According to experimental data, the first ionization energy (PIE) of a hydrogen molecule is 1.494 kJ/mol. As a result of the gap of an electron with a hydrogen molecule, a positive hydrogen ion (H 2 +) is formed. In order to compare the calculated data with the experimental data, we need to calculate the energy of the positive hydrogen ion according to the same scheme that we used to determine the energy of the hydrogen molecule. When using this scheme, we come to the conclusion that the energy of a positive hydrogen ion is equal to the energy of not a helium-like, but a hydrogen-like atom with a charge Z equal to the reduced charge at point E, while Z can be calculated using the following formula:
Z = (N 2 /2n) [(4n/N) 2/3 - 1] 3/2 - S n ,
where N is the nuclear charge in proton units; n is the number of bonding electrons; S n - designation of interelectronic repulsion. In the case of one electron (H 2 +) S n is zero. A detailed proof of this formula is given in the monograph.
When calculated using this equation, we find that:
Z = (1 2 /2) [(4/1) 2/3 - 1] 3/2 = 0.5 (40.666 - 1) 1.5 = 0.93
Accordingly, the energy of H 2 + is determined by the formula:
E H2 + = 1 317 . 0.932 = 1 150 kJ/mol
The H 2 + molecule can be represented as a molecule formed from a hydrogen atom and a proton. The total electronic energy of the initial components is equal to the PIE of the hydrogen atom, i.e. 1317 kJ/mol. That is, according to calculations, the formation of an H 2 + ion does not release energy, but, on the contrary, it loss value of 167 kJ/mol. Thus, according to the calculations, the H 2 + molecule is extremely unstable. [This fact is mentioned in the Encyclopedia of Inorganic Chemistry (1994) on page 1463.] Accordingly, when one electron is removed from a hydrogen molecule, it decays into a hydrogen atom and a proton. The total energy in this case is 1317 kJ/mol. Thus, the experimentally calculated electronic energy of the hydrogen molecule (E H2) is determined by the formula:
E H2 = 1317 kJ/mol + 1494 kJ/mol = 2811 kJ/mol,
where 1.317 kJ/mol is the energy value of the hydrogen atom and 1.494 kJ/mol is the PIE of the hydrogen atom (FIE H 2). The energy of the hydrogen molecule, calculated using the equations, was 2.900 kJ/mol. The discrepancy between the experimental and calculated data was 3.06%.
Thus, (2.900 kJ/mol - 2.811 kJ/mol) / 2.900 kJ/mol = 0.0306. That is, the value of the energy of the hydrogen molecule, calculated using the equations, turned out to be 3.06% higher than the value obtained using experimental data.
As already mentioned in this section, the energy of a hydrogen molecule can be calculated in the same way as the energy of a helium-like atom (a nucleus surrounded by two electrons). Based on the calculation for helium-like atoms, we obtain:
E gel = 1.317 (Z - 0.25) 2 2
The energies of helium-like atoms with nuclear charges equal to 1, 2 and 3 proton units were 1.485; 8.025 and 19.825 kJ/mol, respectively. For comparison, the experimentally calculated energy of these atoms (the sum of the ionization energies of H¯; He; and Li +) was 1.395; 7.607 and 19.090 kJ/mol, respectively.
In other words, the experimentally calculated energy values for H¯; He; and Li + turned out to be less than the calculated data by 6.1%; 5.2% and 3.7% respectively.
As noted above, the experimentally determined value of the energy of the hydrogen molecule turned out to be 3.06% less than the value calculated on the basis of the model, which quite convincingly proves that the model is absolutely accurate.
Hydrogen ion energy H 2 +
From formula (66.2), which combines both Faraday's laws, it follows that if the charge is numerically equal to the Faraday constant, then the mass is equal to, i.e. when a charge equal to 96,484 C passes through the electrolyte, [kg] of any substance is released, i.e. praying for this substance. In other words, to release one mole of a substance, a charge must flow through the electrolyte, numerically equal to [C]. Thus, during the release of a mole of a monovalent substance (1.008 g of hydrogen, 22.99 g of sodium, 107.87 g of silver, etc.), a charge numerically equal to C passes through the electrolyte; when a mole of a divalent substance is released (16.00 g of oxygen, 65.38 g of zinc, 63.55 g of copper, etc.), a charge numerically equal to C passes through the electrolyte, etc.
But we know that one mole of any substance contains the same number of atoms, equal to the Avogadro constant 
mol-1. Thus, each ion of a monovalent substance released at the electrode carries a charge
Cl. (69.1)
With the release of each atom of a divalent substance, a charge passes through the electrolyte 
C, twice as large, etc. In general, when each atom of a -valence substance is released, a charge [C] is transferred through the electrolyte.
