Atomic Structure

A valence shell of $4s^2$ means the outer shell is the 4th with two s electrons. Taking the atom as calcium, the full configuration is

$1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2$

Atomic number $= 2+2+6+2+6+2 = 20$ (Ca).

Number of p-electrons $= 6\,(2p) + 6\,(3p) = 12$.

Controlled nuclear fission is a fission chain reaction in which the number of neutrons available is regulated (using control rods of cadmium or boron that absorb extra neutrons) so that the reaction proceeds at a steady, safe rate and does not explode.

Important application: it is used in nuclear reactors to generate electricity (nuclear power plants).

The magnetic quantum number ($m$) gives the orientation (spatial direction) of an orbital in space. For a given $l$ it has $(2l+1)$ values from $-l$ to $+l$, which is the number of orbitals in that subshell.

For the $3p_x$ orbital: $n=3$, so $l = 1$ (p subshell). The allowed values are $m = -1, 0, +1$ (the three p orbitals); $p_x$ is oriented along the $x$-axis and is one of these three.

Experiment: a narrow beam of $\alpha$-particles from a radioactive source was directed at a very thin gold foil (about $10^{-4}$ mm thick), surrounded by a circular zinc sulphide screen that flashes when struck.

Observations

1. Most $\alpha$-particles passed straight through the foil undeflected.
2. A small number were deflected through small angles.
3. Very few (about 1 in 20,000) were deflected by large angles, and some rebounded almost straight back (about $180^\circ$).

Conclusions

1. Most of the atom is empty space, since most particles went undeflected.
2. All the positive charge and almost all the mass are concentrated in a very small, dense central core called the nucleus (size about $10^{-15}$ m, against $10^{-10}$ m for the atom), because large deflection or rebounds occur only on a close approach or head-on collision with it.
3. The nucleus is positively charged, as it repels the positive $\alpha$-particles.
4. Negatively charged electrons revolve around the nucleus in circular paths (orbits), so the atom as a whole is electrically neutral.

(i) The principal quantum number ($n$) specifies the energy of an electron (mainly its shell and energy level).

(ii) An orbital with $n = 3$ and $l = 2$ is the 3d orbital.

Nuclear fission is the splitting of a heavy nucleus into two (or more) lighter nuclei, with the release of a large amount of energy and neutrons, when it is bombarded by neutrons.

Example: $\mathrm{^{235}_{92}U + ^{1}_{0}n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3\,^{1}_{0}n + energy}$

Postulates of Bohr's model:

1. An electron revolves around the nucleus in certain fixed circular paths called orbits or stationary states, without radiating energy.
2. Only those orbits are allowed in which the angular momentum of the electron is an integral multiple of $\frac{h}{2\pi}$: $mvr = \frac{nh}{2\pi},\quad n = 1,2,3,\ldots$
3. Each orbit has a fixed energy. The energy increases with distance from the nucleus.
4. Energy is absorbed or emitted only when an electron jumps between orbits, and its amount is the difference in energy of the two levels: $\Delta E = E_2 - E_1 = h\nu$

Explanation of stability: Classical physics says an accelerating charged electron should radiate energy continuously and spiral into the nucleus. Bohr's postulate states that an electron in a stationary orbit does not radiate energy. Its energy stays constant, so it does not fall into the nucleus, and the atom remains stable.

For p, $l=1$ so $m$ can take $2l+1=3$ values; for s, $l=0$ so only $m=0$.

(i) Mg (Z = 12) has 12 electrons; Mg²⁺ has 10: $1s^2\,2s^2\,2p^6$

(ii) Cl (Z = 17) has 17 electrons; Cl⁻ has 18: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6$

Nuclear isotopes (isotopes) are atoms of the same element having the same atomic number (same number of protons) but different mass numbers (different numbers of neutrons).

Example: protium ${}^{1}_{1}\mathrm{H}$, deuterium ${}^{2}_{1}\mathrm{H}$ and tritium ${}^{3}_{1}\mathrm{H}$ (or ${}^{35}\mathrm{Cl}$ and ${}^{37}\mathrm{Cl}$).

Differences between orbit and orbital

Shapes

• s orbital: spherical, symmetrical about the nucleus; no directional character, and size increases with $n$ (1s < 2s < 3s).

        .---.
       /  o  \     o = nucleus
       \     /
        '---'

• p orbitals: dumb-bell shaped, two lobes on opposite sides of the nucleus with a node at the nucleus. There are three, $p_x$, $p_y$ and $p_z$, lying along the x, y and z axes at right angles.

   p_x:  (  ) o (  )   lobes along x axis

Conclusion. de Broglie concluded that matter, like radiation, has dual nature: a moving particle such as an electron also has wave properties. Every moving particle is associated with a wave, called a matter wave.

Relationship.

$\lambda = \frac{h}{mv} = \frac{h}{p}$

where $\lambda$ is the wavelength of the matter wave, $h$ is Planck's constant, $m$ is the mass and $v$ is the velocity of the particle.

The element with $Z = 24$ is chromium, whose ground-state configuration is an exception (half-filled $d$-subshell):

$1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^1\,3d^5$

The first 18 electrons fill up to $3p^6$. The 19th electron enters the $4s$ orbital, so its quantum numbers are:

$n = 4,\quad l = 0,\quad m = 0,\quad s = +\tfrac{1}{2}$

Postulates of Bohr's theory.

1. The electron in a hydrogen-like atom revolves around the nucleus in certain fixed circular paths called stationary orbits (energy levels). An electron does not radiate energy while in such an orbit.
2. Only those orbits are allowed in which the angular momentum of the electron is a whole-number multiple of $\dfrac{h}{2\pi}$: $mvr = \frac{nh}{2\pi},\qquad n = 1,2,3,\ldots$
3. The centrifugal force on the electron is balanced by the electrostatic attraction of the nucleus.
4. Energy is emitted or absorbed only when the electron jumps from one orbit to another; the energy change equals the energy difference of the orbits: $\Delta E = E_2 - E_1 = h\nu$

Origin of spectral lines in hydrogen. The energy of the electron in the $n$th orbit is $E_n = -\dfrac{13.6}{n^2}\ \text{eV}$. At ordinary conditions the electron is in the ground state ($n = 1$). When the atom is excited, the electron moves to an outer orbit $n_2$. It is unstable there and falls back to a lower orbit $n_1$, emitting a photon of energy

$\Delta E = 13.6\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\ \text{eV}\qquad\text{or}\qquad \frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

with $R_H = 1.097\times10^{7}\ \text{m}^{-1}$. Since the orbits have fixed energies, only photons of definite frequencies are emitted, which gives sharp lines. Transitions to $n_1 = 1$ give the Lyman series (UV), $n_1 = 2$ the Balmer series (visible), $n_1 = 3$ the Paschen series (IR), $n_1 = 4$ Brackett and $n_1 = 5$ Pfund (IR), in agreement with the observed spectrum.

Element X: Phosphorus (P)

Reasoning:

• K shell (n = 1) holds 2 electrons
• L shell (n = 2) holds 8 electrons
• M shell (n = 3) holds 5 electrons

Total number of electrons = 2 + 8 + 5 = 15

Since the atom is neutral, number of protons = number of electrons = 15

The electronic configuration is 2, 8, 5. The element with atomic number 15 is Phosphorus.

Answer:

• Element X = Phosphorus (P)
• Number of protons = 15
• Number of electrons = 15