JEE Chemistry States of Matter Notes Set 01

Ideal and Real Gases

An ideal gas has no attraction between its molecules and the molecules have negligible size. Such a gas obeys the kinetic theory, the gas laws and \( PV=nRT \). Gases that fail these conditions are called real gases.

Diffusion and Effusion

Diffusion is the spontaneous mixing of non-reacting gases by movement of molecules from a region of higher concentration to lower concentration, with no external pressure applied.
Effusion is the escape of a gas through a very small hole (pin hole) because of a pressure difference.
Rate of diffusion is the distance moved, the volume or the number of moles of gas that diffuses per unit time, at constant T and P.

Graham's Law

At constant temperature and pressure the rate of diffusion of a gas is inversely proportional to the square root of its density, and so to the square root of its molar mass.
\[ \frac{r_1}{r_2}=\frac{V_1}{V_2}\times\frac{t_2}{t_1}=\sqrt{\frac{d_2}{d_1}}=\sqrt{\frac{M_2}{M_1}} \]
When the pressures differ, the rate is proportional to \( P/\sqrt{d} \), so \( \frac{r_1}{r_2}=\frac{P_1}{P_2}\sqrt{\frac{M_2}{M_1}} \).
The law is for non-reacting gases only. A lighter gas is faster: H₂ diffuses about 1.4 times as fast as He.

Van der Waals Equation

Two assumptions of the kinetic theory fail for real gases: there is some attraction between molecules and the molecules occupy some volume. Van der Waals corrected for both.
Attraction lowers the measured pressure, so the ideal pressure is \( P+\frac{an^2}{V^2} \). Molecules occupy a volume, so the free volume is \( V-nb \), where \( b \) is the excluded volume per mole (four times the actual volume of the molecules).
\[ \left(P+\frac{an^2}{V^2}\right)(V-nb)=nRT \]
a measures the attraction between molecules: unit atm L² mol^-2. A larger a means easier liquefaction.
b measures the molecular volume: unit L mol^-1. Both constants depend on the gas but not on temperature and pressure.

Compressibility Factor

The deviation from ideal behaviour is measured by \( Z=\frac{PV_m}{RT}=\frac{V_{real}}{V_{ideal}} \).
Z = 1 for an ideal gas. Z > 1 (positive deviation) means repulsion dominates and the gas is hard to compress or liquefy. Z < 1 (negative deviation) means attraction dominates and the gas is easier to liquefy.
For one mole: at very low pressure \( Z\approx1 \); at low to moderate pressure \( Z=1-\frac{a}{VRT} \) (Z < 1); at high pressure \( Z=1+\frac{Pb}{RT} \) (Z > 1).
Real gases behave ideally at high temperature and low pressure.

Critical Constants and Boyle Temperature

Below the critical temperature a gas can be liquefied by pressure alone. The critical constants in terms of the van der Waals constants for one mole are:
\[ T_c=\frac{8a}{27Rb},\qquad P_c=\frac{a}{27b^2},\qquad V_c=3b \]
Hence \( Z_c=\frac{P_cV_c}{RT_c}=\frac38 \).
The Boyle temperature is the temperature at which a real gas follows the ideal gas law over a wide range of pressure: \( T_b=\frac{a}{bR} \).

Typical van der Waals Constants

Worked Examples

Example 1. How long will 60 mL of oxygen take to effuse if 60 mL of hydrogen takes 15 s under the same conditions?
Answer:
Rate is volume divided by time, and \( \frac{r_{H_2}}{r_{O_2}}=\sqrt{\frac{32}{2}}=4 \).
So \( t_{O_2}=4\times15=60 \) s.

Example 2. One mole of a gas has a = 3.6 atm L² mol^-2 and b = 0.043 L mol^-1. Find its critical temperature (R = 0.0821 L atm K^-1 mol^-1).
Answer:
\( T_c=\frac{8a}{27Rb}=\frac{8\times3.6}{27\times0.0821\times0.043} \).
Denominator = 0.0953, so \( T_c=\frac{28.8}{0.0953}\approx302 \) K.