We see that the charges transferred during electrolysis with each ion are integer multiples of some minimum amount of electricity equal to C. Any monovalent ion (potassium, silver, etc.) carries one such charge. Any divalent ion (an ion of zinc, mercury, etc.) carries two such charges. Cases never occur during electrolysis when a charge containing a fractional part of C is transferred with the ion. The German physicist and physiologist Hermann Helmholtz (1821-1894), who drew attention to this consequence of Faraday's law, concluded from this that the indicated amount of electricity Kl represents the smallest amount of electricity that exists in nature; this minimum charge is called the elementary charge. Monovalent anions (ions of chlorine, iodine, etc.) carry one negative elementary charge, monovalent cations (ions of hydrogen, sodium, potassium, silver, etc.) - one positive elementary charge, divalent anions - two negative elementary charges charge, divalent cations - two positive elementary charges, etc.
Thus, in the phenomena of electrolysis, researchers for the first time encountered manifestations of the discrete (discontinuous) nature of electricity (§ 5) and were able to determine the elementary electric charge. Later, other phenomena were discovered in which the discrete nature of electricity is manifested, and other ways were found to measure the elementary negative charge - the charge of the electron. All these measurements gave the same value for the electron charge as we just got from Faraday's law. This is the best confirmation of the correctness of the ionic mechanism for the passage of current through electrolytes, which we outlined in the previous paragraph.
Ions are usually denoted by the signs "+" or "-" near the corresponding formulas (usually at the top right). The number of signs "+" or "-" is equal to the valency of the ion (for example, copper ions are or, chlorine ions are only, etc.).
§ 79. Charge and mass of the ion.
From what has been said in the previous paragraphs, it follows first of all that the charges carried by positive and negative ions, being opposite in sign, must be identical in absolute value, since they are formed, generally speaking, by splitting neutral molecules of a substance. The first quantitative determinations of quantities that make it possible to judge the mass of ions of various categories were made by J. J. Thomson and W. Wiiom, and the first approximate determinations of the charge of an ion were made by J. J. Thomson.
The main series of studies were devoted to determining the ion charge ratio e to its weight m. In one of the methods used by J. J. Thomson in 1897, he operated on the so-called cathode rays, discovered by Crookes and consisting of a stream of some very peculiar particles that carry negative charges. As is known, cathode rays were observed by Crookes in a very clearly expressed form inside a glass vessel with a very rarefied space in which two electrodes were located: a flat or slightly concave cathode and an anode of some kind. With a sufficiently high potential difference between these electrodes, the aforementioned cathode rays, which have a number of special properties, emanate from the surface of the negative electrode, approximately perpendicular to it. A beam of cathode rays is deflected by the action of a transverse magnetic field, which can be detected using either the fluorescence of gas residues in the tube, or the fluorescence of a special screen on which the rays fall. The same deviation can be obtained by passing cathode rays between the plates of the capacitor, located
placed inside the tube and charged from some constant source. In both cases, the direction of deflection corresponds exactly to the negative electrification of the particles that form the cathode rays. Similar observations can be made, for example, using a tube with a very rarefied gas, shown in Figure 132.
Here C is the cathode, BUT - an anode with a gap of about 2 - 3 millimeters, AT - a metal disk connected to the ground and having a gap about one millimeter wide, D 1 and D 2 - capacitor plates, F - fluorescent screen deposited on the inner surface of the glass tube. The cathode rays emanating from the surface of the cathode C pass through the slots in BUT and AT in the direction OR and give a luminous trace on the screen R. Imagine now that the tube is located in a uniform magnetic field perpendicular to the plane of figure 132, i.e., perpendicular to OP. In this case, the cathode beam will turn from a straight line into a curved one. (OR") along the arc of a circle whose radius will depend on the magnetic induction AT, from charge e particles forming cathode rays, on their mass t and from their speed v. Indeed, the radius of curvature of the ion trajectory will be determined by the condition of equality in absolute value of the centrifugal force, on the one hand, and the force deflecting the particle to the center of curvature, on the other hand. The centrifugal force will be mv 2 /r. deflecting particle
force will be equal to the product of magnetic induction AT and quantities ev, which is nothing more than a measure of the strength of the current due to the movement of the charge e with speed v (the angle between the direction of the vector AT is equal in this case to 90°). Therefore, we can write:
mv 2 / r=Bev.
On the other hand, informing the plates D 1 and D 2 some potential difference, we can also cause the deflection of the cathode beam by applying a transverse electric field to the moving charged elements of the beam. Denoting the electrical force between the plates D 1 and D 2 through E, we can express the mechanical force of this action on each individual particle through Her. In this case, the sign of the potential difference between the plates D 1 and d 2
can be taken such that the deflecting actions on the cathode beam from the electric and magnetic fields are opposite to each other. By setting some specific value of electric force E, we will then change the magnetic induction accordingly AT and in this way we can achieve the elimination of the deviation of the cathode beam, which can be judged by the return of the fluorescent trace of the beam to the point R. When this is achieved, we shall be free to write:
Her=Vev.
