Gaseous State (JEE Advanced)

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1. Ideal Gas Equation The ideal gas equation describes the state of a hypothetical ideal gas. Equation: $PV = nRT$ $P$: Pressure (atm, Pa, bar) $V$: Volume (L, $m^3$) $n$: Number of moles $R$: Universal Gas Constant ($0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}$, $8.314 \text{ J mol}^{-1} \text{ K}^{-1}$, $2 \text{ cal mol}^{-1} \text{ K}^{-1}$) $T$: Temperature (Kelvin) Density of Gas: $d = \frac{PM}{RT}$ Molar Mass: $M = \frac{dRT}{P}$ 2. Gas Laws 2.1 Boyle's Law At constant temperature and number of moles, pressure is inversely proportional to volume. Statement: $P \propto \frac{1}{V}$ (at constant $T, n$) Mathematical Form: $P_1V_1 = P_2V_2$ Graphical Representation: Isotherms (P vs V, PV vs P/V) 2.2 Charles's Law At constant pressure and number of moles, volume is directly proportional to absolute temperature. Statement: $V \propto T$ (at constant $P, n$) Mathematical Form: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ Graphical Representation: Isobars (V vs T) 2.3 Gay-Lussac's Law At constant volume and number of moles, pressure is directly proportional to absolute temperature. Statement: $P \propto T$ (at constant $V, n$) Mathematical Form: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ Graphical Representation: Isochores (P vs T) 2.4 Avogadro's Law At constant temperature and pressure, volume is directly proportional to the number of moles. Statement: $V \propto n$ (at constant $T, P$) Mathematical Form: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$ Molar Volume at STP: $22.4 \text{ L}$ ($0^\circ C, 1 \text{ atm}$), $22.7 \text{ L}$ ($0^\circ C, 1 \text{ bar}$) 3. Dalton's Law of Partial Pressures The total pressure exerted by a mixture of non-reacting gases is the sum of the partial pressures of the individual gases. Equation: $P_{\text{total}} = P_1 + P_2 + P_3 + ...$ Partial Pressure of a gas: $P_i = X_i P_{\text{total}}$ $X_i$: Mole fraction of gas $i$ ($X_i = \frac{n_i}{n_{\text{total}}}$) Application: Collection of gases over water (vapor pressure of water) $P_{\text{dry gas}} = P_{\text{total}} - P_{\text{water vapor}}$ 4. Graham's Law of Diffusion/Effusion The rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. Equation: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$ $r$: Rate of diffusion/effusion (volume/time, moles/time, distance/time) $M$: Molar mass of the gas For mixtures: $\frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}}$ (where $d$ is density) 5. Kinetic Molecular Theory of Gases (KMT) Postulates explaining the behavior of ideal gases: Gases consist of large numbers of identical particles (atoms or molecules) that are in continuous, random motion. The volume occupied by the gas particles themselves is negligible compared to the total volume of the container. There are no attractive or repulsive forces between gas particles. Collisions between gas particles and with the container walls are perfectly elastic. The average kinetic energy of gas particles is directly proportional to the absolute temperature. 5.1 Kinetic Gas Equation Equation: $PV = \frac{1}{3} m N u_{\text{rms}}^2$ $m$: mass of one molecule, $N$: total number of molecules $u_{\text{rms}}$: Root Mean Square velocity Derivation: Relating KMT to Ideal Gas Eq: $PV = nRT = \frac{1}{3} m N u_{\text{rms}}^2 \Rightarrow RT = \frac{1}{3} M u_{\text{rms}}^2$ 5.2 Molecular Speeds Based on Maxwell-Boltzmann distribution: Most Probable Speed ($u_{mp}$): Speed possessed by maximum number of molecules. $u_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2kT}{m}}$ Average Speed ($u_{avg}$): Arithmetic mean of the speeds of all molecules. $u_{avg} = \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8kT}{\pi m}}$ Root Mean Square Speed ($u_{rms}$): Square root of the average of the squares of the speeds. $u_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Relationship: $u_{mp} : u_{avg} : u_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1 : 1.128 : 1.224$ $k$: Boltzmann constant ($R/N_A$) 5.3 Average Kinetic Energy Average K.E. per molecule: $\text{K.E.