Physical and Mathematical Society of High School Students - Problems of increased complexity *. Atomic and Nuclear Physics

Radiation and absorption of energy by atoms. The structure of the energy levels of atoms. Optical spectra of a hydrogen atom and spectra of complex atoms.

The structure of the energy levels of complex molecules. Molecular Spectra.

Emission and absorption spectral analysis, its medical use. Spectroscopes, spectrographs, monochromators, spectrophotometers and their use in medicine.

Luminescence, its types. Characteristics of luminescence (spectrum, duration, quantum yield). Laws Vavilova and Stokes. Luminescent analysis. Fluorescent tags and probes. Medical use of fluorescent research methods.

Radiation and absorption of energy by atoms and molecules.

In 1913, Bohr proposed the theory of light emission, which is based on two postulates.

The internal energy of an atom is discrete, that is, it can take only certain permitted values ​​or levels that are multiples of quantities characteristic of a given atom, or energy quanta. The atomic states corresponding to these energy levels are stationary: in this state, the atom does not emit electromagnetic waves, despite the movement of electrons in it.

The emission (or absorption) of electromagnetic radiation occurs when an atom transitions from one stationary state to another. When this is emitted (or absorbed) photon of monochromatic radiation with an energy equal to the difference in energy levels E m   and E n  corresponding to these conditions:

h   = E m   - E n ,

where E m  and E n - energy of the system in the first and second states.

On the basis of these postulates, Bor developed a theory of the emission and absorption of light energy by a hydrogen atom. He suggested that, of all possible electron orbits, only those are performed for which the angular momentum (angular momentum) is equal to an integer multiple of Planck’s constant divided by 2  :

(n= 1, 2, 3, . . .) . (1.)

Number n, is called the principal quantum number, corresponds to the ordinal number of the orbit.

In the case of a hydrogen atom, the Coulomb force of mutual attraction of a proton and an electron is the centripetal force holding the electron in orbit, that is:

, (2)

where mand   e- mass and electron charge, r- radius of the orbit.

Excluding vfrom (1) and (2), we obtain that the radius of the electron orbits in an atom can take only a series of discrete values:

of (1)

substitute in (2)

from where

,

(n= 1, 2, 3, ...).

Thus, the radii of the orbits of stationary hydrogen atoms are directly proportional to the square of the quantum number   n 2 .

For the first main orbit n= 1 and radius

, that is, the magnitude of the order of the gas-kinetic size of the atom.

Of the relationships considered, the electron velocity in a stationary orbit is also found.

. For the main orbit of the hydrogen atom ( n= 1):

2.310 8 cm / sec. This is the order of the velocity of the electron in its orbit.

The energy level of the atom is due to the total energy of the electron, which is composed of the kinetic energy of the electron (the nucleus is stationary) and the interaction energy of the electron with the nucleus (potential energy). Potential energy of an electron (in sign it is negative as the potential energy of attractive forces):

Total internal energy:

E Uh =

considering that

,

we find: E Uh =

Thus, the energy of an electron bound in an atom with a nucleus is negative. The energy of a free electron is zero.

Substituting the values ​​of the electron velocity into this expression, we find:

E Uh =

(3)

For the hydrogen atom, the following values ​​are obtained:

main (zero) level,   n= 1, Е 1 = - 13.55 eV;

n= 2, Е 2 = - 3.38 eV;

n= 3, Е 3 = - 1.5 eV.

The diagram of the energy levels of the hydrogen atom is shown in the figure.

E electron = 0

Fig. 1. The structure of the energy levels of the hydrogen atom.

Since the energy levels are inversely proportional to the square of the quantum number n 2   , the difference between every two adjacent levels as the number increases and in absolute value decreases. Thus, as the distance from the nucleus increases, the difference between two adjacent atomic energy levels decreases:

E 2   - E 1   \u003e E 3   - E 2   \u003e E 4   - E 3 ,. . .

The stationary level with the lowest energy is called the ground level; it corresponds to the state of the atom, which is not exposed to any external influences. The remaining stationary levels are called excited.

The excitation of an atom, that is, the transition of an electron to a larger radius orbit (Fig. 1., path 1), requires the message of additional energy to the atom and, therefore, occurs as a result of any external influences, for example, when particles collide during intense thermal motion or during electrical discharge in gases, when a photon is absorbed by electromagnetic radiation, as a result of recombination of ions in a gas or electrons and holes in a semiconductor, under the action of radioactive particles on an atom, etc.

The excited state is unstable, after about 10-8 seconds, the electron returns to the main orbit, while a photon is emitted, which carries away the additional energy produced by the excitation, and the atom transfers to the ground state (Fig. 1., path 2).

