Ch 02 - Structure of Atom11th Chemistry

• Experimental Setup: A Cathode Ray Discharge Tube (partially evacuated glass tube with two metal electrodes) operated at very low pressure and very high voltage.
• Observation: A stream of particles flowed from the negative electrode (cathode) to the positive electrode (anode).
• Conclusion: Since these rays deflected towards the positive field in electric/magnetic fields and their properties were independent of the gas or electrode material, it was concluded that they consist of fundamental, negatively charged particles called electrons.

• The defining characteristic of cathode rays is that their behavior and physical properties do not depend upon the material of the electrodes.
• Furthermore, they do not depend on the nature of the gas present in the cathode ray tube.
• This universality proved conclusively that electrons are a basic, fundamental constituent of all atoms, regardless of the element.

Variables & Units:

• e = Magnitude of the charge on the electron (C)
• $m_e$ = Mass of the electron (kg)

Conditions/Notes:

• Discovered by J.J. Thomson in 1897 by applying perpendicular electric and magnetic fields to cathode rays.
• Thomson balanced the fields so the electron hit the screen undeflected, allowing him to isolate this ratio.

1. Quantization of Charge: He found that the electrical charge ($q$) on the suspended oil droplets was always an exact integral multiple of a fundamental charge unit ($q = n e$).
2. Exact Charge Value: He empirically determined the exact charge of a single electron to be $-1.6022 \times 10^{-19}\ C$.

• Bonus: By combining this newly found charge with Thomson's $e/m_e$ ratio, physicists could finally calculate the exact absolute mass of the electron ($9.1094 \times 10^{-31}\ kg$).

• Gas Dependence: Unlike cathode rays (which are purely electrons), the mass and charge-to-mass ratio of Canal Rays depend heavily on the nature of the gas present in the discharge tube.
• Origin: Canal rays are simply the positively charged gaseous ions left behind after the high-energy electrons are knocked out of the neutral gas atoms.
• Note: The lightest and smallest positive ion obtained during these experiments was from Hydrogen gas, and it was formally named the proton.

• Discoverer: James Chadwick (1932).
• Experiment: He bombarded a thin sheet of Beryllium with high-energy $\alpha$-particles.
• Observation: Highly penetrating, electrically neutral particles with a mass slightly greater than that of a proton ($1.6749 \times 10^{-27}\ kg$) were emitted.

• Structure: The atom is a uniform sphere of positive charge with negatively charged electrons embedded in it to provide a stable electrostatic arrangement.
• Achievement: It successfully explained the overall electrical neutrality of the atom.
• Limitation: It failed completely to explain later experiments, specifically Rutherford's $\alpha$-particle scattering results, because it incorrectly assumed the atomic mass and positive charge were uniformly distributed.

1. Most $\alpha$-particles (~99%) passed through the thin gold foil completely undeflected.
2. A small fraction of the $\alpha$-particles were deflected by small angles.
3. A very few $\alpha$-particles (~1 in 20,000) bounced practically straight back, deflected by nearly $180^\circ$.

1. Empty Space: Most of the atom is entirely empty space (because most particles passed undeflected).
2. The Nucleus: The entire positive charge and almost all of the atom's mass are densely concentrated in an extremely small central region (which caused the severe deflections and rebounds).
3. Scale: The volume of the nucleus ($10^{-15}\ m$ radius) is negligibly small compared to the total volume of the atom ($10^{-10}\ m$ radius).

Variables & Units:

• A = Mass Number (Total number of nucleons: protons + neutrons)
• Z = Atomic Number (Number of protons; also equals electrons in a neutral atom)
• n = Number of neutrons

Conditions/Notes:

• The notation for an element is generally written as: $^{A}_{Z}\text{X}$.
• Chemical properties are dictated entirely by $Z$, not $n$.

• Maxwell's Electromagnetic Theory: Charged particles undergoing acceleration must continuously emit electromagnetic radiation.
• Orbital Acceleration: An electron moving in a circular orbit around the nucleus is constantly changing direction, meaning it is constantly accelerating.
• The Spiral: It would continuously lose energy via radiation, causing its orbit to shrink until it spirals into the nucleus in just $10^{-8}\ s$. Therefore, Rutherford's model could not theoretically explain why atoms are stable.

