B) = nRT_B       (9)

where R is the universal gas constant, and T_B is the boiling temperature in Kelvin. The latent heat of vaporization is then

L_V = n [ 2 π N _A γ _LV(2r)² + RT_B]       (10)

Equations (9) can not be used for high pressures since the ideal gas law is not valid for highly dense gasses.

Equation (8) predicts a linear relationship between the internal energy and the surface tension regardless of the temperature. This predicted temperature independence of the linear correlation was tested to experiments of water with positive result (Fig. 2).

The theoretical relationship in Eq. (10) was tested to the experimental data of monoatomic liquids. The physical properties of the 45 elements used for the investigation are listed in Table 1. Using the experimental latent heat of vaporization values, the radius of the surface area was determined in atomic radius units. The calculated values are reasonably close to the theoretical value of 2r with the exceptions of the elements Si, Ge, and Sn in group 14. The surface radius for these elements is between 2.62-2.88r. Possible explanation for this irregularity could be that the surface of these liquids contains two atomic layers and the maximum surface area could be approximated as:

A = 2 π(3r)² = 18 π r²       (11)

The average of the calculated latent heats is about 10% higher than the experiments. The surface tensions used for the calculations, in many cases, were not measured at boiling but lower temperatures which could explain the higher average of the calculated values. The residuals, the difference between the observed and calculated values are plotted against the atomic radius and the surface tension and given in Fig. 3. Visual inspection indicates only random errors in both cases.

The size of the investigated data set is sufficient for the evaluation of the proposed atomic model. Based on the excellent agreement between the theoretical and experimental values it is suggested that the proposed theoretical model correctly describes the physics of vaporization.

Acknowledgement:

I would like to thank to Jeffrey Joens for reading the manuscript and making comments.

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