Physical Chemistry

L. N. SIDOROV, P. A. AKISHIN

A MASS-SPECTROMETRIC METHOD FOR DETERMINING PARTIAL VAPOR PRESSURES AND RELATIVE IONIZATION CROSS SECTIONS OF MOLECULES FROM ISOTHERMS OF COMPLETE EVAPORATION

(Presented by Academician P. A. Rehbinder, March 4, 1963)

At present, in order to determine the molecular composition of vapor both over individual compounds and over binary systems, the mass-spectrometric method is being used with great success \((^{1})\).

With the aid of this method it is possible to establish which molecules exist in the gas phase and to estimate the partial vapor pressure of each component. However, in the quantitative determination of partial pressures a number of difficulties arise, associated with the overlap of the mass spectra of the molecules contained in the vapor and with the absence of experimental data on the relative ionization cross sections of molecules of refractory compounds.

The method proposed in the present work makes it possible to carry out a complete interpretation of the mass spectrum, to determine relative ionization cross sections, and to determine the partial vapor pressures over multicomponent systems.

For simplicity let us consider a system consisting of components A and B.

1. Let molecules A and B be present in the gas phase at temperature \(T\), giving, upon ionization, the ions \(\mathrm{A}^+\) and \(\mathrm{B}^+\). The experimentally obtained isotherm of complete evaporation of a charge \(q\), containing \(q_1\) of substance A and \(q_2\) of substance B, is the dependence of the ion currents \(\mathrm{A}^+\)—\(I_1\) and \(\mathrm{B}^+\)—\(I_2\) on time.

As is known, the relation between the ion current \(I\) and the pressure \(p\) is expressed by the equation

\[ p = kI/T \tag{1} \]

To determine the proportionality coefficient \(k\), we shall use the basic calculation equation of the Knudsen method \((^{2})\):

\[ q = ap\sqrt{\frac{M}{2\pi RT}}\,t \]

and transform it to the form:

\[ q = a\sqrt{\frac{MT}{2\pi R}}\,k\int_{0}^{t} I\,dt = Bk\int_{0}^{t} I\,dt \tag{2} \]

by replacing the product \(pt\) by \(\int p\,dt\) and substituting expression (1) for \(p\), where \(q\) is the charge; \(a\) is the Clausing coefficient multiplied by the area of the effusion orifice; \(t\) is time; \(T\) is the absolute temperature; \(M\) is the molecular weight; \(R\) is the gas constant.

Applying (2) to the evaporation process of components A and B, we obtain

\[ q_1 = B_1 k_1 \int_{0}^{t} I_1\,dt, \tag{2a} \]

\[ q_2 = B_2 k_2 \int_{0}^{t} I_2\,dt. \tag{2б} \]

The areas under the curves of the dependence of the ion current on time,

\[ s_1=\int_0^t I_1\,dt \]

and

\[ s_2=\int_0^t I_2\,dt \]

can be found by graphical integration, which makes it possible to calculate the proportionality coefficients \(k_1\) and \(k_2\) from equations (2a) and (2b), and the ratio of the ionization cross sections of molecules A and B from the equation

\[ \frac{\sigma_1}{\sigma_2}=\frac{k_2}{k_1}. \tag{3} \]

Using equation (1), one can calculate the partial vapor pressures of each component for any moment of time during the evaporation process. To pass from the time dependence to the concentration dependence and to construct the composition—partial pressure diagram, we divide the total area \(s_1\) into a series of areas \(s'_1;\ s''_1;\ s'''_1 \ldots s^{(k)}_1\) by straight lines parallel to the ordinate axis. From equation (2) it is seen that the ratio \(\frac{s'_1}{s_1}\) is the fraction of substance evaporated by the time \(t_1\), \(\frac{s'_1+s''_1}{s_1}\) is the fraction of substance evaporated by the time \(t_2\), etc. The amount of substance remaining in the charge by the time \(t_1\) is

\[ q'_1=g_1-\frac{s'_1}{s_1}q_1, \]

and by the time \(t_2\)

\[ q''_1=q_1-\frac{s''_1+s'_1}{s_1}q_1, \]

etc. In an analogous manner, the amount of substance B in the charge at the times \(t_1, t_2 \ldots t_n\) is determined and the composition of the condensate is calculated.

Thus, in the case of a two-component system with a simple vapor composition, the isotherm of total evaporation makes it possible to find the dependence of the partial pressures on the composition of the condensate and to determine the ratio of the ionization cross sections of the molecules present in the vapor. The possibility of using azeotropic mixtures with a known vapor composition to determine relative ionization cross sections has already been considered \((^3)\).

