Gaseous State — Revision Notes

⚡ 30-Second Revision

Ideal Gas Equation: — $PV = nRT$\n- Boyle's Law: $P_1V_1 = P_2V_2$ (Constant T, n)\n- Charles's Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ (Constant P, n)\n- Gay-Lussac's Law: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (Constant V, n)\n- Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ (Constant n)\n- Dalton's Law: $P_{total} = P_1 + P_2 + ...$ and $P_i = X_i \cdot P_{total}$\n- Graham's Law: $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$\n- RMS Speed: $u_{rms} = \sqrt{\frac{3RT}{M}}$ (M in kg/mol, R in $J\cdot mol^{-1}\cdot K^{-1}$)\n- Average Kinetic Energy: $KE_{avg} = \frac{3}{2}kT$ or $\frac{3}{2}RT$ (per mole)\n- Van der Waals Equation: $(P + \frac{an^2}{V^2})(V - nb) = nRT$\n- Compressibility Factor: $Z = \frac{PV}{nRT}$ (Z=1 for ideal gas)\n- Temperature Conversion: $T(K) = T(^{\circ}C) + 273.15$

2-Minute Revision

The Gaseous State is defined by particles with large intermolecular distances and weak forces, leading to no fixed shape or volume, high compressibility, and low density. Ideal gases obey the fundamental gas laws: Boyle's ($PV=const$), Charles's ($V/T=const$), Gay-Lussac's ($P/T=const$), and Avogadro's ($V/n=const$).

These combine into the Ideal Gas Equation, $PV=nRT$, where T must always be in Kelvin. Dalton's Law states that total pressure in a mixture is the sum of partial pressures, and partial pressure is mole fraction times total pressure.

Graham's Law explains diffusion/effusion rates as inversely proportional to the square root of molar mass. The Kinetic Molecular Theory (KMT) explains these behaviors based on constantly moving, non-interacting, point-mass particles with elastic collisions, where average kinetic energy is proportional to absolute temperature.

Real gases deviate from ideal behavior at high pressure and low temperature due to finite molecular volume and intermolecular attractive forces, accounted for by the Van der Waals equation and quantified by the compressibility factor (Z).

Critical temperature is the maximum temperature for liquefaction.

5-Minute Revision

Gases are characterized by widely spaced particles, negligible intermolecular forces (for ideal gases), and constant random motion. This results in properties like indefinite shape and volume, high compressibility, and low density.

The behavior of ideal gases is governed by several laws: Boyle's Law ($P_1V_1 = P_2V_2$) describes the inverse relationship between pressure and volume at constant temperature. Charles's Law ($\frac{V_1}{T_1} = \frac{V_2}{T_2}$) shows volume's direct proportionality to absolute temperature at constant pressure.

Gay-Lussac's Law ($\frac{P_1}{T_1} = \frac{P_2}{T_2}$) relates pressure directly to absolute temperature at constant volume. Avogadro's Law ($V \propto n$) states that volume is proportional to moles at constant T and P.

These laws are unified by the Ideal Gas Equation, $PV = nRT$, where R is the universal gas constant and T must be in Kelvin. For gas mixtures, Dalton's Law of Partial Pressures states $P_{total} = \sum P_i$, and $P_i = X_i \cdot P_{total}$.

Graham's Law of Diffusion/Effusion, $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$, explains why lighter gases diffuse faster. The Kinetic Molecular Theory (KMT) provides the theoretical basis, postulating negligible particle volume, no intermolecular forces, elastic collisions, and average kinetic energy proportional to absolute temperature ($KE_{avg} = \frac{3}{2}kT$).

Real gases deviate from ideal behavior at high pressures and low temperatures because molecular volume becomes significant and intermolecular forces become appreciable. The Van der Waals equation, $(P + \frac{an^2}{V^2})(V - nb) = nRT$, corrects for these factors.

