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GATE Chemistry Physical Chemistry Syllabus

The Physical Chemistry syllabus for GATE Chemistry in full: 15 chapters and 91 topics, every one listed below, plus 50 Physical Chemistry flashcards with answers.

15Chapters
91Topics
50Flashcards

Physical Chemistry syllabus: all 15 chapters

Expand any chapter to see its topics and sub-topics.

  1. Postulates of Quantum Mechanics

     4 topics
    • Operators
    • Time Dependent and Time Independent Schrödinger Equations
    • Born Interpretation
    • Dirac Bra-Ket Notation

  2. Particle in a Box

     4 topics
    • Infinite and Finite Square Wells
    • Concept of Tunnelling
    • Particle in 1D, 2D and 3D-Box
    • Applications

  3. Harmonic Oscillator

     2 topics
    • Harmonic and Anharmonic Potentials
    • Hermite Polynomials

  4. Rotational Motion

     2 topics
    • Angular Momentum Operators
    • Rigid Rotor

  5. Hydrogen and Hydrogen-like Atoms

     2 topics
    • Atomic Orbitals
    • Radial Distribution Function

  6. Multi-electron Atoms

     4 topics
    • Orbital Approximation
    • Electron Spin
    • Pauli Exclusion Principle
    • Slater Determinants

  7. Approximation Methods

     2 topics
    • Variation Method and Secular Determinants
    • First Order Perturbation Techniques

  8. Atomic Units

     overview

    No separate topics listed for this chapter.

  9. Molecular Structure and Chemical Bonding

     6 topics
    • Born-Oppenheimer Approximation
    • Valence Bond Theory and Linear Combination of Atomic Orbitals (LCAO-MO) Theory
    • Hybrid Orbitals
    • Applications of LCAO-MO Theory to H2+, H2
    • Orbital Theory (MOT) of Homo- and Heteronuclear Diatomic Molecules
    • Hückel Approximation and its Application to Annular Ï€ – Electron Systems

  10. Group Theory

     5 topics
    • Symmetry Elements and Operations
    • Point Groups and Character Tables
    • Internal Coordinates and Vibrational Modes
    • Symmetry Adapted Linear Combination of Atomic Orbitals (LCAO-MO)
    • Construction of Hybrid Orbitals using Symmetry Aspects

  11. Spectroscopy

     11 topics
    • Atomic Spectroscopy
    • Russell-Saunders Coupling
    • Term Symbols and Spectral Details
    • Origin of Selection Rules
    • Rotational, Vibrational, Electronic and Raman Spectroscopy of Diatomic and Polyatomic Molecules
    • Line Broadening
    • Einstein’s Coefficients
    • Relationship of Transition Moment Integral with Molar Extinction Coefficient and Oscillator Strength
    • Basic Principles of Nuclear Magnetic Resonance
    • Gyromagnetic Ratio
    • Chemical Shift, Nuclear Coupling

  12. Equilibrium

     27 topics
    • Laws of Thermodynamics
    • Standard States
    • Thermochemistry
    • Thermodynamic Functions and their Relationships
    • Criteria of Spontaneity and Equilibrium
    • Absolute Entropy
    • Partial Molar Quantities
    • Thermodynamics of Mixing
    • Chemical Potential
    • Fugacity, Activity and Activity Coefficients
    • Ideal and Non-ideal Solutions
    • Raoult’s Law and Henry’s Law
    • Chemical Equilibria
    • Dependence of Equilibrium Constant on Temperature and Pressure
    • Ionic Mobility and Conductivity
    • Debye-Hückel Limiting Law
    • Debye-Hückel-Onsager Equation
    • Standard Electrode Potentials and Electrochemical Cells
    • Nernst Equation and its Application
    • Relationship between Electrode Potential and Thermodynamic Quantities
    • Potentiometric and Conduct Metric Titrations
    • Phase Rule
    • Clausius-Clapeyron Equation
    • Phase Diagram of One Component Systems: CO2, H2O, S
    • Phase Diagram of Two Component Systems: Liquid-Vapour, Liquid-Liquid, Solid-Liquid
    • Fractional Distillation
    • Azeotropes and Eutectics

  13. Statistical Thermodynamics

     3 topics
    • Microcanonical, Canonical and Grand Canonical Ensembles
    • Boltzmann Distribution
    • Partition Functions and Thermodynamic Properties

