States of Matter (Quantitative)

Time limit: 20 minutes
Total marks: 30

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Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. At constant temperature, if the pressure on a fixed mass of gas is doubled, its volume becomes:
(a) doubled (b) halved (c) unchanged (d) quadrupled

Q2. The ideal gas equation is $PV = nRT$. The SI unit of $R$ is:
(a) $\text{J mol}^{-1}\text{K}^{-1}$ (b) $\text{atm L}$ (c) $\text{K mol}^{-1}$ (d) $\text{J mol}$

Q3. According to Graham's law, the rate of effusion of a gas is proportional to:
(a) $M$ (b) $\sqrt{M}$ (c) $1/M$ (d) $1/\sqrt{M}$

Q4. In the kinetic theory expression $P = \tfrac{1}{3}\rho v_{rms}^2$, the symbol $\rho$ represents the gas:
(a) pressure (b) density (c) temperature (d) molar mass

Q5. For a Maxwell–Boltzmann speed distribution, the correct order of speeds is:
(a) $v_{mp} < v_{avg} < v_{rms}$ (b) $v_{rms} < v_{avg} < v_{mp}$ (c) $v_{avg} < v_{mp} < v_{rms}$ (d) all equal

Q6. The compressibility factor $Z$ for an ideal gas is:
(a) 0 (b) 1 (c) depends on $T$ (d) infinite

Q7. In the van der Waals equation, the constant $a$ accounts for:
(a) molecular volume (b) intermolecular attraction (c) temperature (d) number of moles

Q8. The packing fraction of a body-centred cubic (BCC) unit cell is approximately:
(a) 52% (b) 68% (c) 74% (d) 100%

Q9. The coordination number of Na$^+$ in the NaCl (rock salt) structure is:
(a) 4 (b) 6 (c) 8 (d) 12

Q10. A Frenkel defect involves:
(a) missing cation–anion pairs (b) an ion displaced to an interstitial site (c) extra electrons (d) impurity atoms

Q11. Doping silicon with phosphorus (group 15) produces:
(a) p-type semiconductor (b) n-type semiconductor (c) insulator (d) conductor with no carriers

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Section B — Matching (5 marks)

Q12. Match each item in Column I with the correct entry in Column II. (1 mark each)

Column I		Column II
(i) Charles's law		(P) $P/T$ constant at fixed $V$
(ii) Gay-Lussac's law		(Q) constant $b$ in vdW eqn
(iii) Molecular volume correction		(R) $V/T$ constant at fixed $P$
(iv) Number of tetrahedral voids in FCC (n atoms)		(S) $2n$
(v) Critical temperature		(T) above it, gas cannot be liquefied

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Section C — True / False with Justification (2 marks each: 1 T/F + 1 justification)

Q13. The total pressure of a mixture of non-reacting gases equals the sum of their partial pressures.

Q14. Amorphous solids have sharp melting points and long-range order.

Q15. In the CsCl structure, the coordination number of each ion is 8.

Q16. Surface tension of a liquid increases as temperature increases.

Q17. The most probable speed $v_{mp}$ increases with increasing temperature for a given gas.

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[
  {"claim":"BCC packing fraction ≈ 0.68","code":"pf = sqrt(3)*pi/8; result = abs(float(pf)-0.6802) < 0.001"},
  {"claim":"Speed ratio vmp:vavg:vrms = sqrt2:sqrt(8/pi):sqrt3 ordered ascending","code":"vmp=sqrt(2); vavg=sqrt(Rational(8,1)/pi); vrms=sqrt(3); result = (float(vmp)<float(vavg)) and (float(vavg)<float(vrms))"},
  {"claim":"Boyle: doubling P halves V","code":"P,V=symbols('P V',positive=True); k=P*V; Vnew=solve(Eq(2*P*var('x'),k),var('x'))[0] if False else k/(2*P); result = simplify(Vnew - V/2)==0"},
  {"claim":"Z=PV/nRT equals 1 for ideal gas","code":"P,V,n,R,T=symbols('P V n R T',positive=True); Z=(P*V)/(n*R*T); result = simplify(Z.subs(P*V, n*R*T)) == 1"}
]