}_{\text{avg}} = \frac{3}{2} kT$ Total K.E. for n moles: $\text{K.E.}_{\text{total}} = n \times \frac{3}{2} RT$ Total K.E. for N molecules: $\text{K.E.}_{\text{total}} = N \times \frac{3}{2} kT$ K.E. depends only on temperature, not on the nature of the gas. 6. Real Gases (Deviations from Ideal Behavior) Ideal gas assumptions break down at high pressure and low temperature. Reasons for deviation: Volume of gas molecules is not negligible. Intermolecular forces of attraction exist between gas molecules. 6.1 Compressibility Factor (Z) A measure of deviation from ideal behavior. Equation: $Z = \frac{PV}{nRT}$ For ideal gas, $Z = 1$ at all $T$ and $P$. For real gases: $Z $Z > 1$: Repulsive forces dominate (high P), volume of molecules becomes significant, gas is less compressible. $Z=1$: At Boyle temperature ($T_b$), real gas behaves ideally over a range of pressures. 6.2 Van der Waals Equation A modified ideal gas equation for real gases, accounting for molecular volume and intermolecular forces. Equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ $\frac{an^2}{V^2}$: Pressure correction term (due to attractive forces) $an^2/V^2$ is added to P because observed pressure is less than ideal pressure. $nb$: Volume correction term (due to finite size of molecules) $nb$ is subtracted from V because effective volume available for motion is less than container volume. $a$: Van der Waals constant for intermolecular attractive forces. Units: $\text{L}^2 \text{ atm mol}^{-2}$ or $\text{Pa m}^6 \text{ mol}^{-2}$ Higher $a$ means stronger attractive forces, easier to liquefy. $b$: Van der Waals constant for excluded volume (effective size of molecules). Units: $\text{L mol}^{-1}$ or $\text{m}^3 \text{ mol}^{-1}$ $b \approx 4 \times (\text{volume of one molecule}) \times N_A$ Higher $b$ means larger molecular size. 7. Critical Phenomena and Liquefaction of Gases Critical Temperature ($T_c$): The temperature above which a gas cannot be liquefied, no matter how high the pressure. $T_c = \frac{8a}{27Rb}$ Critical Pressure ($P_c$): The minimum pressure required to liquefy a gas at its critical temperature. $P_c = \frac{a}{27b^2}$ Critical Volume ($V_c$): The volume occupied by one mole of a gas at its critical temperature and pressure. $V_c = 3b$ Inversion Temperature ($T_i$): The temperature at which a real gas shows neither heating nor cooling effect upon adiabatic expansion (Joule-Thomson effect). $T_i = \frac{2a}{Rb}$ Below $T_i$, gas shows cooling on expansion. Above $T_i$, gas shows heating. 8. Solved Problems Q1: Ideal Gas Density A gas has a density of $1.8 \text{ g/L}$ at $27^\circ C$ and $750 \text{ mmHg}$. What is its molar mass? Solution: $d = 1.8 \text{ g/L}$ $T = 27^\circ C = 300 \text{ K}$ $P = 750 \text{ mmHg} = \frac{750}{760} \text{ atm}$ $R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}$ Using $M = \frac{dRT}{P}$ $M = \frac{1.8 \times 0.0821 \times 300}{750/760} = \frac{1.8 \times 0.0821 \times 300 \times 760}{750} \approx 45.0 \text{ g/mol}$ Q2: Graham's Law A gas diffuses 5 times faster than another gas X. If the molar mass of gas X is $125 \text{ g/mol}$, what is the molar mass of the first gas? Solution: Let the first gas be A, and the second gas be X. $r_A = 5 r_X$ $M_X = 125 \text{ g/mol}$ Using Graham's Law: $\frac{r_A}{r_X} = \sqrt{\frac{M_X}{M_A}}$ $5 = \sqrt{\frac{125}{M_A}}$ Squaring both sides: $25 = \frac{125}{M_A}$ $M_A = \frac{125}{25} = 5 \text{ g/mol}$ Q3: Real Gas Behavior For a real gas, $Z > 1$ at high pressure. What does this indicate? Solution: $Z = \frac{PV}{nRT}$. If $Z > 1$, then $PV > nRT$. This indicates that the actual volume occupied by the gas ($V$) is greater than what an ideal gas would occupy under the same conditions. This happens at high pressures where the finite volume of the gas molecules themselves becomes significant, and repulsive forces dominate over attractive forces. The molecules effectively occupy a larger volume than predicted by the ideal gas law.