1. Let molecules A, B, and AB be present in the gas phase at temperature \(T\), giving ions \(\mathrm{A}^+\), \(\mathrm{B}^+\), and \(\mathrm{AB}^+\) upon ionization. The experimentally obtained isotherm of total evaporation represents the dependence of the ion currents \(\mathrm{A}^+—I_1\), \(\mathrm{B}^+—I_2\), \(\mathrm{AB}^+—I_3\) on time. In this case the ion current \(\mathrm{A}^+\) has a dual origin: \(I_{13}\), from the molecule AB, and \(I_{11}\), from the molecule A. Similarly, \(I_2=I_{22}+I_{23}\), where \(I_{22}\) is the ion current \(\mathrm{B}^+\) from molecule B, and \(I_{23}\) is the ion current \(\mathrm{B}^+\) from molecule AB.

To interpret the mass spectrum, i.e., to determine the coefficients

\[ a_{13}=\frac{I_{13}}{I_3} \quad\text{and}\quad a_{23}=\frac{I_{23}}{I_3}, \]

it is sufficient, as when using a double effusion chamber \((^{4,5})\), to measure the ion currents at constant temperature and at different pressures in the effusion chamber.

For two different compositions of the condensate we write:

\[ I_1=I_{11}+a_{13}I_3 \tag{4} \]

\[ I'_1=I'_{11}+a_{13}I'_3. \]

System (4) contains three unknowns, \(a_{13}\), \(I_{11}\), and \(I'_{11}\); the third equation is obtained from the condition that the equilibrium constant is independent of pressure at constant temperature:

\[ K_p=\frac{p_1p_2}{p_3}=\frac{p'_1p'_2}{p'_3}. \]

or through ion currents

\[ \frac{I_{11}\cdot I_{22}}{I_3}=\frac{I'_{11}\cdot I'_{22}}{I'_3}. \tag{5} \]

In the region rich in component B, \(I_{22}\cong I_2\) and \(I'_{22}=I'_2\), and equation (5) can be written in the form \(I'_{11}=cI_{11}\), where \(c=\dfrac{I_2\cdot I'_3}{I_3\cdot I'_2}\), a quantity determined experimentally. In an analogous way, a system of equations is drawn up for determining the coefficient \(a_{23}\). With the aid of the coefficients \(a_{13}\) and \(a_{23}\), we find the total ion current from the molecule AB and its complete mass spectrum. On the basis of the results of the interpretation of the mass spectrum, total-evaporation isotherms may be constructed for AB molecules from the sum of the ion currents \(I_3\), \(I_{23}\), \(I_{13}\); for A molecules—from the ion current \(I_{11}\); and for B molecules—from the ion current \(I_{22}\), and the quantities are also found

\[ s_{11}=\int_0^t I_{11}dt,\quad s_{22}=\int_0^t I_{22}dt,\quad s_3=\int_0^t (I_3+I_{23}+I_{13})dt. \]

To determine the connection coefficients \(k_1\), \(k_2\), and \(k_3\), we shall use equation (2) and write it, taking into account the fact that components A and B have partially evaporated in the form of AB molecules:

\[ q_1=q_1^0-\frac{M_1}{M_3}q_3=B_1k_1\int_0^t I_{11}\,dt, \]

\[ q_2=q_2^0-\frac{M_2}{M_3}q_3=B_2k_2\int_0^t I_{22}\,dt, \tag{6} \]

\[ q_3=B_3k_3\int_0^t (I_3+I_{23}+I_{13})\,dt, \]

where: \(q_1^0\) \((q_2^0)\) is the amount of substance A (B) in the initial sample; \(q_1(q_2)\) is the amount of substance that evaporated in the form of A (B) molecules; \(q_3\) is the amount of substance that evaporated in the form of AB molecules. System (6) contains a fourth unknown, \(q_3\). Since, upon complete evaporation at the end of the experiment, the less volatile component A should remain, as the fourth equation we write: \(p_1=k_1I_{11}T\), where \(p_1\) is the saturated-vapor pressure of component A. If the investigated two-component system forms an azeotrope of known composition, then the quantities \(q_1\), \(q_2\), and \(q_3\) are determined from the system of equations:

\[ q_1+q_2+q_3=q_1^0+q_2^0; \tag{7} \]

