Physical Chemistry Study Notes

Cheatsheet Content

### 1. Thermodynamics: Basic Concepts - **System:** Part of the universe under study. - **Open:** Exchanges matter and energy. - **Closed:** Exchanges energy, not matter. - **Isolated:** Exchanges neither matter nor energy. - **Surroundings:** Everything else. - **Boundary:** Separates system from surroundings. - **State Functions:** Properties that depend only on the current state of the system, not on how that state was reached (e.g., $P, V, T, U, H, S, G$). - **Path Functions:** Properties that depend on the path taken (e.g., $q, w$). - **Extensive Properties:** Depend on the amount of substance (e.g., $V, U, H, S, G$). - **Intensive Properties:** Independent of the amount of substance (e.g., $P, T, \rho$). ### 2. First Law of Thermodynamics - **Statement:** Energy cannot be created or destroyed, only transferred or transformed. - **Mathematical Form:** $\Delta U = q + w$ - $\Delta U$: Change in internal energy. - $q$: Heat absorbed by the system (positive if absorbed, negative if released). - $w$: Work done on the system (positive if done on, negative if done by). - **Work (Pressure-Volume Work):** $w = -\int P_{ext} dV$ - **Expansion:** $w 0$ (work done on system). - **Reversible Isothermal Expansion (Ideal Gas):** $w = -nRT \ln(V_f/V_i)$ - **Enthalpy (H):** $H = U + PV$ - At constant pressure, $\Delta H = q_p$. - For reactions involving gases, $\Delta H = \Delta U + \Delta n_g RT$. - **Heat Capacity:** - **Constant Volume ($C_V$):** $C_V = (\partial U / \partial T)_V$. For ideal gas, $C_V = (3/2)R$ (monatomic), $C_V = (5/2)R$ (diatomic). - **Constant Pressure ($C_P$):** $C_P = (\partial H / \partial T)_P$. For ideal gas, $C_P = C_V + R$. ### 3. Second Law of Thermodynamics - **Statement:** The entropy of an isolated system always increases in a spontaneous process. - **Clausius Statement:** Heat cannot spontaneously flow from a colder body to a hotter body. - **Kelvin-Planck Statement:** It is impossible to construct a device that operates in a cycle and produces no effect other than the extraction of heat from a reservoir and the performance of an equal amount of work. - **Entropy (S):** A measure of disorder or randomness. - **Change in Entropy:** $dS = dq_{rev}/T$ - **Isothermal Process:** $\Delta S = q_{rev}/T = nR \ln(V_f/V_i)$ (for ideal gas) - **Temperature Change:** $\Delta S = nC_P \ln(T_f/T_i)$ (constant P), $\Delta S = nC_V \ln(T_f/T_i)$ (constant V) - **Phase Transition:** $\Delta S_{trs} = \Delta H_{trs}/T_{trs}$ - **Entropy of the Universe:** $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} \ge 0$ - Spontaneous process: $\Delta S_{univ} > 0$ - Reversible process: $\Delta S_{univ} = 0$ - Impossible process: $\Delta S_{univ} ### 4. Third Law of Thermodynamics - **Statement:** The entropy of a perfect crystalline substance at absolute zero (0 K) is zero. - **Absolute Entropy:** $S(T) = \int_0^T (C_P/T') dT'$ (for a pure substance) - Allows for the calculation of absolute entropies. ### 5. Gibbs and Helmholtz Energies - **Helmholtz Energy (A):** $A = U - TS$ - For a process at constant $T$ and $V$: $\Delta A \le 0$ (spontaneous if $\Delta A ### 6. Chemical Potential - **Definition:** $\mu_i = (\partial G / \partial n_i)_{P,T,n_{j \ne i}}$. The partial molar Gibbs energy of component $i$. - **Driving Force:** Substances tend to move from regions of higher chemical potential to lower chemical potential. - **For an Ideal Gas:** $\mu = \mu^\circ + RT \ln(P/P^\circ)$ - **For a component in a mixture:** $\mu_i = \mu_i^\circ + RT \ln a_i$, where $a_i$ is the activity. - **Equilibrium:** At equilibrium, the chemical potential of each component is uniform throughout the system. ### 7. Phase Equilibria - **Phase:** A region of uniform physical and chemical