  14. Kinetics

     13 topics
    • Elementary, Parallel, Opposing and Consecutive Reactions
    • Steady State Approximation
    • Mechanisms of Complex Reactions
    • Unimolecular Reactions
    • Potential Energy Surfaces and Classical Trajectories
    • Concept of Saddle Points
    • Transition State Theory: Eyring Equation, Thermodynamic Aspects
    • Kinetics of Polymerization
    • Catalysis Concepts and Enzyme Catalysis
    • Kinetic Isotope Effects
    • Fast Reaction Kinetics: Relaxation and Flow Methods
    • Diffusion Controlled Reactions
    • Kinetics of Photochemical and Photophysical Processes

  15. Surfaces and Interfaces

     6 topics
    • Physisorption and Chemisorption
    • Langmuir, Freundlich and Brunauer–Emmett–Teller (BET) Isotherms
    • Surface Catalysis: Langmuir-Hinshelwood Mechanism
    • Surface Tension, Viscosity
    • Self-assembly
    • Physical Chemistry of Colloids, Micelles and Macromolecules

GATE Chemistry Physical Chemistry flashcards — 50 questions with answers

  1. In quantum mechanics, what is an operator, and what defines an observable's operator?

    An operator is a mathematical instruction that acts on a function to produce another function. A physical observable is represented by a linear, Hermitian operator whose eigenvalues are the possible measurable (real) values.

  2. Write the quantum-mechanical operators for position $x$ and linear momentum $p_x$ in one dimension.

    $$\hat{x} = x, \qquad \hat{p}_x = -i\hbar\frac{\partial}{\partial x}$$

  3. State the commutator of position and momentum, $[\hat{x},\hat{p}_x]$.

    $$[\hat{x},\hat{p}_x] = i\hbar$$ Because it is nonzero, $x$ and $p_x$ cannot be measured simultaneously to arbitrary precision (Heisenberg uncertainty).

  4. What does it mean for an operator $\hat{A}$ to be Hermitian, and why is it required?

    Hermitian means $\int \psi_i^{*}\hat{A}\psi_j\,d\tau = \left(\int \psi_j^{*}\hat{A}\psi_i\,d\tau\right)^{*}$. It guarantees real eigenvalues and orthogonal eigenfunctions, which is essential for physical observables.

  5. Write the one-dimensional time-independent Schrödinger equation.

    $$\hat{H}\psi = E\psi \;\Rightarrow\; -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}} + V(x)\psi = E\psi$$

  6. Write the time-dependent Schrödinger equation.

    $$i\hbar\frac{\partial \Psi(x,t)}{\partial t} = \hat{H}\Psi(x,t) = \left[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}} + V\right]\Psi(x,t)$$

  7. For a time-independent potential, how does the full wavefunction depend on time?

    It separates as $\Psi(x,t) = \psi(x)\,e^{-iEt/\hbar}$. The state is stationary because $|\Psi|^{2} = |\psi|^{2}$ is time-independent.

  8. State the Born interpretation of the wavefunction.

    $|\Psi(x,t)|^{2}\,dx$ is the probability of finding the particle between $x$ and $x+dx$. $\Psi$ itself is a probability amplitude; only $|\Psi|^{2}$ has physical (probability density) meaning.

  9. What is the normalization condition for a wavefunction in one dimension?

    $$\int_{-\infty}^{\infty} |\Psi(x)|^{2}\,dx = 1$$ The total probability of finding the particle somewhere must equal 1.

  10. List the conditions a physically acceptable wavefunction must satisfy.

    It must be single-valued, finite, continuous, and have a continuous first derivative; it must be normalizable (square-integrable). These follow from the Born interpretation.

  11. In Dirac bra-ket notation, what do the ket $|\psi\rangle$ and bra $\langle\psi|$ represent?

    The ket $|\psi\rangle$ represents the state vector (wavefunction $\psi$); the bra $\langle\psi|$ is its conjugate (dual) vector, corresponding to $\psi^{*}$.

  12. Express the overlap integral and an expectation value in Dirac notation.

    Overlap: $\langle\phi|\psi\rangle = \int \phi^{*}\psi\,d\tau$. Expectation value: $\langle \hat{A}\rangle = \dfrac{\langle\psi|\hat{A}|\psi\rangle}{\langle\psi|\psi\rangle}$, which equals $\langle\psi|\hat{A}|\psi\rangle$ for normalized $\psi$.

  13. What is the orthonormality condition for a set of states $\{|n\rangle\}$ in bra-ket notation?

    $$\langle m|n\rangle = \delta_{mn}$$ where $\delta_{mn}=1$ if $m=n$ and $0$ otherwise.

