Structure of Atom - JEE Mains

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1. Early Atomic Models & Their Limitations Dalton's Atomic Theory (1808) Postulates: Matter consists of indivisible and indestructible atoms. Atoms of a given element are identical in mass and properties. Compounds are formed by a combination of two or more different kinds of atoms in simple whole-number ratios. A chemical reaction is a rearrangement of atoms. Atoms are neither created nor destroyed in a chemical reaction. Limitations: Could not explain the existence of subatomic particles (electrons, protons, neutrons). Failed to account for isotopes (atoms of same element with different masses) and isobars (atoms of different elements with same mass). Did not explain why atoms of different elements have different masses, sizes, and valencies. Thomson's Model (Plum Pudding Model, 1898) Postulates: An atom consists of a uniform sphere of positive charge. Electrons are embedded within this sphere like plums in a pudding, or seeds in a watermelon. The total positive charge is equal to the total negative charge of the electrons, making the atom electrically neutral. Limitations: Failed to explain the results of Rutherford's $\alpha$-scattering experiment, which showed that most of the atom is empty space. Could not account for the stability of an atom. Rutherford's Nuclear Model (1911) Based on $\alpha$-scattering experiment: A beam of high-energy $\alpha$-particles (He$^{2+}$ ions) was directed at a thin gold foil. Observations: Most $\alpha$-particles passed straight through the foil undeflected. A small fraction of $\alpha$-particles were deflected by small angles. Very few $\alpha$-particles (about 1 in 20,000) were deflected back by nearly $180^\circ$. Conclusions & Postulates: Most of the space in an atom is empty, explaining why most $\alpha$-particles passed through. A tiny, dense, positively charged region called the nucleus is present at the center of the atom. This accounts for the deflection of some $\alpha$-particles. The size of the nucleus is very small compared to the size of the atom ($r_{nucleus} \approx 10^{-15} \text{ m}$, $r_{atom} \approx 10^{-10} \text{ m}$). Electrons revolve around the nucleus in well-defined circular paths (orbits). The atom as a whole is electrically neutral, with the number of electrons equal to the number of protons in the nucleus. Limitations: Stability of Atom: According to classical electromagnetic theory, an accelerating charged particle (like an electron revolving around the nucleus) should continuously emit radiation and lose energy. This would cause the electron to spiral into the nucleus, making the atom unstable. However, atoms are stable. Line Spectrum: Could not explain the discrete line spectra observed for atoms, particularly for hydrogen. If electrons continuously lost energy, they should produce a continuous spectrum. 2. Electromagnetic Radiation (EMR) Wave Nature of EMR: EMR travels in waves and has associated properties like wavelength ($\lambda$), frequency ($\nu$), and speed ($c$). Relationship: $c = \lambda \nu$, where $c = 3.0 \times 10^8 \text{ m/s}$ (speed of light in vacuum). Units: $\lambda$ in meters (m), nanometers (nm), Ångstroms (Å); $\nu$ in hertz (Hz or s$^{-1}$). Particle Nature of EMR (Planck's Quantum Theory): EMR is emitted or absorbed in discrete packets of energy called quanta or photons . Energy of a single photon: $E = h\nu = \frac{hc}{\lambda}$ $h$ (Planck's constant) = $6.626 \times 10^{-34} \text{ Js}$. Photoelectric Effect: The phenomenon of ejection of electrons from the surface of a metal when light of suitable frequency (above a certain threshold) strikes it. Key Observations: Electrons are ejected only if the incident light has frequency greater than a certain minimum frequency, $\nu_0$ (threshold frequency). The number of ejected electrons is proportional to the intensity of incident light. The kinetic energy of the ejected electrons is directly proportional to the frequency of incident light (above $\nu_0$). There is no time lag between the incidence of light and the ejection of electrons. Einstein's Equation: $h\nu = h\nu_0 + KE_{max}$ or $h\nu = W_0 + \frac{1}{2}mv^2$ $h\nu$: Energy of incident photon. $h\nu_0$ or $W_0$: Work function (minimum energy required to eject an electron). $KE_{max}$: Maximum kinetic energy of the ejected electron. 