Considering the value AT, thus selected, and combining the obtained two ratios, we get:
The magnitude of the charge itself e was, as we shall see below, directly determined from other observations.
Attitude e to m and speed value v were obtained by J. J. Thomson and another method, in which, among other things, the magnitude of the amount of negative electricity carried by a certain portion of the cathode stream was determined by the Perrin method (Fig. 133).
It is in the path of the cathode beam emanating from the negative electrode C that a hollow metal cylinder is located AT with a hole in the bottom facing electrode C. This cylinder AT very carefully insulated and placed inside a protective metal chamber to prevent any kind of electrical influences BUT, playing the role of the anode at the same time. Cylinder AT attached to a specially calibrated electrometer, with which you can measure the electric charge acquired by the cylinder. As Perrin showed, the cathode beam, getting inside the cylinder AT, charges it with negative electricity, and the magnitude of this charge under given unchanged conditions is strictly proportional to the time during which the cathode beam acts. Making experience for some
a certain period of time, J. J. Thomson measured the charge Q, acquired during this time by the cylinder AT. Denoting through N the number of negative electricity carriers entering the cylinder AT, we get:
Ne= Q.
Then J. J. Thomson measured the amount of kinetic energy that these N particles, causing the same cathode beam to fall on a specially made thermocouple in the same time interval, located for this purpose in the path of the cathode beam, instead of a cylinder AT, and graduated like a calorimeter. Denoting through W the amount of energy acquired by a calorimetric thermocouple due to bombardment of it N particles with a mass m each and rushing with speed v, and assuming that the kinetic energy of each particle is completely converted into heat when it hits the surface of the thermocouple, we obtain the second relation:
1 / 2 Nmv 2 =M.
Finally, making the experiment described above with the deflection of the cathode beam by a magnetic field, we add the third relation:
mv 2 / r= Bev.
From these three ratios, you get:
Thus, J. J. Thomson could determine the charge-to-mass ratio and the velocity of the particles that make up the cathode beam in various ways. Speed value v over a wide range depends on the potential difference applied to the electrodes of the tube. Under the operating conditions of J. J. Thomson at voltages up to 10,000 volts and slightly higher, v reached 3.6 10 9 centimeters per second, i.e., to a value somewhat exceeding one tenth of the speed of light. As for the magnitude of the ratio e/ m, then completely independent of any attendant circumstances (voltage, nature of the gas in the tube, the substance of the negative electrode, etc.), this ratio turns out to be invariably of the same order. J. J. Thomson obtained in the experiments described:
e/ m= about 10 7 in abs. el.-mag. units.
We now know, from the results of later, more advanced experiments, that a more precise value for this ratio should be:
e/ m\u003d 1.76 10 7 in abs. el.-mag. units.
The indicated small discrepancy, explained by a number of sources of errors in the initial experiments, however, has no significant significance in substantiating those extremely important and fundamental conclusions that J. J. Thomson arrived at when analyzing his results. In this regard, it is only necessary to know the order of magnitude - , and J.J. Thomson determined it quite accurately, and then compared the obtained value with what is obtained for the charge-to-mass ratio in the case of ordinary material ions. He calculated that in the case of the lightest ion that we deal with when passing current through electrolytes, namely in the case of a hydrogen ion, the ratio of interest to us will be about 10 4 (its more accurate value is 0.96 10 4). As we shall see later, J. J. Thomson showed that the magnitude of the charge of the elements of the cathode beam and of the electrolytic ions must be recognized as the same. From this he deduced the conclusion that the mass of the cathode flux particle is many times (more than a thousand times) lighter than the lightest atom, the hydrogen atom. We now know that the mass of a hydrogen atom is approximately 1840 times that of electron, which name, proposed by Johnston Stoney, finally established itself in science to designate those carriers of negative electricity, which we meet, generally speaking, always in the case of current passing through gases and emptiness. The greatest merit of J. J. Thomson lies precisely in the fact that he was the first to establish the basic physical characteristics of the lightest material particles, which are carriers of the smallest electric charge that we encounter in experience. These lightest particles, whose mass is 1840 times less than the mass of a hydrogen atom, we now consider with good reason as atoms of electricity. A careful theoretical and experimental study of the question of the mass of an electron shows that it is not constant, but turns out to be a function of velocity. Denoting the mass of an electron moving slowly compared to the speed of light, through m 0 , based on the latest experience, we can accept:
where v is the speed of the electron, and with - the speed of light, we can theoretically substantiate the following expression for the mass of an electron moving at a speed v:
As a result, the idea arose electromagnetic nature of the electron mass.
