JEE Chemistry States Of Matter Notes Set 02

Properties of Gases

A gas has no fixed shape or volume; it fills its container and exerts equal pressure on all walls. Gas molecules are far apart, move randomly and have the weakest intermolecular forces of the three states, so thermal energy dominates over attraction.
Pressure units: 1 atm = 760 mm Hg = 760 torr = 101325 Pa = 1.013 bar. Temperature must be in kelvin: T(K) = t(°C) + 273.

Gas Laws

Boyle's law (constant n, T): \( PV = \text{constant} \), so \( P_1V_1 = P_2V_2 \).
Charles' law (constant n, P): \( V/T = \text{constant} \). Volume changes by 1/273 of its volume at 0 °C for every 1 °C.
Gay-Lussac's law (constant n, V): \( P/T = \text{constant} \).
Avogadro's law (constant P, T): \( V \propto n \). One mole of any gas occupies 22.4 L at STP (273 K, 1 atm).
Combined law: \( \dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2} \).

Ideal Gas Equation

\( PV = nRT \), where R = 0.0821 L atm K\(^{-1}\) mol\(^{-1}\) = 8.314 J K\(^{-1}\) mol\(^{-1}\) = 1.987 cal K\(^{-1}\) mol\(^{-1}\).
Density and molar mass: \( d = \dfrac{PM}{RT} \), so at constant pressure density is inversely proportional to T.
Dalton's law of partial pressures: for non-reacting gases \( P_{total} = p_1 + p_2 + \dots \) and \( p_i = x_i P_{total} \).
Graham's law: at constant T and P, \( \dfrac{r_1}{r_2} = \sqrt{\dfrac{M_2}{M_1}} = \dfrac{d_2^{1/2}}{d_1^{1/2}} \).

Kinetic Theory of Gases

Assumptions: gas molecules are point masses, in continuous random motion, with no attraction or repulsion between them and perfectly elastic collisions.
Kinetic gas equation: \( PV = \tfrac13 mN u^{2} \). Average kinetic energy per mole = \( \tfrac32 RT \) and per molecule = \( \tfrac32 kT \), which depends only on temperature.
Types of molecular speed: \( u_{mp} = \sqrt{2RT/M} \), \( u_{av} = \sqrt{8RT/\pi M} \), \( u_{rms} = \sqrt{3RT/M} \), in the ratio 1 : 1.128 : 1.224, so \( u_{rms}/u_{av} = 1.086 \).
Raising the temperature shifts the Maxwell speed distribution to higher speeds, lowers and broadens the peak, and keeps the area under the curve the same.

Real Gases

Real gases deviate from ideal behaviour at high pressure and low temperature. Compressibility factor \( Z = PV/nRT \) equals 1 for an ideal gas, is below 1 when attractions dominate and above 1 when molecular volume dominates.
van der Waals equation for one mole: \( \left(P + \dfrac{a}{V^{2}}\right)(V - b) = RT \). The constant a measures attraction and b the excluded volume.
Joule-Thomson effect: a gas cools on expansion below its inversion temperature and warms above it.

Worked Examples

Example 1. A gas occupies 500 mL at 27 °C and 1 atm. Find its volume at 127 °C and 2 atm.
Answer:
\( V_2 = 500\times\dfrac{1}{2}\times\dfrac{400}{300} = 333.3 \) mL

Example 2. Find the density of oxygen at 27 °C and 2 atm.
Answer:
\( d = \dfrac{PM}{RT} = \dfrac{2\times32}{0.0821\times300} = 2.60 \) g/L

Example 3. Compare the rates of diffusion of hydrogen and oxygen.
Answer:
\( r_{H_2}/r_{O_2} = \sqrt{32/2} = 4 \), so hydrogen diffuses four times as fast.