An electron can return to the main orbit not only by a single transition, but also by steps through intermediate levels. In this case, during the transition several photons will be emitted (Fig. 1., path 3) with frequencies corresponding to the difference of the energies of these levels.

The dependence of the amount of energy emitted by atoms or molecules on the wavelength or frequency of the light wave is called emission spectrum , and absorbed - absorption spectrum . The intensity of the spectral lines is determined by the number of identical transitions occurring per unit time, and therefore depends on the number of emitting (absorbing) atoms and the probability of the corresponding transition.

Atomic spectra are called as emission spectra, and absorption spectra, which arise during quantum transitions between the levels of free or weakly interacting atoms. Atomic spectra are linearly.

Advanced tasks *. Physics of the atom and nucleus.

1 ) The radius of the first orbit in a hydrogen atom r  1 = 5.3-10 -11 m. Find the electric field of the nucleus at this distance and the kinetic energy of the electron in this orbit.

Decision:

The electric field will find the formula

where

After substituting numeric values ​​we getE = 5.1 · 10 11 V / m.

To find the second parameter, we write Newton's second law, taking into account the fact that the Coulomb attractive force acts on the electronwhich tells him centripetal acceleration  Where   v- the speed of the electron. So, we get the equation

Therefore, the kinetic energy of an electron


Answer:   W  k = 13.6 eV.

2) How many times will the radius of the electron orbit at the hydrogen atom, which is in the ground energy state, is absorbed by the atom of a photon with an energy of 12. 09 eV?

Decision:

According to Bohr's theory, the total energy of a hydrogen atom is equal to the sum of the kinetic energy of an electron and the potential energy of an electrostatic interaction between an electron and a proton of a nucleus, and in any energy state

In basic condition   When a photon is absorbed, a hydrogen atom goes into a state with greater energy. On the other hand,   Where r  n   - the radius of the electron orbit.

Then

The radius of the electron orbit will increase by 9 times.
Answer: 9 times.

3) Initially, the unexcited hydrogen will start emitting photons if a beam of electrons passing through the potential difference is passed through it  and 0 —10. 2 V. What is the minimum accelerating potential difference that a proton beam must pass so that when passing them through an initially unexcited hydrogen, the latter begins to emit photons? Assume that the electron mass is much less than the proton mass, and the hydrogen atom before the collision rested.

Decision:

Let the mass of the incident particle  t, her speed v  0, mass of hydrogen atom Mparticle velocity after collision  v 1 and   v 2 . According to the law of conservation of momentum .   In a collision, a hydrogen atom absorbs energy W n, h,  so according to the law of conservation of energy

We obtain the system of equations:

Let us prove that the energy of the incident particle will be minimal if, after the collision, the particle and the hydrogen atom move with the same speeds (absolutely inelastic collision). Express from the first equation of the system   and substitute in the second equation, then we get:

Thus, the energy of the incident particle   W 0  expressed as a quadratic function of v  1 with the first positive coefficient

Such a function has a minimum at the point where So, the speed corresponding to the minimum  W 0:

Wherein

So, it is proved that the energy of the incident particle is minimal if after a collision the particles move with the same speed, i.e. v 1 =v  2 The minimum energy of the incident particle

  In the case when electrons are passed through hydrogen,   so   those. all the energy of electrons is absorbed by hydrogen, which goes into the excited state. Electrons acquire energy by passing an accelerating potential difference. and  0, i.e.

In the case when protons are passed through hydrogen, the ratiot / m = 1, therefore the minimum energy of the incident protons

Thus, the minimum accelerating potential difference for protons is equal to 20.4 V.
Answer: 20.4 V.

4) What spectral lines will appear when atomic hydrogen is excited by electrons with energy  W=12. 1 eV?

Decision:

All the energy of electrons is absorbed by hydrogen, which is excited and passes from the ground energy state withp = 1 to some state characterized by a natural number k . According to the law of conservation of energy   On the other hand: where   W and he= 13. 6 eV. Get the equation

Ie state with   k= 3 direct transition to the state withn = 1 or n = 2, as well as the transition from the second energy level to the first. Thus, we get three spectral lines. Calculate the corresponding wavelengths using the formula

Transition from state to   k = 3 to state with n = 1 corresponds to a spectral line with a wavelength

Transition from state to k= 3 to state with n=2   corresponds to the spectral line with a wavelength

Transition from state to  k = 2 to state with n= 1 corresponds to the spectral line with a wavelength

5) Electric field strength in an electromagnetic wave with a frequency   amplitude modulated frequency   changes over time according to the law:  Where BUT- constant. Determine the energy of electrons knocked out by this wave from gaseous hydrogen atoms with ionization energy  W ion = 13.6 eV.

