गैस नियम (Gas law) : CHEMISTRY

These laws describe the relationship between two variables while keeping others constant.

Statement: For a given mass of gas at constant temperature, the volume is inversely proportional to its pressure.
Formula: \( V \propto \frac{1}{P} \) (when T and n are constant)
Equation: \( P_1V_1 = P_2V_2 \)
Graph: Plot of P vs V is a hyperbola. Plot of V vs 1/P is a straight line.

Statement: For a given mass of gas at constant pressure, the volume is directly proportional to its absolute temperature.
Formula: \( V \propto T \) (when P and n are constant)
Equation: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)
Note: Temperature must be in Kelvin (K). \( 0^\circ \text{C} = 273 \text{K} \)

Statement: For a fixed volume, the pressure of a given mass of gas is directly proportional to its absolute temperature.
Formula: \( P \propto T \) (when V and n are constant)
Equation: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \)

Statement: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.
Formula: \( V \propto n \) (when P and T are constant)
Implication: Equal volumes of all gases, under the same conditions of T & P, contain an equal number of molecules.
Molar Volume: At STP (0°C, 1 atm), 1 mole of any gas occupies 22.4 Liters.

Combines all four fundamental laws into one equation.

Formula: \( PV = nRT \)
Where:
- P = Pressure
- V = Volume
- n = Number of moles
- R = Universal Gas Constant
- T = Temperature (in Kelvin)
Derived Forms:
- \( PM = dRT \) (where M = Molar Mass, d = Density)
- \( PV = \frac{m}{M}RT \) (where m = mass of the gas)

Density (d): \( d = \frac{\text{Mass}}{\text{Volume}} = \frac{m}{V} \)
From Ideal Gas Equation: \( PV = \frac{m}{M}RT \)
Rearranging for density: \( P \times M = \frac{m}{V}RT = dRT \)
Key Formula: \( M = \frac{dRT}{P} \)
Important: At constant T and P, density is directly proportional to molar mass (\( d \propto M \)).

Statement: The total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of the individual gases.
Formula: \( P_{\text{total}} = P_1 + P_2 + P_3 + … \)
Partial Pressure: The pressure exerted by an individual gas in a mixture. \( P_1 = x_1 \times P_{\text{total}} \)
Mole Fraction (x₁): \( x_1 = \frac{n_1}{n_{\text{total}}} \)
Application: Used to calculate pressure of dry gas collected over water: \( P_{\text{dry}} = P_{\text{total}} – P_{\text{water vapour}} \)

Diffusion: Spreading and mixing of gases spontaneously.
Effusion: Escape of gas molecules through a tiny hole.
Statement: The rate of diffusion/effusion of a gas is inversely proportional to the square root of its density or molar mass.
Formula:
- \( \frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}} = \sqrt{\frac{M_2}{M_1}} \)
- \( \frac{\text{Time taken for diffusion}}{\text{Rate of diffusion}} \propto \sqrt{M} \)
Example: Lighter gases (like H₂, He) diffuse/effuse faster than heavier gases (like O₂, CO₂).

Universal Gas Constant (R):
- \( R = 0.0821 \text{L atm mol}^{-1} \text{K}^{-1} \) (Most common)
- \( R = 8.314 \text{J mol}^{-1} \text{K}^{-1} \) (SI units)
- \( R = 8.314 \text{Pa m}^3 \text{mol}^{-1} \text{K}^{-1} \)
- \( R = 1.987 \text{cal mol}^{-1} \text{K}^{-1} \)
Standard Temperature and Pressure (STP):
- Temperature = 0°C = 273 K
- Pressure = 1 atm = 760 mm Hg = 760 torr
- Molar Volume at STP = 22.4 L mol⁻¹
Normal Temperature and Pressure (NTP):
- Temperature = 20°C = 293 K
- Pressure = 1 atm
Absolute Zero: 0 K = -273.15°C (The temperature where volume and pressure of an ideal gas become zero).

Boyle’s Law: \( P_1V_1 = P_2V_2 \)
Charles’ Law: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)
Gay-Lussac’s Law: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \)
Avogadro’s Law: \( V \propto n \)
Ideal Gas Equation: \( PV = nRT \)
Density Relation: \( M = \frac{dRT}{P} \), \( d = \frac{PM}{RT} \)
Dalton’s Law: \( P_{\text{total}} = P_1 + P_2 + P_3 + … \), \( P_1 = x_1 \cdot P_{\text{total}} \)
Graham’s Law: \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \)
Mole Fraction: \( x_1 = \frac{n_1}{n_{\text{total}}} \)
Number of Moles (n): \( n = \frac{\text{mass}}{\text{molar mass}} = \frac{m}{M} \)