  14. Give the normalized energy eigenfunctions and energy levels of a particle in a 1D infinite square well of width $L$.

    $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right), \qquad E_n = \frac{n^{2}h^{2}}{8mL^{2}}, \quad n=1,2,3,\dots$$

  15. What is the zero-point energy of a particle in a 1D infinite square well, and why can't it be zero?

    The lowest energy is $E_1 = \dfrac{h^{2}}{8mL^{2}}$ ($n=1$). $n=0$ would make $\psi=0$ everywhere (no particle), and zero energy would violate the uncertainty principle.

  16. How does a finite square well differ from an infinite square well regarding the wavefunction and energy levels?

    In a finite well the wavefunction penetrates (decays exponentially) into the barrier regions, energies are slightly lower than the infinite-well values, and only a finite number of bound states exist (plus possible continuum states above the well).

  17. What is quantum mechanical tunnelling?

    Tunnelling is the penetration of a particle through a potential barrier higher than its energy—classically forbidden. The wavefunction decays exponentially inside the barrier but has nonzero amplitude beyond it, giving a finite transmission probability.

  18. How does the tunnelling probability depend on particle mass, barrier width, and barrier height?

    Transmission $T \approx e^{-2\kappa a}$ with $\kappa = \dfrac{\sqrt{2m(V-E)}}{\hbar}$. Tunnelling decreases (exponentially) with greater mass $m$, wider barrier $a$, and higher barrier $(V-E)$.

  19. Give the energy levels for a particle in a 2D box with sides $L_x$ and $L_y$.

    $$E_{n_x,n_y} = \frac{h^{2}}{8m}\left(\frac{n_x^{2}}{L_x^{2}} + \frac{n_y^{2}}{L_y^{2}}\right), \quad n_x,n_y = 1,2,\dots$$

  20. Give the energy levels for a particle in a 3D cubic box of side $L$.

    $$E_{n_x,n_y,n_z} = \frac{h^{2}}{8mL^{2}}\left(n_x^{2}+n_y^{2}+n_z^{2}\right)$$

Show the other 30 Physical Chemistry flashcards

  1. What is degeneracy in a particle in a box, illustrated with a cubic 3D box?

    Degeneracy is when different quantum-number sets give the same energy. In a cubic box, states like $(1,1,2),(1,2,1),(2,1,1)$ all have $n_x^2+n_y^2+n_z^2=6$ and the same energy—triply degenerate. Degeneracy arises from the cubic symmetry.

  2. Name two physical applications modeled by the particle in a box.

    The absorption spectra of conjugated polyenes/cyanine dyes (electrons as particles in a 1D box) and the qualitative electronic structure of quantum dots/nanoparticles (3D box confinement).

  3. Write the potential energy and Schrödinger equation for the 1D quantum harmonic oscillator.

    $$V(x) = \tfrac{1}{2}kx^{2} = \tfrac{1}{2}m\omega^{2}x^{2}, \qquad -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}} + \tfrac{1}{2}kx^{2}\psi = E\psi$$ with $\omega=\sqrt{k/m}$.

  4. State the energy levels of the quantum harmonic oscillator and its zero-point energy.

    $$E_v = \left(v+\tfrac{1}{2}\right)\hbar\omega, \quad v=0,1,2,\dots$$ Levels are equally spaced by $\hbar\omega$; zero-point energy is $E_0=\tfrac{1}{2}\hbar\omega$.

  5. What form do harmonic oscillator wavefunctions take, and what role do Hermite polynomials play?

    $$\psi_v(x) = N_v\,H_v(\alpha^{1/2}x)\,e^{-\alpha x^{2}/2}$$ The Hermite polynomials $H_v$ provide the nodal structure (degree $v$ gives $v$ nodes); the Gaussian ensures decay at large $|x|$.

  6. Give the first three Hermite polynomials $H_0, H_1, H_2$.

    $$H_0 = 1, \quad H_1 = 2y, \quad H_2 = 4y^{2}-2$$ (in the variable $y=\alpha^{1/2}x$).

  7. State the Hermite polynomial recurrence relation.

    $$H_{v+1}(y) = 2y\,H_v(y) - 2v\,H_{v-1}(y)$$

  8. How does an anharmonic (Morse) potential differ from the harmonic potential, and what are the spectral consequences?

    The Morse potential $V=D_e(1-e^{-a(r-r_e)})^{2}$ is asymmetric and allows bond dissociation. Energy levels converge (spacing decreases at high $v$), there are a finite number of levels, and overtone transitions ($\Delta v=\pm2,\pm3$) become allowed, unlike the equally-spaced harmonic levels.

