JEE Chemistry States Of Matter Notes Set 05

Rate of Diffusion and Graham's Law

1. Rate of diffusion (or effusion) is the volume, moles or distance of gas passing per unit time: r = V/t = n/t = d/t.
2. Graham's law (for non-reacting gases at the same T and P): r is proportional to \( \frac{1}{\sqrt{d}} \), where d is density, so \( \frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}} \).
3. At constant temperature, r is also proportional to \( \frac{P}{\sqrt{d}} \), so \( \frac{r_1}{r_2}=\frac{P_1}{P_2}\sqrt{\frac{M_2}{M_1}} \).
4. For equal volumes, the times taken are in the ratio \( \frac{t_1}{t_2}=\sqrt{\frac{M_1}{M_2}} \): the heavier gas takes longer.
5. Example: if 4 g of oxygen diffuses through a narrow hole, the mass w of hydrogen diffusing in the same time satisfies \( \frac{4/32}{w/2}=\sqrt{\frac{2}{32}} \), giving w = 1 g.

Real Gases and the van der Waals Equation

1. A real gas obeys PV = nRT only at low pressure and high temperature. Two kinetic-theory assumptions fail for real gases: molecules do attract each other, and their own volume is not negligible.
2. Pressure correction: attraction reduces the wall pressure, so the ideal pressure is \( P+\frac{an^2}{V^2} \).
3. Volume correction: the free volume is V − nb, where b is the excluded volume (four times the actual volume of one mole of molecules).
4. The van der Waals equation is \( \left(P+\frac{an^2}{V^2}\right)(V-nb)=nRT \).
5. Constant a measures the strength of intermolecular attraction (unit atm L² mol⁻²). A larger a means easier liquefaction.
6. Constant b measures the effective molecular size (unit L mol⁻¹). A larger b means a larger incompressible volume.

Compressibility Factor

1. \( Z=\frac{V_{real}}{V_{ideal}}=\frac{PV}{nRT} \).
2. Z = 1 for an ideal gas. Z < 1 means attraction dominates (gas is easily compressed and liquefied). Z > 1 means repulsion dominates (gas is hard to compress).
3. At very low pressure, Z is close to 1. At moderate pressure, \( Z=1-\frac{na}{VRT} \) (below 1). At very high pressure, \( Z=1+\frac{Pb}{RT} \) (above 1).
4. Hydrogen and helium have Z > 1 at almost all pressures at room temperature because their a is very small.

Critical Constants and Boyle Temperature

1. Critical temperature T_c is the highest temperature at which a gas can be liquefied by pressure alone: \( T_c=\frac{8a}{27Rb} \).
2. Critical pressure \( P_c=\frac{a}{27b^2} \) is the pressure needed to liquefy the gas at T_c.
3. Critical volume \( V_c=3b \) is the volume of one mole at T_c and P_c.
4. Boyle temperature \( T_b=\frac{a}{Rb} \) is the temperature at which a real gas obeys Boyle's law over a wide range of pressure.
5. The relation \( T_b=\frac{27}{8}T_c \) follows from the two formulas.