Decision:

We transform the expression for the electric field strength:

Thus, the amplitude-modulated wave is the sum of three monochromatic waves with frequencies. In accordance with the postulates of Bohr, a hydrogen atom can emit and absorb electromagnetic energy only in certain portions (quanta). Let us calculate the energy quanta corresponding to the monochromatic waves found.

For a wave with a frequency

For a wave with a frequency

For a wave with a frequency

  The ionization energy of a hydrogen atom and, as you can see, it is greater than  Therefore, quanta with frequencies   cannot ionize a hydrogen atom. Ionization is caused only by quantum with frequency  and according to the law of conservation of energy, the energy of the electrons knocked out by this quantum

Answer:W e = 0.88 eV.

6) Three elements are located side by side in the periodic table.   X, Y, S.   Radioactive isotope element Xturns into an element Have, and that, in turn, - in the element   S. The latter is converted into an isotope of the original α -decay.

7) Neptunium core after α   and β -decay turns into a bismuth nucleus. What number α   and β -decay occurs during this?

Decision:

As a result of decays, the mass number changes by 237 - 209 = 28 atomic units. As you know, when β - the decay of the mass of the nucleus does not change, therefore, the change in the mass number occurs only due to α decays. With one α - the decay of the mass number changes to 4 atomic units, which means that 28/4 = 7 should occur α decays. In this case, the nuclear charge should decrease by 7 · 2 = 14 elementary charges. In our case, the decrease in the nuclear charge is 93 - 83 = 10. Therefore, as a result β -decay charge increased by 14 - 10 = 4 elementary charges. As with one β - the decay of the nuclear charge is increased by 1 elementary charge, then 4/1 = 4 should occur β -decay.

Bohr's postulates. Atomic emission and absorption

  A 1 What is the energy of a photon emitted during the transition of an atom from an excited state with energy to the ground state with energy? 1) 2) 3)4)   A 2   What is the energy of a photon absorbed during the transition of an atom from the ground state with energy to an excited one with energy? 1) 2) 3)4)   A 3   The frequency of a photon emitted during the transition of an atom from an excited state with energy to the ground state with energy is calculated by the formula 1) 2) 3) 4)   A 4   The frequency of a photon absorbed by an atom during the transition of an atom from the ground state with energy to an excited one with energy is equal to 1) 2) 3) 4)   A 5   The wavelength of a photon emitted by an atom when an atom transitions from an excited state with energy to the main one with energy is 1) 2) 3)4)   A 6   The wavelength of a photon absorbed by an atom during the transition of an atom from the ground state with energy to an excited one with energy is equal to 1) 2) 3)4)   A 7   When a photon is emitted with an energy of 6 eV, the atomic charge   1) does not change   2) increases by   3) increases by   4) decreases by   IN 1   Find the change in the energy of the hydrogen atom when it emits waves of frequency. Answer round to two significant digits, multiply by.   AT 2   How much has the energy of an atom changed when it emitted a photon wavelength? Answer round to two significant digits, multiply by.   IN 3   How much has the energy of an atom decreased when it emits a photon wavelength? Answer round to two significant digits, multiply by.   A 8   Resting atom absorbed photon with energy. In this case, the momentum of the atom   1) has not changed   2) became equal   3) became equal   4) became equal   A 9   An atom emitted a photon with energy. What impulse has an atom gained? 1) 0 2) 3) 4)   A 10   What is the momentum produced by an atom when a photon is absorbed by a frequency? 1) 2)3) 4)   A 11   The electron of the outer shell of an atom first passes from a stationary state with energy to a stationary state with energy, absorbing a photon with a frequency. Then it goes from a state to a stationary state with energy, absorbing a photon with a frequency. What happens when an electron transitions from a state to a state?   1) Light emission frequency 2) Light absorption frequency 3) Light emission frequency 4) Light absorption frequency   A 12   The emission of photons occurs during the transition from excited states with energies to the ground state. For the frequencies of the corresponding photons,, the ratio 1) 2) 3) 4)   A 13   The figure shows a diagram of the energy levels of the atom. Which of the transitions between energy levels marked by arrows is accompanied by the emission of a quantum of the minimum frequency? 6 5 4 3 2 1 1) 1 2) 4 3) 6 4) 7   A 14 The figure shows a diagram of the energy levels of the atom. Which of the transitions between energy levels marked by arrows is accompanied by the absorption of a quantum of the minimum frequency? 7 6 5 4 3 2 1 1) 2 2) 3 3) 4 4) 7   A 15   The figure shows a diagram of the energy levels of the atom. Which of the transitions between energy levels marked by arrows is accompanied by the emission of a quantum of minimum wavelength? 7 6 5 4 3 2 1 1) 1 2) 4 3) 64) 7   A 16   The figure shows a diagram of the energy levels of the atom. Which of the transitions between energy levels marked by arrows is accompanied by the absorption of a quantum of the minimum wavelength? 7 6 5 4 3 2 1 1) 2 2) 3 3) 64) 7   A 17   How many photons with different frequencies can emit hydrogen atoms in the second excited state? 1) 1 2) 2 3) 34) 4   A 18   Atoms of a gas can be in three states with energies: -2.5 eV, -3.2 eV, -4.6 eV. If they are in a state with an energy of 3.2 eV, then photons, what energy can the atoms of this gas emit?   1) Only 0.7 eV   2) 1.4 eV and 0.7 eV   3) 2.5 eV, 3.2 eV, 4.6 eV   4) Only 1.4 eV   A 19   The figure shows a diagram of possible values ​​of the energy of a gas atom. Atoms are in a state with energy. What energy photons can contain light emitted by a gas?   1) Only 2 eV 2) Only 2.5 eV 3) Any, but less than 2.5 eV 4) Any in the range from 2.5 to 4.5 eV   A 20