Of great interest is the comparison of values - for electron and for positive gas ions, and for this purpose you can use the results of the experiments of V. Wien, who determined this ratio in the case of positive ions forming the so-called sunset rays, first observed by Goldstein. If an electric discharge occurs between some anode and cathode in a highly rarefied gas, and the cathode consists of a metal plate with a large number of small holes, then behind the cathode, i.e., from the side opposite to the anode, very weakly luminous beams are observed that penetrate through the holes and causing a noticeable fluorescence of the glass at the place of their fall on the walls of the vessel. Wien showed, firstly, that Goldstein's sunset rays consist of positively charged ions, which acquired very high velocities in the electric field on the other side of the cathode and, due to this, were able, so to speak, to slip through the holes by inertia. By exposing a beam of sunset rays to an electric and magnetic field and using the same method that was described above for cathode rays, Win
could determine the value - for sunset rays and received: e/ m= about 300 in abs. el.-mag. units
v - about 3 10 7 centimeters per second.
So, the speed turned out to be 100 times less than the speeds observed for electrons in conditions of similar electric fields. Since, further, there is no doubt that the charges carried by both positive and negative ions in gases must be identical, then, obviously, the mass of positive ions in Wien's experiments turned out to be approximately 30,000 times greater than the mass of an electron. For reference, we can indicate that for iron during the electrolysis of solutions of iron salts, we get
e/ m= about 400.
In other words, positive gas ions have masses of the same order as heavy electrolytic ions, that is, they are one or another, sometimes very heavy, combinations of ordinary atoms and molecules of matter.
Turning now to the question of the charges carried by gaseous ions, let us first dwell on the work of J. J. Thomson, who was the first to determine the charge of the electron. He took advantage of the property of water vapor to condense around ions and form fog droplets. This property was discovered by Wilson, who showed that in the case of adiabatic expansion of saturated water vapor in the presence of gas cones, fog arises and at a lower degree of expansion than is required if the air does not contain ions at all. Wilson found that in air purified from dust and free from ionization, saturated water vapor produces fog only when the sudden increase in gas volume is not less than 1.38 times. When expanding by 1.25 times, fog is formed only in the presence of negative ions that condense water droplets on themselves. This is also observed with a further increase in the degree of expansion up to the limit equal to 1.31, upon reaching which water and positive ions begin to condense. With a degree of expansion from 1.31 to 1.38, water vapor will condense on ions of both signs. Starting with an expansion of 1.38 times, the formation of fog occurs, as mentioned above, regardless of the presence of ions. J. J. Thomson ionized air saturated with water vapor using X-rays, and then produced an adiabatic (almost very fast) expansion of it by a factor of 1.25. A cloud of fog, formed from droplets condensed around negative ions, falls under the action of gravity, and, using the relationships given by Stokes, it was possible to determine the size and mass of individual droplets from the rate of fall. J. J. Thomson calculated the total amount of condensed water based on thermodynamic data and divided it by the mass of a single droplet. In this way the number of all the droplets that made up the mist was determined. To obtain the value of the total charge carried by a combination of negative ions participating in the formation of fog, an electric field was applied, under the action of which ions of the same sign settled on an electrode connected to a specially calibrated electrometer. By dividing this total charge by the number of droplets, J. J. Thomson obtained the charge of each ion. And in this case, his great achievement was a fairly accurate determination of the order of magnitude of the charge of a gas ion. Namely, he got:
e= about 4 10 -10 abs. el.-stat. units.
J. J. Thomson compared this amount of electricity with the charge of an electrolytic ion, for example, hydrogen. If a N is the number of molecules per cubic meter. centimeter of hydrogen at a pressure of 760 mm mercury column and at a temperature of 0 ° C, and e is the charge of the hydrogen ion, which we deal with in the electrolysis of solutions, then on the basis of direct experiments we can put:
Ne"= 1.22 10 10 abs. el.-stat. units.
1,29 10 -10 <e"< 6,1 10 -10 ,
whence it follows that the charge carried by the gas ion is equal to the charge possessed by the hydrogen ion during the electrolysis of solutions. This result of the classical experiments of J. J. Thomson is fully justified by the totality of modern data, which undoubtedly testify that in the most diverse cases we invariably meet with the same elementary electric charge. Later and more advanced observation methods made it possible to very accurately (with an accuracy of four decimal places) determine the magnitude of the charge e. In this regard, the experiments of Millikan, who observed the behavior of individual tiny droplets of oil and mercury, charged by a very small number of ions, are of particular importance. Determining the charges of the droplets, Millikan found that they invariably turn out to be multiples of some specific amount of electricity. (e) and thereby showed by direct experience the atomic nature of electricity. Currently value e, obtained by Millikan is considered very reliable and, therefore, on the basis of his research, they accept:
e=4.774 10 -10 abs. el.-stat. units = 1.592 10 -20 abs. el.-mag. units.
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