  9. Write the anharmonic oscillator energy expression including the anharmonicity constant.

    $$E_v = \left(v+\tfrac{1}{2}\right)\hbar\omega - \left(v+\tfrac{1}{2}\right)^{2}\hbar\omega\,x_e + \cdots$$ where $x_e$ is the anharmonicity constant; the negative term causes level convergence.

  10. Write the operators for the square of total angular momentum $\hat{L}^{2}$ and its $z$-component $\hat{L}_z$ in spherical coordinates.

    $$\hat{L}_z = -i\hbar\frac{\partial}{\partial \phi}, \qquad \hat{L}^{2} = -\hbar^{2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]$$

  11. State the eigenvalues of $\hat{L}^{2}$ and $\hat{L}_z$ acting on spherical harmonics $Y_l^{m}$.

    $$\hat{L}^{2}Y_l^{m} = l(l+1)\hbar^{2}\,Y_l^{m}, \qquad \hat{L}_z Y_l^{m} = m\hbar\,Y_l^{m}$$ with $l=0,1,2,\dots$ and $m=-l,\dots,+l$.

  12. State the commutation relations among the angular momentum components.

    $$[\hat{L}_x,\hat{L}_y]=i\hbar\hat{L}_z,\;[\hat{L}_y,\hat{L}_z]=i\hbar\hat{L}_x,\;[\hat{L}_z,\hat{L}_x]=i\hbar\hat{L}_y$$ and $[\hat{L}^{2},\hat{L}_i]=0$. So $L^2$ and one component (e.g. $L_z$) can be known simultaneously, but not two components.

  13. Define the angular momentum raising and lowering (ladder) operators and their action.

    $$\hat{L}_{\pm} = \hat{L}_x \pm i\hat{L}_y$$ They raise/lower $m$: $\hat{L}_{\pm}Y_l^{m} \propto Y_l^{m\pm1}$, while leaving $l$ unchanged.

  14. Give the energy levels and degeneracy of a rigid rotor (rotational energy levels).

    $$E_J = \frac{\hbar^{2}}{2I}J(J+1) = hcB\,J(J+1), \quad J=0,1,2,\dots$$ Each level has degeneracy $g_J=2J+1$; $I=\mu r^{2}$ and $B=\dfrac{h}{8\pi^{2}cI}$.

  15. What is the rotational constant $B$ and how does it relate to bond length?

    $B=\dfrac{h}{8\pi^{2}cI}$ (in cm$^{-1}$), with $I=\mu r^{2}$. A larger moment of inertia (longer bond or heavier atoms) gives a smaller $B$, so $B$ lets one determine bond lengths from rotational spectra.

  16. What is the form of a hydrogenic atomic orbital wavefunction?

    $$\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r)\,Y_l^{m}(\theta,\phi)$$ It factors into a radial part $R_{n,l}(r)$ and an angular part (spherical harmonic) $Y_l^{m}$.

  17. Give the allowed values and meaning of the quantum numbers $n$, $l$, and $m_l$.

    $n=1,2,3,\dots$ (principal, size/energy); $l=0,1,\dots,n-1$ (azimuthal, shape, gives s,p,d,f); $m_l=-l,\dots,+l$ (magnetic, orientation), giving $2l+1$ orbitals per subshell.

  18. State the energy of a hydrogen-like atom and how many nodes an orbital has.

    $$E_n = -\frac{Z^{2}me^{4}}{8\varepsilon_0^{2}h^{2}n^{2}} = -13.6\,\frac{Z^{2}}{n^{2}}\;\text{eV}$$ Total nodes $=n-1$: angular nodes $=l$, radial nodes $=n-l-1$.

  19. Define the radial distribution function (RDF) and write its expression.

    The RDF gives the probability of finding the electron in a thin shell at radius $r$: $$P(r) = r^{2}R_{n,l}^{2}(r) \quad (\text{or } 4\pi r^{2}\psi^{2}\text{ for s})$$ The $r^{2}$ factor accounts for the increasing shell volume with $r$.

  20. For the hydrogen 1s orbital, where does the radial distribution function peak?

    The RDF $P(r)=4r^{2}a_0^{-3}e^{-2r/a_0}$ peaks at $r=a_0$ (the Bohr radius, $\approx 52.9$ pm)—the most probable radius despite $\psi^{2}$ being maximum at the nucleus.