• Isotopes: Atoms of the same element with the same Atomic Number ($Z$) but different Mass Number ($A$). They possess different numbers of neutrons. Because they have identical electron configurations, their chemical properties are identical (e.g., $^{12}C, ^{13}C, ^{14}C$).
• Isobars: Atoms of different elements with the same Mass Number ($A$) but different Atomic Number ($Z$). Because their proton and electron counts differ, they are completely different elements with entirely different chemical properties (e.g., $^{14}C$ and $^{14}N$).

Variables & Units:

• $c$ = Speed of light ($3.0 \times 10^8\ m/s$)
• $\nu$ = Frequency ($Hz$ or $s^{-1}$)
• $\lambda$ = Wavelength ($m$)
• $\bar{\nu}$ = Wavenumber ($m^{-1}$ or $cm^{-1}$)

Conditions/Notes:

• Wavenumber represents the number of wavelengths per unit length.

• Protium ($^1_1H$): 0 neutrons. It is the most abundant isotope (~99.985%).
• Deuterium ($^2_1D$): 1 neutron. It has trace abundance (0.015%) and is used in heavy water.
• Tritium ($^3_1T$): 2 neutrons. It is extremely rare and radioactive, found only in trace amounts on Earth.

• Definition: Atoms or ions of different elements that have exactly the same total number of electrons.
• Example Series (10 electrons): $N^{3-}, O^{2-}, F^-, Ne, Na^+, Mg^{2+}, Al^{3+}$.
• Note: Even though they have the exact same number of electrons, their ionic radii and chemical properties differ greatly because their effective nuclear charges (number of protons pulling on those electrons) are fundamentally different.

• $\alpha$-rays: High-energy Helium nuclei ($He^{2+}$). They possess the largest mass and charge, resulting in the least penetrating power.
• $\beta$-rays: Negatively charged particles similar to electrons. Their penetrating power is roughly 100 times that of $\alpha$-particles.
• $\gamma$-rays: High-energy electromagnetic radiations (neutral, no rest mass, no particles). They possess the highest penetrating power, roughly 1000 times that of $\alpha$-particles.

Variables & Units:

• E = Energy of a single quantum/photon (J)
• h = Planck's constant ($6.626 \times 10^{-34}\ \text{J s}$)
• $\nu$ = Frequency of the radiation ($Hz$ or $s^{-1}$)

Conditions/Notes:

• Energy can only be emitted or absorbed in integral multiples of this quantum ($E = nh\nu$). It is not continuous.

1. Radio Waves (~$10^6$ Hz)
2. Microwaves (~$10^{10}$ Hz)
3. Infrared (IR) (~$10^{13}$ Hz)
4. Visible Light (~$10^{15}$ Hz)
5. Ultraviolet (UV) (~$10^{16}$ Hz)
6. X-rays (Higher frequency)
7. Gamma Rays (Highest frequency)

• Note on Visible Light: Red has the lowest frequency and longest wavelength (~$750\ nm$); Violet has the highest frequency and shortest wavelength (~$400\ nm$).

Variables & Units:

• $h\nu$ = Energy of the striking photon (J)
• $h\nu_0$ = Work Function ($W_0$), the minimum binding energy to remove the electron (J)
• $\frac{1}{2}m_ev^2$ = Kinetic energy of the ejected electron (J)

Conditions/Notes:

• The incoming photon must act as a particle, transferring its entire energy instantaneously in a 1-to-1 collision with an electron.

• Definition: An ideal body that is a perfect absorber and perfect emitter of radiation of all frequencies.
• The Curve: The exact frequency distribution of the emitted radiation depends only on the body's temperature. As temperature increases, the peak intensity shifts noticeably to shorter wavelengths (higher frequencies).
• Classical Failure: Classical physics predicted that intensity would continue to rise infinitely at shorter wavelengths (known as the Ultraviolet catastrophe). Max Planck eventually solved this paradox by introducing quantized energy.

Variables & Units:

• $\bar{\nu}$ = Wavenumber ($cm^{-1}$)
• $109677$ = Rydberg Constant ($R_H$) in $cm^{-1}$
• $n_1$ = Lower energy level integer ($1, 2, 3...$)
• $n_2$ = Higher energy level integer ($n_1+1, n_1+2...$)

Conditions/Notes:

• Strictly valid only for Hydrogen. For hydrogen-like species (He+, Li2+), multiply the RHS by $Z^2$.