3. Bohr's Model of Hydrogen Atom (1913) Postulates: Electrons revolve around the nucleus in specific, stable circular paths called orbits or stationary states . Each orbit has a fixed radius and energy. Electrons in these stationary orbits do not radiate energy. The angular momentum of an electron in a given orbit is quantized, meaning it can only take certain discrete values: $mvr = n\frac{h}{2\pi}$, where $n = 1, 2, 3, ...$ (principal quantum number). Energy is absorbed or emitted only when an electron jumps from one stationary orbit to another. Absorption: Electron moves from a lower to a higher energy orbit. Emission: Electron moves from a higher to a lower energy orbit. The energy difference is given by $\Delta E = E_{final} - E_{initial} = h\nu$. Formulas for H-atom and H-like Species (Bohr Model) Radius of $n^{th}$ orbit ($r_n$): $r_n = 0.529 \times \frac{n^2}{Z} \text{ Å}$ $n$: principal quantum number ($1, 2, 3, ...$) $Z$: atomic number (for H, $Z=1$; for He$^+$, $Z=2$; for Li$^{2+}$, $Z=3$) $r_1$ for H-atom (Bohr radius) $\approx 0.529 \text{ Å}$ Energy of $n^{th}$ orbit ($E_n$): $E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV/atom}$ $E_n = -2.18 \times 10^{-18} \times \frac{Z^2}{n^2} \text{ J/atom}$ The negative sign indicates that the electron is bound to the nucleus. As $n$ increases, energy becomes less negative (i.e., increases). Velocity of electron in $n^{th}$ orbit ($v_n$): $v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s}$ Hydrogen Spectrum & Rydberg Formula When an electron in a hydrogen atom jumps from a higher energy level ($n_2$) to a lower energy level ($n_1$), it emits a photon of specific energy, resulting in a line spectrum. The wavenumber ($\bar{\nu} = 1/\lambda$) of the emitted radiation is given by the Rydberg formula : $$ \bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) $$ $R_H$ (Rydberg constant) = $1.097 \times 10^7 \text{ m}^{-1}$ or $109677 \text{ cm}^{-1}$. $n_1$ is the lower energy level, $n_2$ is the higher energy level ($n_2 > n_1$). Spectral Series: Series $n_1$ (Lower Level) $n_2$ (Higher Levels) Region of EMR Lyman 1 2, 3, 4, ... Ultraviolet (UV) Balmer 2 3, 4, 5, ... Visible Paschen 3 4, 5, 6, ... Infrared (IR) Brackett 4 5, 6, 7, ... Infrared (IR) Pfund 5 6, 7, 8, ... Infrared (IR) Humphrey 6 7, 8, 9, ... Far Infrared (FIR) Limitations of Bohr's Model: Applicable only to single-electron species (H, He$^+$, Li$^{2+}$). Failed to explain the spectra of multi-electron atoms. Could not explain the splitting of spectral lines in magnetic field (Zeeman effect) or electric field (Stark effect). Could not explain the ability of atoms to form molecules by chemical bonds. Violates Heisenberg's Uncertainty Principle (electron's exact position and momentum are determined). 4. Dual Nature of Matter (De Broglie Principle) Louis de Broglie (1924) proposed that, like light, matter also exhibits both particle and wave properties. Every moving particle has a wave associated with it, called a matter wave or de Broglie wave . De Broglie Wavelength ($\lambda$): $$ \lambda = \frac{h}{p} = \frac{h}{mv} $$ $h$: Planck's constant $p$: momentum of the particle $m$: mass of the particle $v$: velocity of the particle For a charged particle (e.g., electron) accelerated through a potential difference $V$: Kinetic Energy ($KE$) = $eV$ $KE = \frac{1}{2}mv^2 = \frac{p^2}{2m} \Rightarrow p = \sqrt{2mKE} = \sqrt{2meV}$ So, $\lambda = \frac{h}{\sqrt{2meV}}$ For an electron: $\lambda \approx \frac{12.27}{\sqrt{V}} \text{ Å}$ Experimental Verification: Davisson and Germer experiment (1927) confirmed the wave nature of electrons by observing electron diffraction. 5. Heisenberg's Uncertainty Principle Proposed by Werner Heisenberg (1927). It is fundamentally impossible to determine simultaneously and precisely both the position and momentum of a microscopic particle (like an electron). Mathematical Formulation: $$ \Delta x \cdot \Delta p \ge \frac{h}{4\pi} $$ or $$ \Delta x \cdot m\Delta v \ge \frac{h}{4\pi} $$ $\Delta x$: uncertainty in position $\Delta p$: uncertainty in momentum $\Delta v$: uncertainty in velocity A similar uncertainty relationship exists for energy and time: $$ \Delta E \cdot \Delta t \ge \frac{h}{4\pi} $$ $\Delta E$: uncertainty in energy $\Delta t$: uncertainty in time Implication: This principle disproves the concept of definite electron orbits (as in Bohr's model) because if an electron moves in a definite orbit, its position and velocity would be precisely known at all times. 