  21. What is the orbital approximation in many-electron atoms?

    It approximates the total wavefunction as a product of one-electron orbitals: $\Psi(1,2,\dots,n)\approx \psi_1(1)\psi_2(2)\cdots\psi_n(n)$. Each electron is assigned its own hydrogen-like orbital, ignoring instantaneous electron-electron correlation.

  22. What is electron spin, and what are the spin quantum number values for an electron?

    Spin is an intrinsic angular momentum of the electron with spin quantum number $s=\tfrac{1}{2}$. Its $z$-projection $m_s=\pm\tfrac{1}{2}$, giving spin states $\alpha$ (spin up) and $\beta$ (spin down).

  23. Give the eigenvalue equations for the spin operators $\hat{S}^{2}$ and $\hat{S}_z$.

    $$\hat{S}^{2}\chi = s(s+1)\hbar^{2}\chi = \tfrac{3}{4}\hbar^{2}\chi, \qquad \hat{S}_z\chi = m_s\hbar\chi = \pm\tfrac{1}{2}\hbar\chi$$

  24. State the Pauli exclusion principle in both its forms.

    No two electrons in an atom can have the same set of all four quantum numbers $(n,l,m_l,m_s)$. Equivalently (antisymmetry principle): the total wavefunction of fermions must be antisymmetric under exchange of any two electrons.

  25. What does the antisymmetry requirement mean physically, and what does it forbid?

    Exchanging two electrons must change the sign of the total wavefunction: $\Psi(1,2)=-\Psi(2,1)$. It forbids two electrons from occupying the same spin-orbital, since $\Psi$ would then vanish—the basis of the exclusion principle.

  26. For two electrons in helium, write the spatial-spin wavefunction respecting antisymmetry for the ground state.

    Ground state: symmetric spatial $\times$ antisymmetric (singlet) spin: $$\Psi = 1s(1)1s(2)\cdot\frac{1}{\sqrt{2}}\big[\alpha(1)\beta(2)-\beta(1)\alpha(2)\big]$$

  27. What is a Slater determinant and why is it used?

    It is an antisymmetrized product of spin-orbitals written as a determinant. Because swapping two rows (electrons) or columns changes its sign, it automatically satisfies the Pauli antisymmetry principle, and it vanishes if two spin-orbitals are identical.

  28. Write the Slater determinant for a two-electron system with spin-orbitals $\chi_a$ and $\chi_b$.

    $$\Psi(1,2) = \frac{1}{\sqrt{2}}\begin{vmatrix} \chi_a(1) & \chi_b(1) \\ \chi_a(2) & \chi_b(2) \end{vmatrix} = \frac{1}{\sqrt{2}}\big[\chi_a(1)\chi_b(2)-\chi_b(1)\chi_a(2)\big]$$

  29. What is the normalization factor of an $N$-electron Slater determinant, and what do its rows and columns represent?

    The prefactor is $\dfrac{1}{\sqrt{N!}}$. Each column corresponds to one spin-orbital and each row to one electron (or vice versa); interchanging two electrons swaps two rows, flipping the sign as required by antisymmetry.

  30. Compare the energy-level spacing of the particle in a box, harmonic oscillator, and rigid rotor.

    Particle in a box: spacing increases with $n$ ($E\propto n^{2}$). Harmonic oscillator: equally spaced ($\Delta E=\hbar\omega$). Rigid rotor: spacing increases linearly with $J$ ($\Delta E = 2B(J+1)$, since $E\propto J(J+1)$).

GATE Chemistry Physical Chemistry FAQ

What is in the GATE Chemistry Physical Chemistry syllabus?

GATE Chemistry Physical Chemistry is split into 15 chapters: Postulates of Quantum Mechanics, Particle in a Box, Harmonic Oscillator, Rotational Motion, Hydrogen and Hydrogen-like Atoms, Multi-electron Atoms, Approximation Methods, Atomic Units, Molecular Structure and Chemical Bonding, Group Theory, Spectroscopy, Equilibrium, Statistical Thermodynamics, Kinetics and Surfaces and Interfaces. Together they contain 91 topics.

How many chapters are in GATE Chemistry Physical Chemistry?

15 chapters, containing 91 topics. The largest chapter is Equilibrium, with 27 topics.

Are there flashcards for GATE Chemistry Physical Chemistry?

Yes — a 50-card Physical Chemistry deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.