Isotope atoms may have some different physical, chemical, geological, and biological properties. In addition, the isotopes are usually present not as free atoms, but in com­pounds, participating in chemical bonds. This means that there are isotope compounds or isotope molecules in which one atom (or perhaps more atoms) is substituted by another isotope. For example, the very simple hydrogen molecule represents six different isotope molecules, which can be written using two different symbolisms:

1H2; lH2H; lH3H
2H2; 2H3H
3H2

and

H2; HD; HT

D2; DT

T2

where D and T mean the isotope of hydrogen with mass number 2 and 3, namely, deuterium and tritium, respectively.

A similar situation exists for oxygen molecules, as follows:

16O2,16O170,16O18O
17O2,17O18O
18O2

The compound of these elements, water, may have 18 different isotope mole­cules. Of course, the relative amount of the isotope molecules is very different, determined by the natural abundance of the isotopes.

The thermodynamic properties of the substances can be characterized by the partition function, combining translation, rotation, vibration, and electron excita­tion. At a constant temperature, the translation energy of the isotope atoms or mole­cules is the same.

The rotation energy (Er) of a diatomic molecule can be expressed by the Schrodinger equation for a rigid rotor:

@2Ф @2Ф @2Ф 8п2р т „

—- :т 1 ——— :т 1 ——— т — 1 ——- ;;— £гФ — 0

@x2 @y2 @z2 h2

The solution of Eq. (3.1) is:

where Ф is the wave function, I is the moment of inertia, ш is the angular speed, L is the moment of impulse, and J is the rotation quantum number. The moment of inertia of a diatomic molecule is expressed as follows:

I — теУ 1 m2r| — ^ (3.3)

where T and r2 are the distance of the center of mass from the atoms with mj and m2 mass and p is the reduced mass, i. e.,

m1 m2

P —

m1 1 m2

The reduced mass can be very different for the isotope molecules, and this differ­ence will affect the chemical properties. For example, the masses of the TH and D2 molecules are very similar, but the reduced masses are rather different: 3/4 and 1 for TH and D2, respectively.

The vibration energy of a diatomic molecule (Ev) can be expressed by the SchrOdinger equation of a harmonic oscillator:

where

where v is the vibration quantum number, ш can be defined as:

А, і

2nc і

In this equation, k is a constant (a spring constant in classical physics), x = r — re, and r and re are the mean and the shortest distance between the two atoms, respectively.

When comparing the ratio of the vibration energies for two isotope molecules, look at the following equation:

Ev1 = /і2

Ev2 І1

The electronic excitation can be characterized by the wave number (v*) of spec­trum lines. It can be described by Moseley’s law. For the hydrogen atom, it is:

where Ma and me are the masses of the nucleus and the electron, n1 and n2 are the main quantum numbers of the electron shells involved in the excitation process, and Ry is the Rydberg constant. As seen, the reduced mass of the atom appears in Eq. (3.9), which may be different when the isotope is not the same because of the different masses of the nuclei.

All expressions of the rotation, vibration, and electronic excitation energies con­tain the reduced masses, which are different for isotope atoms and molecules. This difference in the reduced masses is responsible for the isotopic effects, namely, the different physical, chemical, and other properties of the isotopes and isotope molecules.

1.1.1 Physical Isotope Effects

At a given temperature, the thermal (kinetic) energy of ideal gases is the same, independent of the chemical identity of the gas. So, the kinetic energy (Ekin) of the different molecules of hydrogen isotopes (H, D,T) is:

Ekin = 2 RT = 2 mHvH = 2 movD = 2 mTvT

Since the ratio of the masses of the isotopes is mH:mD:mT = 1:2:3,

. . =,. 1 . 1 V”VDVT = ‘.pf. pf

This difference in the velocity of the isotope molecules influences all the proper­ties involving the movement of gases, for example, diffusion and viscosity.

In gas columns, such as the atmosphere, the isotopes separate because of their different masses. This separation can be calculated by the following barometric formula:

Mgh

ph = po e RT

where p0 and ph are the pressure at the level of a reference level (zero level) and at the height h, respectively, M is the molar mass of the gas, g is the gravitational constant, h is the height related to the reference level, R is the gas constant, and T is the temperature (in kelvin).

For two isotopes/isotope molecules with different mass numbers (Mi and M2):

The partial pressures, of course, are proportional to the concentrations of the iso — topes/isotope molecules.

A similar expression can be deduced for the centrifugation of the isotope mole­cules, substituting g X h with (шг)2, where ш is the angular speed and r is the dis­tance from the rotation axis:

p2 p20 («7-«1)(шг)2

p1 p10

As seen in Eqs. (3.13) and (3.14), the degree of the isotope effects is determined by the difference of the masses. It means that these effects are observed for all iso­topes, including heavy elements. Therefore, the centrifugation can be applied to the

235 238

separation of isotopes of heavy elements, for example, U and U.

In electric and magnetic fields, the charged particles move along a curved path. The deviation from the initial direction is proportional to the specific charge of the moving particle.

In electric fields,

where X is the deviation, k is a constant, E is the strength of the electric field, v, e, and m are the speed, the charge, and the mass of the particle, and e/m is the specific charge (mass-to-charge ratio).

In magnetic fields,

He

Km

vm where Y is the deviation, Km is a constant, and H is the strength of the magnetic field.

The specific charge of isotopes with different masses and the same charge is dif­ferent; therefore, they move along differently curved paths in the same electric or magnetic field. The mass spectrometers utilize this process for determining the mass of particles. Isotopes can also be separated in macroscopic quantities using the deviation from the straight line in electric and magnetic fields.