The compressibility factor, $Z = \frac{PV}{nRT}$, quantifies deviation, with $Z=1$ for ideal gases. Understanding critical temperature ($T_c$) and critical pressure ($P_c$) is essential for gas liquefaction.

For NEET, focus on applying these laws to numerical problems, understanding conceptual differences between ideal and real gases, and interpreting graphs.

Prelims Revision Notes

1
Gas Laws & Ideal Gas Equation:\n * Boyle's Law: $P \propto \frac{1}{V}$ (T, n constant) $\implies P_1V_1 = P_2V_2$.\n * Charles's Law: $V \propto T$ (P, n constant) $\implies \frac{V_1}{T_1} = \frac{V_2}{T_2}$.\n * Gay-Lussac's Law: $P \propto T$ (V, n constant) $\implies \frac{P_1}{T_1} = \frac{P_2}{T_2}$.\n * Avogadro's Law: $V \propto n$ (P, T constant) $\implies \frac{V_1}{n_1} = \frac{V_2}{n_2}$.\n * Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$.\n * Ideal Gas Equation: $PV = nRT$. Remember $T$ in Kelvin ($T(K) = T(^{\circ}C) + 273.15$). Common R values: $0.0821\,L\cdot atm\cdot mol^{-1}\cdot K^{-1}$ or $8.314\,J\cdot mol^{-1}\cdot K^{-1}$.\n * Density relation: $PM = dRT$, where $d = \frac{m}{V}$ and $M$ is molar mass.\n2. Dalton's Law of Partial Pressures:\n * For a mixture of non-reacting gases: $P_{total} = P_A + P_B + P_C + ...$\n * Partial pressure of a gas $P_A = X_A \cdot P_{total}$, where $X_A$ is the mole fraction of gas A ($X_A = \frac{n_A}{n_{total}}$).\n * Crucial for gases collected over water: $P_{gas} = P_{total} - P_{water\,vapor}$.\n3. Graham's Law of Diffusion/Effusion:\n * Rate of diffusion/effusion is inversely proportional to the square root of molar mass: $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$.\n4. Kinetic Molecular Theory (KMT):\n * Postulates: Gases are tiny particles in constant random motion; negligible particle volume; no intermolecular forces; elastic collisions; average KE $\propto$ absolute T.\n * Average Kinetic Energy: $KE_{avg} = \frac{3}{2}kT$ (per molecule) or $KE_{avg} = \frac{3}{2}RT$ (per mole). Independent of gas identity.\n * Molecular Speeds: $u_{mp} < u_{avg} < u_{rms}$.\n * $u_{rms} = \sqrt{\frac{3RT}{M}}$ (M in $kg\cdot mol^{-1}$, R in $J\cdot mol^{-1}\cdot K^{-1}$)\n5. Real Gases & Deviations:\n * Deviate from ideal behavior at high pressure and low temperature.\n * Reasons: Finite molecular volume (molecules occupy space) and intermolecular attractive forces (molecules attract each other).\n * Van der Waals Equation: $(P + \frac{an^2}{V^2})(V - nb) = nRT$.\n * $\frac{an^2}{V^2}$ corrects for attractive forces (a is related to attraction strength).\n * $nb$ corrects for molecular volume (b is related to molecular size).\n * Compressibility Factor (Z): $Z = \frac{PV}{nRT}$.\n * For ideal gas, $Z=1$.\n * For real gases, $Z<1$ (attractive forces dominate, moderate P) or $Z>1$ (molecular volume/repulsion dominates, high P).\n6. Liquefaction of Gases:\n * **Critical Temperature ($T_c$):** Max temperature above which a gas cannot be liquefied.\n * **Critical Pressure ($P_c$):** Min pressure required to liquefy a gas at $T_c$.

Vyyuha Quick Recall

Perfect Volume Never Reaches True Pressure. (PV=nRT, P=Partial Pressure, T=Total Pressure for Dalton's Law). Or, for Real Gas deviations: High Pressure, Low Temperature, Deviation Increases (HPLTDI).