CUET UG Chemistry: Physical Chemistry - States of Matter, Gas Laws, Ideal Gas Equation, van der Waals

Must-Know (15–20 detailed bullets)

• The ideal gas equation is ( PV = nRT ), where ( P ) = pressure (in atm), ( V ) = volume (in L), ( n ) = number of moles, ( R ) = 0.0821 L·atm·K?¹·mol?¹, and ( T ) = temperature (in K).
• At STP (Standard Temperature and Pressure: 273.15 K and 1 atm), one mole of an ideal gas occupies 22.4 L.
• Boyle’s Law: At constant temperature, ( P \propto \frac{1}{V} )? ( P_1V_1 = P_2V_2 ); e.g., compressing a gas syringe increases pressure.
• Charles’s Law: At constant pressure, ( V \propto T )? ( \frac{V_1}{T_1} = \frac{V_2}{T_2} ); e.g., balloon expands on heating.
• Avogadro’s Law: At constant ( T ) and ( P ), ( V \propto n ); 1 mol gas at same ( T ) and ( P ) has same volume as another 1 mol gas.
• Gay-Lussac’s Law: At constant volume, ( P \propto T )? ( \frac{P_1}{T_1} = \frac{P_2}{T_2} ); e.g., pressure in a sealed container increases with temperature.
• The combined gas law is ( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} ), derived from Boyle’s, Charles’s, and Gay-Lussac’s laws.
• Real gases deviate from ideal behavior at high pressure and low temperature due to intermolecular forces and finite molecular volume.
• The van der Waals equation is ( \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT ), where ( a ) and ( b ) are gas-specific constants.
• The constant ( a ) in van der Waals equation accounts for intermolecular attraction; higher ( a ) means greater deviation (e.g., CO? has higher ( a ) than He).
• The constant ( b ) represents excluded volume per mole of gas molecules; for He, ( b \approx 0.0237 \, \text{L/mol} ), for CO?, ( b \approx 0.0427 \, \text{L/mol} ) (verify from NCERT).
• Boyle temperature is the temperature at which a real gas behaves ideally over a wide pressure range; it is given by ( T_B = \frac{a}{Rb} ).
• Critical temperature (( T_c )) is the highest temperature at which a gas can be liquefied by pressure alone; for CO?, ( T_c = 304.15 \, \text{K} ).
• Critical pressure (( P_c )) is the pressure required to liquefy a gas at its critical temperature; for CO?, ( P_c = 73.9 \, \text{atm} ).
• Critical volume (( V_c )) is the volume occupied by one mole of gas at ( T_c ) and ( P_c ); related to van der Waals constants as ( V_c = 3nb ).
• The compressibility factor ( Z = \frac{PV}{nRT} ); for ideal gas, ( Z = 1 ); for real gases, ( Z < 1 ) at low pressure (attractive forces dominate), ( Z > 1 ) at high pressure (repulsive forces dominate).
• Gases like H? and He have ( Z > 1 ) even at low pressures due to weak intermolecular forces and low ( a ) values.
• The SI unit of pressure is pascal (Pa); 1 atm = 101325 Pa = 760 mm Hg = 760 torr.
• The value of ( R ) in SI units is 8.314 J·K?¹·mol?¹.
• Dalton’s Law of Partial Pressures: Total pressure of a gaseous mixture is sum of partial pressures; ( P_{\text{total}} = P_1 + P_2 + \dots ); partial pressure ( P_i = x_i P_{\text{total}} ), where ( x_i ) is mole fraction.

Difficulty Level

Intermediate — requires understanding of both conceptual behavior and mathematical manipulation of gas laws and deviations.

Common CUET Traps

• Trap: Assuming all gases obey ideal gas law at STP.
  Avoid: Only low molecular mass gases like H? and He show near-ideal behavior at STP; others (e.g., CO?, NH?) deviate slightly.

• Trap: Using Celsius instead of Kelvin in gas law calculations.
  Avoid: Always convert temperature to Kelvin (K = °C + 273.15); e.g., 27°C = 300 K.