6. Quantum Mechanical Model of Atom Developed by Erwin Schrödinger (1926) based on de Broglie's dual nature and Heisenberg's uncertainty principle. Describes the behavior of electrons in atoms using wave mechanics. Does not define the exact path of an electron but rather the probability of finding an electron in a 3D region of space around the nucleus. Schrödinger Wave Equation: $$ \hat{H}\psi = E\psi $$ $\hat{H}$: Hamiltonian operator (represents total energy of the system). $\psi$ (psi): Wave function, a mathematical function whose value depends on the coordinates of the electron and represents the amplitude of the electron wave. $E$: Total energy of the electron. Significance of $\psi$ and $\psi^2$: $\psi$ itself has no physical meaning. $|\psi|^2$ (square of the wave function) represents the probability density of finding an electron at a particular point in space. It is always positive. Atomic Orbital: A three-dimensional region around the nucleus where the probability of finding an electron is maximum (typically 90-95%). Quantum Numbers A set of four numbers that completely describe the position, energy, and spin of an electron in an atom. 1. Principal Quantum Number ($n$): Values: Positive integers $1, 2, 3, ...$ (corresponding to K, L, M, ... shells). Significance: Determines the main energy level or shell in which the electron resides. Determines the size and energy of the orbital. Higher $n$ means larger orbital and higher energy. Maximum number of electrons in a shell $= 2n^2$. 2. Azimuthal (Angular Momentum) Quantum Number ($l$): Values: Integers from $0$ to $(n-1)$. Significance: Determines the shape of the subshell/orbital. Also known as the orbital angular momentum quantum number. Values of $l$ correspond to subshells: $l=0 \rightarrow s$ subshell (spherical shape) $l=1 \rightarrow p$ subshell (dumbbell shape) $l=2 \rightarrow d$ subshell (double dumbbell/complex shape) $l=3 \rightarrow f$ subshell (more complex shape) Number of subshells in a main shell $= n$. 3. Magnetic Quantum Number ($m_l$): Values: Integers from $-l$ to $+l$, including $0$. Significance: Determines the orientation of the orbital in space. For a given $l$, there are $(2l+1)$ possible values of $m_l$, which means $(2l+1)$ orbitals in a subshell. Examples: If $l=0 (s)$, $m_l=0$ (1 s-orbital). If $l=1 (p)$, $m_l=-1, 0, +1$ (3 p-orbitals: $p_x, p_y, p_z$). If $l=2 (d)$, $m_l=-2, -1, 0, +1, +2$ (5 d-orbitals: $d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2}$). 4. Spin Quantum Number ($m_s$): Values: Only two possible values: $+\frac{1}{2}$ (spin up, $\uparrow$) or $-\frac{1}{2}$ (spin down, $\downarrow$). Significance: Describes the intrinsic angular momentum (spin) of an electron, which creates a magnetic field. It indicates the direction of the electron's spin. 7. Rules for Filling Electrons in Orbitals Aufbau Principle (Building Up Principle): Electrons occupy the lowest energy orbitals available first. The order of increasing energy of orbitals is generally determined by the $(n+l)$ rule: Orbitals with lower $(n+l)$ values have lower energy. If two orbitals have the same $(n+l)$ value, the orbital with the lower $n$ value has lower energy. Energy Order: $1s Pauli's Exclusion Principle: No two electrons in an atom can have all four quantum numbers identical ($n, l, m_l, m_s$). This implies that an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins (one with $m_s=+\frac{1}{2}$ and the other with $m_s=-\frac{1}{2}$). Hund's Rule of Maximum Multiplicity: For degenerate orbitals (orbitals of the same energy, e.g., the three $p$ orbitals or five $d$ orbitals), electron pairing does not occur until each orbital in the subshell is singly occupied. Furthermore, the electrons occupying the singly filled orbitals must have parallel spins (e.g., all spin up). This rule maximizes the total spin and thus increases the stability of the atom. 8. Electronic Configuration The arrangement of electrons in the various atomic orbitals of an atom. Notation: $n l^x$, where $n$ is the principal quantum number, $l$ is the subshell (s, p, d, f), and $x$ is the number of electrons in that subshell. Examples: Hydrogen ($Z=1$): $1s^1$ Helium ($Z=2$): $1s^2$ Carbon ($Z=6$): $1s^2 2s^2 2p^2$ Neon ($Z=10$): $1s^2 2s^2 2p^6$ Sodium ($Z=11$): $1s^2 2s^2 2p^6 3s^1$ or $[Ne] 3s^1$ (using noble gas core) Exceptions to Aufbau Principle (due to extra stability of half-filled and completely filled orbitals): Chromium ($Z=24$): Expected $[Ar] 3d^4 4s^2$, but actual is $[Ar] 3d^5 4s^1$. (Half-filled $d$-subshell is more stable). Copper ($Z=29$): Expected $[Ar] 3d^9 4s^2$, but actual is $[Ar] 3d^{10} 4s^1$. (Completely filled $d$-subshell is more stable).