1 In actuality, the eye's response to different light levels does not behave exactly in a logarithmic way according to the Weber-Fechner law—see 1997 Am. J. Phys. 65 1003 and 1998 Mercury p 8, May/June.
The brightness of a first magnitude star is, therefore, 100 times greater than that of a sixth magnitude star. This is a very useful figure to remember.
Example 15.2. When a telescope is pointed to two stars in turn, the received power is 5-3 x 10-14 W and 3-9 x 10-14 W. What is the difference in apparent magnitude of these stars?
Now the energy received is proportional to brightness. Therefore, by equation (15.11),
Note that, in this case, the magnitude difference (m 1 - m2) is a negative quantity, indicating that m2 > m 1, this being so since B1 > B2.
Example 15.3. At an observing site the brightness of the night sky background per □ " is equivalent to a 21st magnitude star. In the search for a faint object a telescope scans the sky with a field of view limited to 200 □ ". Calculate the equivalent magnitude of the star matching the total effective brightness of the sky background.
The ratio of recorded energy in the experiment to that from 1 □ " is 200:1. Using equation (15.11)
15.7 Spectral lines 15.7.1 Introduction
Soon after the turn of the 20th century, experiments were performed which revealed that atoms had component particles. As a result of the investigations of electrical discharges through gases, a 'radiation' was discovered which caused a fluorescence on the walls of the glass discharge tube opposite the end at which the negative voltage was applied. The radiation was at first given the name cathode rays, as it appeared to emanate from the negative terminal or cathode. Experiments with electric and magnetic fields demonstrated that the rays consisted of negatively charged particles and the name electron was given to them. Determination of the ratio of their charge to their mass (e/me) showed that it was about 1840 times the same ratio (e/M) obtained by Faraday for the hydrogen ion. As the charges on the two types of particle were found to be of the same value (but different signs), it follows that the mass of the electron is only 1/1840 times the mass of the hydrogen ion. It was immediately obvious that with such a low mass, the electron could not take its place in the periodic table of chemical elements and it was suggested that it constituted one of the fundamental parts of an atom. An experiment by Millikan in 1905 gave a measure of electronic charge (1-6 x 10-19 C) and this allowed determination of the mass me of the electron (9-1 x 10-31 kg).
In a further experiment with the discharge tube, Goldstein punctured the cathode with small holes and discovered what he called canal rays which appeared to flow from the anode of the tube. Again these rays were found to have a particle nature but the e/M ratio for the particles depended on the gas contained in the tube. The highest value for e/M is obtained when the canal rays are produced in a hydrogen discharge tube. This was indicative of the hydrogen ion being the fundamental unit of positive charge and it was, therefore, called a proton.
With the discovery of two of the fundamental particles—the electron and the proton—the problem of understanding the nature of atoms was tackled. A further experiment by Lord Rutherford in 1911,
his now famous a-particle scattering experiment, indicated that the positive charge in an atom is concentrated at its centre, the nucleus occupying only a small space in relation to the distances between atoms. In order to keep the atom electrically neutral, he proposed that the correct number of electrons should surround the nucleus and revolve around it in orbits in a similar way to the planetary orbits round the Sun. This explanation caused difficulties as it immediately contradicted the laws of classical physics. If the electrons are revolving in circular or elliptical orbits round the positive nucleus, they are subject to a constant acceleration along the line joining the electron and the nucleus. According to the classical laws of Maxwell and Lorentz, a charge which suffers an acceleration will radiate electromagnetic waves with energy which is proportional to the square of the acceleration. Classical laws predict that electrons could not occupy such stable orbits as suggested by Rutherford and these rapidly lose their orbital energies by radiation and spiral into the nucleus. However, this difficulty was overcome by Bohr in 1913 who applied a quantum concept based on the principle suggested by Planck. His theory was first applied to the hydrogen atom because of its apparent simplicity.
According to Bohr's theory, the hydrogen atom consists of the heavy, positively charged nucleus around which the electron performs orbits under a central force provided by the electrostatic force which normally exists between charged bodies. (As the mass of the proton, M, is very much greater than the mass of the electron, me, it can be assumed that the proton is at a fixed centre of the electron's orbit). The orbit is illustrated in figure 15.6. For simplicity, we shall consider the simplest case of an electron orbit which is circular.
Suppose that the electron is at a distance r from the proton, that its velocity is v and that its energy is E. Now the electron's energy is made up of two parts: it has kinetic energy (KE) and potential energy (PE), i.e.
According to classical dynamics, the kinetic energy of the electron is me v2/2. Coulomb's law, relating the force which exists between charged bodies, shows that the electrostatic force, F, between the proton and electron is given by
where e0 is the permittivity of free space and that for the orbit to be stable this must be balanced by the centrifugal force, mev2/r. Hence,
The kinetic energy is, therefore, given by
The potential energy of the atom can be assessed by considering the electron not to be in motion but at a distance from the proton. If the separation is altered, work must be performed in taking the electron to a new distance from the proton: by increasing the distance, positive work must be done against the force of attraction between the two particles; by decreasing the distance, a negative amount of work must be performed. The amount of work applied in altering the distance corresponds to the change of potential energy in the system. In a stable orbit, the potential energy is given by the work done in taking the electron from infinity to the distance from the proton given by the radius of the orbit and, as the force between the particles is one of attraction, it will be seen that this is a negative quantity. Now the work done is the force multiplied by the distance moved and, as the force depends on the separation of the particles, the total work done in moving of particle from infinity to the orbital distance is evaluated by integration. Hence, the potential energy of the electron in its orbit is given by r e2
4tt£o Joo r2 dr and, therefore,
By using the equations (15.15) and (15.16), equation (15.14) reduces to
The angular momentum, H, of the electron in its orbit is given by the classical expression
Bohr suggested that the radii of the orbits could not take on any value as they might according to classical mechanics. He proposed that the electron could only revolve around the proton in preferred orbits and he proposed that the size of such orbits is given by allowing the angular momentum of the electron about the nucleus to have a value given by an integral multiple of the unit h/2n where h is Planck's constant. His concept is, therefore, one of the quantization of angular momentum.
By this principle it was postulated that while the electron occupies a quantized orbit it does not radiate energy. As in the case of classical mechanics where a change in orbit corresponds to a change energy, a change in energy occurs when an electron jumps from one quantized orbit to another. At the time of a jump, the transfer is achieved by the emission or absorption of electromagnetic radiation but, as the orbits are quantized, so must be the amounts of radiation that are involved. In other words, the radiation emitted or absorbed by atoms is quantized.
Following Bohr's postulate, equation (15.18) may be written as nh f mer \1/2 2n \4nfi0/
The radii of the Bohr orbits can, therefore, be expressed as r=S-4^ (15.19)
n e2me and by putting n = 1 and inserting the values of e0, h, e and me, the size of the smallest Bohr orbit is obtained, giving a value equal to 5-3 x 10-11 m or 0-53 A.
The energies of the Bohr orbits are given by substituting equation (15.19) into (15.17) which gives e4me
The energy of the first orbit, known as the ground state, is obtained by letting n = 1, giving a value equal to -2-17 x 10-18 J.
It is more convenient to describe the energies of electron orbits in units of electron volts (eV) rather than in joules. One electron volt is defined as the energy acquired by an electron after it has been accelerated by a potential difference of 1 volt. Thus,
1 eV = charge x potential difference.
Now the charge on an electron is 1-6 x 10-19 C and, therefore,
The energy of the ground state in the hydrogen atom in units of eV is given by
and the energies of the other orbits or excited states are, therefore, given by
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