Physical Chemistry for the Chemistry Olympiad — NSEC and INChO

Weightage: Physical chemistry is the most quantitative of the three chemistry areas and a large share of NSEC and INChO marks. The syllabus is described as building on Classes XI and XII with extension beyond it, so confirm the current syllabus and format with HBCSE. Method matters: derive the relation you need from a few fundamentals and carry units through.

1. Thermodynamics and equilibrium

The master relation is $\Delta G=\Delta H-T\Delta S$. A process is spontaneous at constant $T$ and $P$ when $\Delta G<0$. The link to equilibrium is:

$\Delta G^{\circ }=-RT\ln K,\Delta G=\Delta G^{\circ }+RT\ln Q$

so a reaction proceeds in the direction that lowers $\Delta G$, and sits at equilibrium when $Q=K$.

Worked example. For $K=100$ at $298$ K, $\Delta G^{\circ }=-(8.314)(298)\ln 100=-11.4$ kJ/mol.

The van't Hoff equation gives the effect of temperature:

$\ln \frac{K_2}{K_1}=-\frac{\Delta H^{\circ }}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$

An endothermic reaction ($\Delta H>0$) has a $K$ that rises with temperature. Le Chatelier's principle predicts shifts qualitatively. For gases, $K_p=K_c(RT{)}^{\Delta n}$, and $K_p$ depends on temperature only.

Heat and work. For an ideal gas, $\Delta U=n{C}_v\Delta T$ and $\Delta H=n{C}_p\Delta T$, with $C_p-C_v=R$. Hess's law lets you add reactions. A reaction's enthalpy can be estimated from bond enthalpies as bonds broken minus bonds formed.

2. Chemical kinetics

The rate law is found from experiment, not from the stoichiometry. For a first-order reaction:

$\ln [A]=\ln [A{]}_0-kt,t_{1/2}=\frac{0.693}{k}$

For second order in one reactant, $\frac{1}{[A]}=\frac{1}{[A{]}_0}+kt$ and $t_{1/2}=\frac{1}{k[A{]}_0}$. For zero order, $[A]=[A{]}_0-kt$.

The Arrhenius equation is $k=A{e}^{-E_a/RT}$, so

$\ln \frac{k_2}{k_1}=\frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right).$

Worked example. With $E_a=50$ kJ/mol, raising $T$ from $300$ K to $310$ K gives $\ln (k_2/k_1)=6014\times 1.075\times 10^{-4}=0.647$, so the rate rises by a factor of $1.9$, close to the rule that rates double for a $10$ K rise.

Mechanisms. The rate law reflects the slowest (rate-determining) step. Use the steady-state approximation for reactive intermediates, setting their net rate of change to zero. A catalyst lowers the activation energy and does not change $K$.

3. Electrochemistry

For a redox reaction in a cell, $\Delta G=-nFE$ with $F=96485$ C/mol, and under standard conditions $\Delta G^{\circ }=-nF{E}^{\circ }=-RT\ln K$. The Nernst equation at $298$ K is:

$E=E^{\circ }-\frac{0.0592}{n}\log Q$

Worked example. For the Daniell cell $E^{\circ }=1.10$ V with $n=2$. With $[{\text{Zn}}^{2+}]=0.1$ M and $[{\text{Cu}}^{2+}]=0.001$ M, $Q=100$, so $E=1.10-0.0296\times 2=1.04$ V.

Electrolysis: the mass deposited follows Faraday's law, $m=\frac{MIt}{nF}$. A concentration cell has $E^{\circ }=0$ and runs on the difference in concentration.

4. Gases and colligative properties

The ideal gas law $PV=nRT$ is corrected for real gases by the van der Waals equation $\left(P+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$, where $a$ reflects attraction and $b$ the molecular volume. Real gases approach ideal behaviour at high temperature and low pressure.

Colligative properties depend on the number of solute particles. With the van't Hoff factor $i$:

• Boiling point elevation: $\Delta T_b=i{K}_{b}m$, and freezing point depression: $\Delta T_f=i{K}_{f}m$.
• Osmotic pressure: $\pi =iCRT$.
• Relative lowering of vapour pressure: $\frac{\Delta P}{P^{\circ }}=x_{solute}$ (Raoult's law).

Worked example. A $0.1$ M glucose solution at $298$ K has $\pi =0.1\times 0.0821\times 298=2.45$ atm.

5. Chemical bonding, structure and spectra

Quantum ideas appear in the particle in a box and in atomic spectra. The Bohr energy $E_n=-13.6Z^2/n^2$ eV gives the hydrogen-like spectrum, and the Rydberg formula gives line wavelengths. The de Broglie relation is $\lambda =h/p$.

6. Solving olympiad problems

1. Write the governing equation from a short list of fundamentals.
2. Keep units through the calculation, such as J versus kJ for $R=8.314$ J mol${}^{-1}$ K${}^{-1}$.
3. Convert temperature to kelvin.
4. Use logarithms carefully, distinguishing $\ln$ and ${\log }_{10}$.
5. Check the sign and the size of the answer against chemical sense.

Common traps

• Mixing kJ and J in $\Delta G=-RT\ln K$.
• Reading the rate law from the stoichiometry.
• Using $\ln$ where the formula needs ${\log }_{10}$ in the Nernst equation.
• Forgetting the van't Hoff factor for ionic solutes.
• Assuming a catalyst changes the equilibrium constant.

Memory aids

• "Delta G is minus RT ln K": the central link.
• "Slowest step sets the rate law."
• "0.0592 over n times log Q": Nernst at 298 K.

Summary

Thermodynamics links free energy to the equilibrium constant, and the van't Hoff and Arrhenius equations show how temperature changes $K$ and $k$. Kinetics uses integrated laws and the rate-determining step.

Electrochemistry connects $E$, $\Delta G$ and $K$ through the Nernst equation, and colligative properties follow from particle counts with the van't Hoff factor.

Exam protocol

• State the governing equation and the units of each symbol.
• Convert to kelvin and consistent energy units.
• Distinguish natural and common logarithms.
• Confirm the current syllabus and format with HBCSE.