1. Instantaneous Emission: Electrons are ejected with absolutely no time lag as soon as light strikes the metal surface.
2. Threshold Frequency ($u_0$): No electrons are ejected below a certain minimum threshold frequency, completely regardless of the light's intensity (brightness).
3. Kinetic Energy Independence: The kinetic energy of the ejected electrons depends only on the frequency of the incident light, NOT on its intensity. Intensity only increases the number of electrons ejected, not their energy.

• Emission Spectrum: Produced when an excited atom or molecule gives up previously absorbed energy. Visually, it appears as discrete bright coloured lines set against a dark background.
• Absorption Spectrum: Produced when a continuum of broad radiation passes through a sample that selectively absorbs specific, discrete wavelengths. Visually, it appears as dark gaps or lines superimposed continuously on a bright continuous spectrum.
• Note: For a given element, the emission and absorption spectra are photographic negatives of each other.

Variables & Units:

• $m_e$ = Mass of electron
• $v$ = Velocity of electron
• $r$ = Radius of orbit
• $n$ = Principal quantum number (1, 2, 3...)
• $h$ = Planck's constant

Conditions/Notes:

• This postulate explains why only certain fixed orbits are allowed. Maxwell's electromagnetic theory does not apply to these specific quantized states.

1. Lyman: $n_1=1$ (Ultraviolet)
2. Balmer: $n_1=2$ (Visible)
3. Paschen: $n_1=3$ (Infrared)
4. Brackett: $n_1=4$ (Infrared)
5. Pfund: $n_1=5$ (Infrared)

• Note: The Balmer series is unique because it is the only series that appears in the visible light region.

Variables & Units:

• $r_n$ = Radius of the $n^{th}$ orbit
• $n$ = Principal quantum number
• $Z$ = Atomic number ($1$ for H, $2$ for He+, etc.)
• $a_0$ = Bohr radius, exactly $52.9\ \text{pm}$ ($0.0529\ \text{nm}$)

Conditions/Notes:

• The first orbit of Hydrogen ($n=1, Z=1$) has a radius of $52.9\ \text{pm}$.
• As $Z$ increases, the radius shrinks. As $n$ increases, the radius grows quadratically.

Variables & Units:

• $E_n$ = Energy of the $n^{th}$ state (Joules)
• $Z$ = Atomic Number
• $n$ = Principal quantum number
• $2.18 \times 10^{-18}\ \text{J}$ = Rydberg constant for energy ($R_H$)

Conditions/Notes:

• Negative Sign Significance: The energy of an electron bound to the nucleus is lower (more stable) than a free electron at infinite distance, which is assigned an energy of $0$.

• Definition: Electrons strictly move around the central nucleus in well-defined circular paths of fixed radius and energy called orbits or stationary states.
• Energy Conservation: Contrary to classical Maxwell predictions, an electron residing in a given stationary state does not continuously radiate or lose energy with time.
• Transitions: Energy is only physically emitted or absorbed in discrete amounts (quanta) when the electron instantly jumps between these specifically allowed stationary states.

Variables & Units:

• $\lambda$ = Wavelength ($m$)
• $h$ = Planck's constant ($6.626 \times 10^{-34}\ \text{J s}$)
• $p$ = Momentum ($kg\ m\ s^{-1}$)
• $m, v$ = Mass of the particle ($kg$) and its velocity ($m\ s^{-1}$)

Conditions/Notes:

• The wave nature is only detectable for microscopic particles (like electrons). For macroscopic objects, the mass is so large that the resulting wavelength is imperceptibly small.

Variables & Units:

• $\Delta x$ = Uncertainty in position ($m$)
• $\Delta p_x$ = Uncertainty in momentum ($kg\ m\ s^{-1}$) (Note: $\Delta p_x = m\Delta v_x$)
• $h$ = Planck's constant ($6.626 \times 10^{-34}\ \text{J s}$)

Conditions/Notes:

• Statement: It is fundamentally impossible to determine simultaneously the exact position and exact momentum of an electron.
• Significance: It conclusively rules out the existence of definite paths or trajectories (like Bohr's fixed orbits) for subatomic particles.