properties. - **Phase Rule (Gibbs):** $F = C - P + 2$ - $F$: Degrees of freedom (number of intensive variables that can be varied independently). - $C$: Number of components. - $P$: Number of phases. - **Clapeyron Equation:** $dP/dT = \Delta H_{trs} / (T \Delta V_{trs})$ - Describes the slope of phase boundaries. - **Clausius-Clapeyron Equation (for liquid-vapor equilibrium, assuming ideal gas and $\Delta V \approx V_g$):** $\ln(P_2/P_1) = -(\Delta H_{vap}/R)(1/T_2 - 1/T_1)$ - **Phase Diagrams:** Graphical representation of the stable phases of a substance as a function of $P$ and $T$. - **Triple Point:** All three phases coexist ($F=0$). - **Critical Point:** Above this, liquid and gas phases are indistinguishable (supercritical fluid). ### 8. Solutions - **Ideal Solution:** Follows Raoult's Law. $\Delta H_{mix} = 0$, $\Delta V_{mix} = 0$. - **Raoult's Law:** $P_A = x_A P_A^\circ$ (vapor pressure of component A in solution). - **Non-Ideal Solutions:** Deviations from Raoult's Law. - **Positive Deviation:** Weaker A-B interactions, higher vapor pressure than ideal. - **Negative Deviation:** Stronger A-B interactions, lower vapor pressure than ideal. - **Colligative Properties:** Properties that depend only on the concentration of solute particles, not their identity. - **Vapor Pressure Lowering:** $\Delta P = x_B P_A^\circ$ - **Boiling Point Elevation:** $\Delta T_b = i K_b m$ - **Freezing Point Depression:** $\Delta T_f = -i K_f m$ - **Osmotic Pressure ($\Pi$):** $\Pi = i c RT$ - $i$: Van't Hoff factor (number of particles per formula unit). - $K_b, K_f$: Molal boiling/freezing point constants. - $m$: Molality. - $c$: Molar concentration. ### 9. Chemical Kinetics: Rate Laws - **Reaction Rate:** Change in concentration of reactants or products over time. - Rate $= -(1/a) d[A]/dt = -(1/b) d[B]/dt = (1/c) d[C]/dt$ for $aA + bB \to cC$. - **Rate Law:** Expresses rate as a function of reactant concentrations. Rate $= k[A]^x[B]^y$. - $k$: Rate constant. - $x, y$: Reaction orders (determined experimentally). - **Overall Order:** $x+y$. - **Integrated Rate Laws:** - **Zero Order:** $[A]_t = [A]_0 - kt$. Half-life $t_{1/2} = [A]_0 / (2k)$. - **First Order:** $\ln[A]_t = \ln[A]_0 - kt$. Half-life $t_{1/2} = \ln(2) / k$. - **Second Order:** $1/[A]_t = 1/[A]_0 + kt$. Half-life $t_{1/2} = 1 / (k[A]_0)$. - **Arrhenius Equation:** $k = A e^{-E_a/RT}$ - $A$: Pre-exponential factor (frequency factor). - $E_a$: Activation energy. - $\ln(k_2/k_1) = -(E_a/R)(1/T_2 - 1/T_1)$. ### 10. Reaction Mechanisms - **Elementary Steps:** Individual steps in a reaction mechanism. - **Molecularity:** Number of molecules involved in an elementary step (unimolecular, bimolecular, termolecular). - **Rate-Determining Step (RDS):** The slowest step in a mechanism, which limits the overall reaction rate. - **Steady-State Approximation:** Assumes that the concentration of an intermediate remains constant during the reaction ($d[Intermediate]/dt = 0$). - **Pre-Equilibrium Approximation:** Assumes that an initial fast, reversible step reaches equilibrium rapidly. ### 11. Catalysis - **Catalyst:** A substance that increases the rate of a reaction without being consumed. - **Mechanism:** Lowers the activation energy ($E_a$) by providing an alternative reaction pathway. - **Types:** - **Homogeneous:** Catalyst and reactants are in the same phase. - **Heterogeneous:** Catalyst and reactants are in different phases. - **Enzyme:** Biological catalysts (proteins). - **Michaelis-Menten Kinetics (Enzyme Catalysis):** - $E + S \rightleftharpoons ES \to E + P$ - Rate $v = (V_{max}[S]) / (K_M + [S])$ - $V_{max}$: Maximum rate. - $K_M$: Michaelis constant (substrate concentration at $V_{max}/2$). ### 12. Quantum Mechanics: Basic Concepts - **Wave-Particle Duality:** Matter exhibits