  Suppose that atoms of a certain gas can only be in states with energy levels shown in the figure. At the initial time, the atoms are in a state with energy E  2 According to Bohr's postulates, the light emitted by such a gas can contain photons with energy

1)   only 3 × 10 –19 J 2)   only 4 × 10 –19 J 3)   only 2 × 10 –19, 5 × 10 –19 and 9 × 10 –19 J 4)   any from 2 × 10 –19 to 9 × 10 –19 J   A 21   The figure shows a diagram of possible values ​​of the energy of a gas atom. Atoms are in a state with energy. What is the energy of photons of light emitted by a gas?   1) Only 0.5 eV 2) Any, but less than 0.5 eV 3) Any in the range of 1.8 to 0.5 eV 4) Only 0.5 eV, 0.8 eV and 1.3 eV   AT 4   How many quanta with different energies can emit atoms that are in a state with energy?   A 22   Suppose that the scheme of energy levels of atoms of a rarefied gas has the form shown in the figure. At the initial moment of time, the atoms are in a state with energy. According to Bohr’s postulates, this gas can absorb photons with energy   1) 0.3 eV, 0.5 eV and any greater than 0.5 eV 2) only 0.3 eV 3) only 0.3 eV and 0.5 eV 4) any in the range from 0 to 0.5 eV   A 23 1) Anyone between 2 and 2) Any, but less than 3) Only 4) Any, greater or equal   A 24   The figure shows a diagram of possible values ​​of the energy of gas atoms. Atoms are in a state with energy. Photons, what energy can a given gas absorb?   1) photons with any energy in the range from to 2) photons with energy and 3) photons with energy, and any more 4) photons with any energy more   From 1   Suppose that the scheme of the energy levels of atoms of some substance has the form shown in the figure, and the atoms are in a state with the energy E (1). An electron moving with a kinetic energy of 1.5 eV collided with one of these atoms and bounced off, acquiring some additional energy. Determine the momentum of the electron after the collision, assuming that the atom was at rest before the collision. The possibility of the emission of light by an atom in a collision with an electron is neglected.   From 2   Suppose that the scheme of the energy levels of atoms of some substance has the form shown in the figure, and the atoms are in a state with the energy E (1). The electron, having collided with one of these atoms, bounced off, acquiring some additional energy. The impulse of an electron after collision with an atom at rest turned out to be equal to 1.2 × 10 –24 kg × m / s. Determine the kinetic energy of the electron before the collision. The possibility of the emission of light by an atom in a collision with an electron is neglected.   From 3   Suppose that the scheme of the energy levels of atoms of some substance has the form shown in the figure, and the atoms are in a state with the energy E (1). The electron, having collided with one of these atoms, bounced off, acquiring some additional energy. The electron impulse after the collision turned out to be 1.2 × 10 –24 kg × m / s. Determine the electron momentum before the collision. The possibility of the emission of light by an atom in a collision with an electron is neglected. Assume that the atom was stationary before the collision.   From 4   Suppose that the scheme of the energy levels of atoms of some substance has the form shown in the figure, and the atoms are in a state with the energy E (1). An electron moving with a kinetic energy of 1.5 eV collided with one of these atoms and bounced back, acquiring some additional energy. Determine the momentum of the atom after the collision, if the atom was at rest. The possibility of the emission of light by an atom in a collision with an electron is neglected.  , if a , ,