Physics 1st Ch 10 Ideal Gas and Kinetics of Gases

Physics – First Paper Chapter 10: Ideal Gas and Kinetics of Gases Curated Questions and Complete Model Solutions (HSC English Version) Chapter 10: Ideal Gas and Kinetics of Gases B. Creative Questions 1. The average translational kinetic energy of a molecule of helium gas is 1.6×10−20 J. Boltz- mann constant is kB = 1.38 × 10−23 J K−1. Helium is considered to behave as an ideal gas. (a) What is relative humidity? (b) State the ideal gas equation for n moles of a gas. (c) Calculate the temperature of the helium gas in Celsius (◦C). (d) Will the average kinetic energy of the molecules double if the Celsius temperature of the gas is doubled from 50 ◦C to 100 ◦C? Explain mathematically. Answer: (a) Relative Humidity: The ratio of the mass of water vapor present in a given volume of air at a particular temperature to the mass of water vapor required to saturate the same volume of air at that same temperature, expressed as a percentage: R = f F × 100% where f is the actual vapor pressure (at the dew point) and F is the saturated vapor pressure at air temperature. (b) Ideal Gas Equation: PV = nRT where P is pressure, V is volume, n is the number of moles, R = 8.314 J/(mol K) is the univer

(d) Effect of doubling Celsius temperature: Average kinetic energy is directly proportional to absolute temperature in Kelvin, not Celsius: ¯Ek ∝T (in Kelvin) When the temperature changes from θ1 = 50 ◦C to θ2 = 100 ◦C: T1 = 50 + 273.15 = 323.15 K T2 = 100 + 273.15 = 373.15 K The ratio of average kinetic energies is: ¯Ek2 ¯Ek1 = T2 T1 = 373.15 323.15 ≈1.155 The average kinetic energy increases by only 15.5%, rather than doubling (100%). To double the kinetic energy, the absolute temperature T must be doubled, which would require heating to 2 × 323.15 = 646.3 K = 373.15 ◦C. 2. While sitting beside a lake, Risala noticed an air bubble rising from the bottom of the lake to the water surface. Upon reaching the surface, the volume of the bubble became 5 times its volume at the bottom. The atmospheric pressure at that time was P0 = 105 N m−2, the density of water was ρ = 103 kg m−3, and g = 9.8 m/s2. The temperature of the water is assumed to be uniform throughout. (a) What is an ideal gas? (b) What is meant by 70% relative humidity at a given location? (c) Using the information from the stimulus, calculate the depth of the lake. (d) Give a clear explanation of Boyle’s law based on the k

Dividing both sides by V1: P0 + hρg = 5P0 =⇒hρg = 4P0 Solving for h: h = 4P0 ρg = 4 × 105 103 × 9.8 = 400000 9800 ≈40.816 m The depth of the lake is 40.82 m. (d) Kinetic interpretation of Boyle’s law: According to the kinetic theory of gases, the pressure exerted by a gas on the container walls is: P = 1 3 mN V c2 rms = 2 3 N V 1 2mc2 rms  The mean kinetic energy 1 2mc2 rms is directly proportional to absolute temperature T. When temperature is kept constant, the average molecular kinetic energy and root mean square speed crms remain constant. Therefore: P ∝1 V =⇒PV = constant When the volume of a gas is decreased, the number of molecules per unit volume increases, leading to a higher rate of collisions per unit wall area per second, which increases the pressure proportionally. 3. The figure below shows two identical cubic containers P and Q. The length of each side of both containers is a = 10 cm. Container P contains 16 g of oxygen (O2) gas and container Q contains 16 g of hydrogen (H2) gas. The pressure in both containers is measured to be 1.2 × 106 N m−2. 16 g O2 Container P 16 g H2 Container Q (a) What is meant by 1 standard atmosphere (1 atm)? (b) Why is the root mean squar