• A free electron at rest (at an infinite distance from the nucleus) feels no electrostatic attraction and is assigned an arbitrary baseline energy value of zero ($E_\infty = 0$).
• When an electron gets closer to the nucleus and becomes bound, attractive electrostatic forces do work, and energy is emitted into the surroundings.
• Therefore, the total energy of a bound electron is always strictly lower than that of a free electron, resulting in a negative mathematical value. The more negative the energy, the more deeply bound and stable the orbit is.

• Zeeman Effect: The splitting of spectral lines into multiple components in the presence of an external magnetic field.
• Stark Effect: The splitting of spectral lines in the presence of an external electric field.
• Note: Bohr's model also completely failed to explain the complex spectra of multi-electron atoms (atoms with more than one electron) and the ability of atoms to form molecules via chemical bonds.

Variables & Units:

• $n$ = Principal quantum number
• $l$ = Azimuthal quantum number

Conditions/Notes:

• Node: A region in space where the probability density ($|\psi|^2$) of finding an electron is exactly zero.
• Angular nodes are planar or conical surfaces, while radial nodes are spherical surfaces.

• Wave Function ($\psi$): A pure mathematical amplitude function describing the electron's quantum state coordinates. Crucially, it has no direct physical or macroscopic meaning on its own.
• Probability Density ($|\psi|^2$): The square of the wave function's absolute magnitude at a given point in space. It represents the actual, observable probability of finding the electron within a small volume around that specific point in space.

• Orbit: A classical, well-defined, and fixed circular 2D physical path around the nucleus where an electron supposedly revolves. (This directly violates Heisenberg's Uncertainty Principle as it assumes exact position and momentum).
• Orbital: A quantum mechanical 3D region of space around the nucleus where the probability of finding an electron is maximum (usually enclosing a ~90% probability region). It represents a likelihood, not a physical track.

• The probability density ($|\psi|^2$) of finding an electron decreases exponentially as the radial distance from the nucleus increases, but it mathematically never reaches exactly zero at any finite physical distance.
• Therefore, drawing a strict boundary surface that guarantees a 100% chance of finding the electron is physically impossible, as the volume would have to be infinitely large.
• To solve this, scientists draw contour surfaces enclosing ~90% of the total electron probability density, providing a practical, finite visual representation of the orbital's 3D shape.

• Identifies the main electron shell (Designated as K, L, M, N... corresponding to $n = 1, 2, 3, 4...$).
• Principally dictates the macroscopic size of the orbital (a larger $n$ equals a larger average radial distance from the nucleus).
• Largely determines the total energy of the orbital.
• Limits: The maximum number of allowed orbitals in a given shell is $n^2$, and the maximum number of electrons that shell can hold is $2n^2$.

• Determines the 3D geometric shape of the orbital and identifies the specific subshell within a main shell.
• Permitted values: Integers ranging from $0$ up to $(n-1)$ for any given principal quantum number $n$.
• Standard Notations:
  • $l=0 \rightarrow s$ subshell (spherical shape)
  • $l=1 \rightarrow p$ subshell (dumbbell shape)
  • $l=2 \rightarrow d$ subshell (double-dumbbell / cloverleaf shape)
  • $l=3 \rightarrow f$ subshell (complex shape)

• Determines the spatial orientation of the orbital with respect to a standard set of coordinate axes (x, y, z) when placed in an external magnetic field.
• Permitted values: Integers from $-l$ through $0$ up to $+l$.
• The total number of valid orientations (which perfectly equals the total number of individual orbitals) for a given subshell is calculated by the formula $2l + 1$.

• Reason: It was introduced to successfully explain the presence of doublet or triplet closely spaced lines in multi-electron high-resolution line spectra. These splittings originate from the electron's own intrinsic spin angular momentum on its own axis.
• Values: Because an electron can only spin in two distinct directions, the values can only be $+\frac{1}{2}$ (spin up, $\uparrow$) or $-\frac{1}{2}$ (spin down, $\downarrow$).