both wave-like and particle-like properties. - **De Broglie Wavelength:** $\lambda = h/p = h/(mv)$ - **Heisenberg Uncertainty Principle:** It is impossible to simultaneously know with perfect accuracy both the position and momentum of a particle: $\Delta x \Delta p \ge h/(4\pi)$ - **Schrödinger Equation:** $H\Psi = E\Psi$ - $H$: Hamiltonian operator (total energy). - $\Psi$: Wavefunction (describes the state of a quantum system). - $E$: Energy of the system. - **Born Interpretation:** $|\Psi|^2$ represents the probability density of finding a particle at a given point in space. - **Operators:** Mathematical representations of physical observables (e.g., momentum $p_x = -i\hbar (\partial/\partial x)$, energy $E = i\hbar (\partial/\partial t)$). ### 13. Particle in a Box - **1D Box:** - **Potential:** $V(x) = 0$ for $0 \le x \le L$, $V(x) = \infty$ otherwise. - **Wavefunctions:** $\Psi_n(x) = \sqrt{2/L} \sin(n\pi x/L)$, where $n=1, 2, 3, ...$ - **Energies:** $E_n = n^2 h^2 / (8mL^2)$ - **Quantization:** Energy levels are discrete. - **Zero-Point Energy:** $E_1 > 0$ (due to uncertainty principle). - **3D Box:** - **Energies:** $E_{n_x,n_y,n_z} = (h^2/(8m)) (n_x^2/L_x^2 + n_y^2/L_y^2 + n_z^2/L_z^2)$ - **Degeneracy:** Multiple wavefunctions (states) having the same energy. ### 14. Harmonic Oscillator - **Model:** Describes vibrational motion. - **Potential Energy:** $V(x) = (1/2)kx^2$ - **Wavefunctions:** Hermite polynomials multiplied by a Gaussian function. - **Energies:** $E_v = (v + 1/2)h\nu$, where $v = 0, 1, 2, ...$ - $\nu = (1/(2\pi))\sqrt{k/\mu}$ (classical frequency). - $\mu$: Reduced mass. - **Zero-Point Energy:** $E_0 = (1/2)h\nu$. - **Selection Rule for IR Spectroscopy:** $\Delta v = \pm 1$. ### 15. Rigid Rotor - **Model:** Describes rotational motion (e.g., diatomic molecules). - **Energies:** $E_J = BJ(J+1)$, where $J = 0, 1, 2, ...$ - $B = h^2/(8\pi^2 I)$ (rotational constant). - $I = \mu r^2$ (moment of inertia). - $\mu$: Reduced mass. - $r$: Bond length. - **Degeneracy:** $g_J = 2J+1$. - **Selection Rule for Rotational (Microwave) Spectroscopy:** $\Delta J = \pm 1$. - **Requires Permanent Dipole Moment:** For absorption/emission. ### 16. Atomic Structure (Hydrogen Atom) - **Schrödinger Equation Solution:** Yields quantum numbers: - **Principal (n):** $1, 2, 3, ...$ (energy and size). - **Angular Momentum (l):** $0, 1, ..., n-1$ (shape, $s, p, d, f$ orbitals). - **Magnetic ($m_l$):** $-l, ..., 0, ..., +l$ (orientation). - **Spin ($m_s$):** $+1/2, -1/2$. - **Orbital Energies:** $E_n = -R_H/n^2$ (for hydrogenic atoms). - **Radial Distribution Function:** $4\pi r^2 |\Psi(r)|^2$ (probability of finding electron at distance $r$). - **Angular Wavefunctions:** Describe the shape of orbitals. ### 17. Molecular Orbital Theory (MOT) - **LCAO Approximation:** Linear Combination of Atomic Orbitals. - $\Psi_{MO} = c_A \phi_A + c_B \phi_B$ - **Bonding Orbitals:** Lower energy, increased electron density between nuclei (constructive interference). - **Antibonding Orbitals:** Higher energy, decreased electron density between nuclei (destructive interference, node). - **Bond Order:** $(1/2)(\text{number of electrons in bonding MOs} - \text{number of electrons in antibonding MOs})$. - **Homonuclear Diatomics (e.g., O2, N2):** - Energy levels: $\sigma_{1s}, \sigma^*_{1s}, \sigma_{2s}, \sigma^*_{2s}, \sigma_{2p}, (\pi_{2p}, \pi_{2p}), (\pi^*_{2p}, \pi^*_{2p}), \sigma^*_{2p}$. (Order of $\sigma_{2p}$ and $\pi_{2p}$ can swap for N2 and lighter). - **Heteronuclear Diatomics:** More complex, atomic orbitals of different energies. ### 18. Spectroscopy: General Principles - **Interaction with Electromagnetic Radiation:** Molecules absorb or emit light at specific frequencies. - **Energy Levels:** Quantized energy levels (rotational, vibrational, electronic). - **Selection Rules:** Govern which transitions are allowed. - **Beer-Lambert Law:** $A = \epsilon b c$ - $A$: Absorbance. - $\epsilon$: Molar absorptivity. - $b$: Path length. - $c$: Concentration. ### 19. Rotational Spectroscopy (Microwave) - **Transitions:** Between rotational energy levels. - **Region:** Microwave. - **Condition:** Molecule must have a permanent electric dipole moment. - **Spectrum:** Series of equally spaced lines with spacing $2B$. - **Information:** Moment of inertia, bond length. ### 20. Vibrational Spectroscopy (Infrared, Raman) - **Transitions:** Between vibrational energy levels. - **Region:** Infrared. - **IR Active:** Change in dipole moment during vibration. - **Raman Active:** Change in polarizability during vibration. - **Spectrum:** Peaks correspond to specific vibrational modes. - **Information:** Functional groups, bond strengths. ### 21. Electronic Spectroscopy (UV-Vis) - **Transitions:** Between electronic energy levels. - **Region:** Ultraviolet-Visible. - **Mechanism:** Absorption of photons promotes electrons to higher energy orbitals. - **Franck-Condon Principle:** Electronic transitions are fast, so nuclei positions remain essentially fixed during the transition. - **Information:** Conjugation, presence of chromophores. ### 22. Statistical Thermodynamics - **Bridge between Microscopic and Macroscopic:** Relates molecular properties to bulk thermodynamic properties. - **Boltzmann Distribution:** $N_i/N = (g_i e^{-E_i/kT}) / Q$ - $N_i$: Number of molecules in state $i$. - $g_i$: Degeneracy of state $i$. - $Q$: Partition function. - **Partition Function (Q):** Sum over all possible states, weighted by Boltzmann factor. $Q = \sum_i g_i e^{-E_i/kT}$ - **Total Partition Function:** $Q_{total} = Q_{trans} Q_{rot} Q_{vib} Q_{elec}$ - **Relationship to Thermodynamic Properties:** - $U = kT^2 (\partial \ln Q / \partial T)_V$ - $S = k \ln Q + k T (\partial \ln Q / \partial T)_V$ - $A = -kT \ln Q$ - $P = kT (\partial \ln Q / \partial V)_T$ - **Translational Partition Function (3D):** $q_{trans} = (2\pi mkT)^{3/2} V / h^3$ ### 23. Electrochemistry: Basic Concepts - **Redox Reactions:** Involve transfer of electrons. - **Oxidation:** Loss of electrons (anode). - **Reduction:** Gain of electrons (cathode). - **Electrochemical Cells:** - **Galvanic (Voltaic) Cell:** Spontaneous reaction generates electrical energy. - **Electrolytic Cell:** Non-spontaneous reaction driven by external electrical energy. - **Standard Electrode Potential ($E^\circ$):** Reduction potential relative to Standard Hydrogen Electrode (SHE, $E^\circ = 0$ V). - **Cell Potential ($E_{cell}$):** $E_{cell} = E_{cathode} - E_{anode}$ ### 24. Nernst Equation - **Relates cell potential to concentrations:** $E = E^\circ - (RT/nF) \ln Q$ - $R$: Gas constant. - $T$: Temperature (K). - $n$: Number of electrons transferred. - $F$: Faraday constant (96485 C/mol). - $Q$: Reaction quotient. - **At 298 K (25 °C):** $E = E^\circ - (0.0592/n) \log Q$ - **Relationship to Gibbs Energy:** $\Delta G = -nFE_{cell}$ - At standard conditions: $\Delta G^\circ = -nFE^\circ$ - **At Equilibrium:** $E_{cell} = 0$, $Q = K_{eq}$. So, $E^\circ = (RT/nF) \ln K_{eq}$. ### 25. Ionic Conductivity - **Conductance (G):** Reciprocal of resistance ($G = 1/R$). Unit: Siemens (S). - **Conductivity ($\kappa$):** $G \times (L/A)$. Unit: S/m. - **Molar Conductivity ($\Lambda_m$):** $\kappa / c$. Unit: S m$^2$/mol. - **Kohlrausch's Law of Independent Migration of Ions:** For strong electrolytes at infinite dilution, $\Lambda_m^\circ = \nu_+ \lambda_+^\circ + \nu_- \lambda_-^\circ$. - $\lambda^\circ$: Limiting ionic molar conductivity. - **Debye-Hückel Theory:** Describes the behavior of ions in dilute solutions, accounting for interionic interactions.