• Trap: Confusing the significance of van der Waals constant ( a ) and ( b ).
  Avoid: ( a ) corrects for intermolecular attraction (higher for polar gases), ( b ) corrects for molecular volume (larger for bigger molecules).

Practice MCQs

1. Question: What is the volume occupied by 2 moles of an ideal gas at STP?
   A) 11.2 L
   B) 22.4 L
   C) 44.8 L
   D) 5.6 L
   Answer: C
   Explanation: 1 mole occupies 22.4 L at STP, so 2 moles occupy 44.8 L.
   Why others fail: Option B is volume for 1 mole, a common mistake if moles are ignored.

2. Question: Which gas has the highest value of van der Waals constant 'a'?
   A) He
   B) H?
   C) CO?
   D) O?
   Answer: C
   Explanation: CO? has stronger intermolecular forces (polarizable) than others, hence higher 'a'.
   Why others fail: He and H? have very low 'a' due to weak forces; students may guess O? due to diatomic nature.

3. Question: A gas obeys the van der Waals equation. If the constant ( b = 0.03 \, \text{L/mol} ), what is the excluded volume per molecule?
   A) 0.03 L
   B) 0.09 L
   C) 0.01 L
   D) 0.06 L
   Answer: D
   Explanation: Excluded volume per mole is ( b ), but total excluded volume for 1 mole is ( 4 \times ) actual molecular volume; ( b \approx 4V_{\text{molecule}} ), so ( V_{\text{excluded}} = 2b = 0.06 \, \text{L} ) (verify from NCERT).
   Why others fail: Option A assumes ( b ) is per molecule, but it's per mole.

4. Question: At 0°C, the compressibility factor ( Z ) of CO? is less than 1 at moderate pressures. What is the reason?
   A) Repulsive forces dominate
   B) Attractive forces dominate
   C) High molecular speed
   D) Low density
   Answer: B
   Explanation: ( Z < 1 ) indicates gas is more compressible due to intermolecular attraction.
   Why others fail: Option A leads to ( Z > 1 ), which occurs at very high pressure.

5. Question: Two gases A and B have van der Waals constants ( a ) as 3.0 L²·atm·mol?² and 5.0 L²·atm·mol?² respectively. Which gas liquefies more easily?
   A) Gas A
   B) Gas B
   C) Both equally
   D) Cannot be predicted
   Answer: B
   Explanation: Higher 'a' means stronger intermolecular forces, so easier liquefaction.
   Why others fail: Students may think 'b' matters more, but 'a' is key for liquefaction tendency.

Last?Minute Revision

• Ideal gas equation: ( PV = nRT ); R = 0.0821 L·atm·K?¹·mol?¹.
• STP: 273.15 K, 1 atm, 1 mol gas = 22.4 L.
• Boyle’s Law: ( P \propto 1/V ) (T constant).
• Charles’s Law: ( V \propto T ) (P constant).
• Avogadro’s Law: ( V \propto n ) (T, P constant).
• Gay-Lussac: ( P \propto T ) (V constant).
• Combined gas law: ( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} ).
• Real gases deviate at high P, low T.
• van der Waals equation: ( \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT ).
• 'a'-intermolecular attraction; 'b'-molecular volume.
• Higher 'a'-easier liquefaction (e.g., NH? > CH?).
• Compressibility factor ( Z = \frac{PV}{nRT} ); Z = 1-ideal.
• Z < 1-attractive forces dominate.
• Z > 1-repulsive forces dominate (e.g., H? at room T).
• Critical temperature ( T_c ): max temp for liquefaction by pressure.
• For CO?, ( T_c = 304.15 \, \text{K} ), ( P_c = 73.9 \, \text{atm} ).
• Dalton’s Law: ( P_{\text{total}} = \sum P_i ), ( P_i = x_i P_{\text{total}} ).
• R = 8.314 J·K?¹·mol?¹ in SI units.
• Always use Kelvin in gas laws.
• Mnemonic: "Boys Prefer Candy"-Boyle (P-V), Charles (V-T), Combined.