• The intrinsic energy of an orbital in a multi-electron atom is primarily dictated by its total $(n + l)$ value.
• A lower $(n+l)$ value definitively signifies a lower energy state.
• The Tie-breaker: If two different orbitals happen to have the identical $(n + l)$ value, the orbital with the lower principal quantum number ($n$) has the lower energy.
• Example: $4s$ ($4+0=4$) fills before $3d$ ($3+2=5$). But comparing $3p$ ($3+1=4$) and $4s$ ($4+0=4$), the $3p$ orbital is lower energy strictly because $3 < 4$.

Variables & Units:

• $\Delta E$ = Energy absorbed or emitted (J)
• $R_H$ = Rydberg constant for energy ($2.18 \times 10^{-18}\ \text{J}$)
• $n_i$ = Initial principal quantum number
• $n_f$ = Final principal quantum number

Conditions/Notes:

• Absorption: $n_f > n_i$ ($\Delta E$ is positive).
• Emission: $n_i > n_f$ ($\Delta E$ is negative, energy is released).

• Statement: In the lowest-energy ground state of atoms, orbitals are filled with electrons in the strict order of their increasing energies.
• Electrons will unconditionally occupy the lowest energy orbital available to them before entering higher energy orbitals. The sequence follows the $(n+l)$ rule: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s...$
• Note: 'Aufbau' is not a scientist's name; it is the German word for 'building up' or 'construction'.

Variables & Units:

• $\hat{H}$ = Hamiltonian operator (represents the total energy operator of the system)
• $\psi$ = Wave function (contains all information about the electron)
• $E$ = Energy eigenvalue (the quantized total energy of the state)

Conditions/Notes:

• The Hamiltonian ($\hat{H}$) accounts for the kinetic energies of all sub-atomic particles and the potential energies (attractive and repulsive) among them.
• Its solution gives the possible energy states ($E$) and corresponding wave functions ($\psi$).

• Formal Statement: No two electrons in the exact same atom can have the exact same set of all four quantum numbers ($n, l, m_l, m_s$).
• Alternative Statement: Only two electrons may exist in the exact same orbital, and these two electrons must have opposite spins ($+\frac{1}{2}$ and $-\frac{1}{2}$).
• This fundamental quantum mechanical rule naturally and strictly restricts any single orbital in the universe to holding a maximum of 2 electrons.

• Statement: Pairing of electrons in degenerate orbitals belonging to the exact same subshell does not take place until each orbital in that subshell has been singly occupied (one electron each).
• Furthermore, all of these singly occupied orbitals must have electrons with parallel spins (pointing in the same direction, usually spin-up) to maximize exchange energy and minimize electrostatic repulsion, ensuring maximum atomic stability.

• Concept: Due to their specific 3D geometric shapes, electrons in different orbitals spend different amounts of time very close to the nucleus. Orbitals that 'penetrate' closer to the nucleus experience a higher effective nuclear charge ($Z_{eff}$) and physically shield outer electrons more effectively from the nucleus's pull.
• Sequence: The extent of penetration and shielding ability within the same shell dictates the sequence: $s > p > d > f$.
• Result: Because an $s$ electron penetrates closer to the nucleus than a $p$ electron, it is more tightly bound, rendering an $s$ orbital lower in energy than a $p$ orbital of the same shell.

• Hydrogen Atom: Orbital energy depends solely on the principal quantum number ($n$). Because there are no other electrons to cause repulsion or shielding, all subshells within the exact same shell are perfectly degenerate (e.g., $2s$ and $2p$ have the exact same energy).
• Multi-Electron Atoms: Orbital energy depends on both $n$ and the azimuthal quantum number ($l$). This is explicitly due to electron-electron mutual electrostatic repulsion and the varying shielding/penetration effects of different subshell shapes, breaking the degeneracy (e.g., $2s < 2p$).

• Chromium ($Cr$): Expected sequence is $[Ar] 4s^2 3d^4$, but the actual stable configuration is $[Ar] 4s^1 3d^5$.
• Copper ($Cu$): Expected sequence is $[Ar] 4s^2 3d^9$, but the actual stable configuration is $[Ar] 4s^1 3d^{10}$.
• Reasoning: Half-filled ($d^5$) and fully-filled ($d^{10}$) degenerate subshells offer tremendous extra thermodynamic stability. This stability arises from a greater symmetrical distribution of electron density around the nucleus and maximized exchange energy (electrons of parallel spins exchanging positions in degenerate orbitals).