\[ \frac{q_1+\dfrac{M_1}{M_3}q_3}{q_1+q_2+q_3} = \frac{q_1^0}{q_1^0+q_2^0} \quad\text{— composition of the initial sample;}\quad \frac{q'_1+\dfrac{M_1}{M_3}q'_3}{q'_1+q'_2+q'_3} = N \quad\text{— composition} \]

of the mixture at the end of evaporation (composition of the azeotrope). Solving system (7) gives the values of the quantities \(q_1\), \(q_2\), and \(q_3\). From the equations of system (6), the coefficients \(k_1\), \(k_2\), and \(k_3\) and the ratios of the ionization cross sections are calculated:

\[ \frac{\sigma_2}{\sigma_1}=\frac{k_1}{k_2} \quad\text{and}\quad \frac{\sigma_3}{\sigma_1}=\frac{k_1}{k_3}. \]

The partial pressures are determined from equation (1), i.e.,

\[ p_1=k_1I_{11}T,\quad p_2=k_2I_{22}T\quad\text{and}\quad p_3=k_3(I_3+I_{13}+I_{23})T. \]

The composition of the condensate is calculated according to the scheme set forth in the first section.

1. In an analogous manner, systems in the gas phase over which the molecules \(A_2, A, AB, B\) are present, giving upon ionization the ions \(A_2^+, A^+, AB^+, B^+\), or \(A_2^+, A^+, B^+\), and certain other systems, for example those containing in the vapor the molecules \(A_2, A, B\) and giving upon ionization the ions \(A_2^+, A^+, B^+\), may be considered. In all cases the isotherms of complete evaporation make it possible to decipher the mass spectrum, find the ratio of ionization cross sections, and determine the partial pressures as a function of the composition of the condensate.

Using the method described, we investigated the system \(\mathrm{NaF}—\mathrm{ZrF}_4\)*. The mass spectrum of the \(\mathrm{NaZrF}_5\) molecule, the ratio of ionization cross sections, and the composition–partial pressure diagram are presented below and in Fig. 1.

Mass spectrum of the positive ions of the \(\mathrm{NaZrF}_5\) molecule at a temperature of \(1167^\circ\mathrm{K}\) and an ionizing-electron energy of 80 eV.

Relative ionization cross sections

Fig. 1. Composition—partial pressure diagram at \(1167^\circ\mathrm{K}\):
1 — pressure of \(\mathrm{NaF}\) vapor, 2 — pressure of \(\mathrm{Na}_2\mathrm{F}_2\) vapor, 3 — pressure of \(\mathrm{NaZrF}_5\) vapor, 4 — pressure of \(\mathrm{ZrF}_4\) vapor.

The dissociation enthalpies of the \(\mathrm{NaZrF}_5\) and \(\mathrm{Na}_2\mathrm{F}_2\) molecules were found from the equation for the isobar of the chemical reaction and are \(62.2 \pm 4.6\ \dfrac{\mathrm{kcal}}{\mathrm{mol}}\) and \(54.9 \pm 2.2\), respectively.

In studying the \(\mathrm{NaF}—\mathrm{ZrF}_4\) system it was established that, under conditions of molecular effusion, evaporation into vacuum without a change in composition occurs at a total pressure over the system different from the extremal one. This is explained by the fact that the extremum of the total pressure, in accordance with Konovalov’s law, is observed at a vapor composition coinciding with the composition of the condensate, whereas distillation at constant composition under conditions of molecular effusion is possible only when the composition of the condensate coincides with the composition of the stream leaving the effusion chamber. As is known, for a viscous character of effusion the compositions of the vapor and of the stream coincide, while for molecular effusion they differ in the ratio of the square roots of the masses.

Moscow State University
named after M. V. Lomonosov

Received
28 II 1963

CITED LITERATURE

1. M. Ingram, J. Drowart, High Temperature Studies, Moscow, 1962, p. 214.
2. A. N. Nesmeyanov, Vapor Pressure of the Chemical Elements, 1961, p. 39.
3. A. W. Searcy, W. S. Williams, P. D. Shissel, J. Chem. Phys., 32, No. 3, 957 (1960).
4. L. N. Gorokhov, Vestn. Moscow Univ., ser. Math., Mech., Astron., Phys., Chem., No. 6, 231 (1958).
5. P. A. Akishin, L. N. Gorokhov, L. N. Sidorov, ZhFKh, 33, No. 12, 2822 (1959).

* V. I. Belousov and V. B. Sholts took part in the experimental work.