States of Matter (Quantitative)

Q1. (4 marks)

• Boyle's law: At constant temperature and amount of gas, the volume is inversely proportional to pressure: $PV = \text{constant}$ (or $P_1V_1=P_2V_2$). Constant held: T, n. (2 marks: statement 1 + form 1)
• Charles' law: At constant pressure, the volume is directly proportional to absolute temperature: $\dfrac{V}{T}=\text{constant}$. Constant held: P, n. (2 marks)

Q2. (4 marks) Combined gas law: $\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}$ (1)

$V_2 = \frac{P_1 V_1 T_2}{T_1 P_2}=\frac{1.5\times2.0\times350}{300\times2.0}$ (2)

$V_2 = \frac{1050}{600}=1.75\ \text{L}$ (1)

Q3. (4 marks) Mole fractions: $x_{N_2}=\dfrac{2}{5}=0.4$, $x_{O_2}=\dfrac{3}{5}=0.6$ (1) Dalton's law: $p_i = x_i P_{total}$ (1) $p_{N_2}=0.4\times5.0 = 2.0\ \text{atm}$ (1) $p_{O_2}=0.6\times5.0 = 3.0\ \text{atm}$ (1)

Q4. (4 marks) Graham's law: at constant T and P, the rate of effusion is inversely proportional to the square root of molar mass: $r\propto\dfrac{1}{\sqrt M}$. (2) $\frac{r_{H_2}}{r_{O_2}}=\sqrt{\frac{M_{O_2}}{M_{H_2}}}=\sqrt{\frac{32}{2}}=\sqrt{16}=4$ (2) Hydrogen effuses 4 times faster than oxygen.

Q5. (5 marks) Steps: (1 mark each, capped at 4 + 1 for final)

• Consider $N$ molecules of mass $m$ in a cube of side $l$; a molecule of velocity component $v_x$ hits a wall and reverses momentum, $\Delta p = 2mv_x$.
• Frequency of collisions with one wall $=\dfrac{v_x}{2l}$; force per molecule $=\dfrac{mv_x^2}{l}$.
• Summing over all molecules and using $\overline{v_x^2}=\tfrac13\overline{v^2}$, total force $=\dfrac{Nm\,\overline{v^2}}{3l}$.
• Pressure $P=\dfrac{F}{l^2}=\dfrac{Nm\overline{v^2}}{3l^3}=\dfrac{Nm\overline{v^2}}{3V}$; with density $\rho=\dfrac{Nm}{V}$: $P=\frac{1}{3}\rho\, v_{rms}^2$ (4)
• Since $PV=\tfrac13 Nm v_{rms}^2 = nRT$: $v_{rms}=\sqrt{\dfrac{3RT}{M}}$. (1)

Q6. (4 marks) $v_{mp}=\sqrt{\frac{2RT}{M}},\quad \bar v=\sqrt{\frac{8RT}{\pi M}},\quad v_{rms}=\sqrt{\frac{3RT}{M}}$ (3, one each) Ratio: $v_{mp}:\bar v:v_{rms}=\sqrt2:\sqrt{8/\pi}:\sqrt3 \approx 1:1.128:1.225$ (1)

Q7. (5 marks) (i) $Z=\dfrac{PV}{nRT}$ (=1 for ideal gas). (1) (ii) $Z<1$: attractive forces dominate (gas more compressible than ideal). $Z>1$: repulsive forces / finite molecular volume dominate (less compressible). (2) (iii) $a$ measures magnitude of intermolecular attractive forces (correction to pressure); $b$ (excluded/co-volume) accounts for the finite volume of the molecules. (2)

Q8. (5 marks) BCC: atoms touch along the body diagonal $=\sqrt3\,a = 4r\Rightarrow r=\dfrac{\sqrt3}{4}a$. (2) Number of atoms per cell $=2$ ($8\times\tfrac18 + 1$). (1) $\text{PE}=\frac{2\times\frac43\pi r^3}{a^3}=\frac{2\times\frac43\pi\left(\frac{\sqrt3}{4}a\right)^3}{a^3}$ (1) $=\frac{\frac{8}{3}\pi\cdot\frac{3\sqrt3}{64}}{1}=\frac{\sqrt3\,\pi}{8}\approx0.680=68\%$ (1)

Q9. (5 marks) (i) Schottky: equal numbers of cations and anions missing from lattice sites → decreases density; seen in ionic solids with high coordination and similar ion sizes (e.g. NaCl). Frenkel: an ion (usually smaller cation) leaves its lattice site and occupies an interstitial site → density unchanged; seen where there is large size difference (e.g. ZnS, AgCl). (3) (ii) Group 15 element (e.g. P) has 5 valence electrons; four form bonds with Si, the fifth is a free/mobile electron. These extra electrons act as majority charge carriers → n-type semiconductor. (2)

[
  {"claim":"Q2 combined gas law gives V2 = 1.75 L","code":"P1,V1,T1,P2,T2=1.5,2.0,300,2.0,350; V2=P1*V1*T2/(T1*P2); result = abs(V2-1.75)<1e-9"},
  {"claim":"Q3 partial pressures 2.0 and 3.0 atm","code":"P=5.0; pN2=Rational(2,5)*P; pO2=Rational(3,5)*P; result = (pN2==2) and (pO2==3)"},
  {"claim":"Q4 rate ratio H2/O2 = 4","code":"ratio=sqrt(Rational(32,2)); result = ratio==4"},
  {"claim":"Q8 BCC packing efficiency = sqrt(3)*pi/8 approx 0.6802","code":"a=symbols('a',positive=True); r=sqrt(3)/4*a; PE=2*Rational(4,3)*pi*r**3/a**3; result = abs(float(simplify(PE))-0.6802)<1e-3"}
]