21. Kinetic Theory of Ideal Gases

(II) At about what pressure would the mean free path of air molecules be

(b) equal to the diameter of air molecules, ≈ 3 x 10⁻¹⁰ m? Assume T = 20° C .

(I) What speed would a 1.5-g paper clip have if it had the same kinetic energy as a molecule at 22°C?

(II) (b) What is the total translational kinetic energy of 1.0 mol of N₂ molecules at 25°C?

(II) A 1.0-mol sample of helium gas has a temperature of 18°C.

(a) What is the total kinetic energy of all the gas atoms in the sample?

(I) Approximately how long would it take for the ammonia of Example 18–9 to be detected 1.0 m from the bottle after it is opened? What does this suggest about the relative importance of diffusion and convection for carrying odors?

(II) A 1.0-mol sample of helium gas has a temperature of 18°C.

(b) How fast would a 65-kg person have to run to have the same kinetic energy?

A 0.40-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 15 m/s?

(II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2 x 10⁻⁹ m² . Assume the concentration of oxygen inside is half what it is outside in the atmosphere.

(b) Calculate the diffusion rate J. Assume the diffusion constant is 1 x 10⁻⁵ m²/s.

(II) In the van der Waals equation of state, the constant b represents the amount of “unavailable volume” (per mole) occupied by the molecules themselves. Thus V is replaced by (V - nb) , where n is the number of moles. For oxygen, b is about 3.2 x 10⁻⁵ m³ /mol . Estimate the diameter of an oxygen molecule.

(II) For oxygen gas, the van der Waals equation of state achieves its best fit for α = 0.14 N·m⁴ /mol and b = 3.2 x 10⁻⁵ m³/ mol. Determine the pressure in 1.0 mol of the gas at 0°C if its volume is 0.70 L, calculated using

(a) the van der Waals equation,

A scuba tank has a volume of 3100 cm³ . For very deep dives, the tank is filled with 50% (by volume) pure oxygen and 50% pure helium.

(b) What is the ratio of the average kinetic energies of the two types of molecule?

For oxygen gas the van der Waals constants are α = 0.14 N·m⁴/mol² and b = 3.2 x 10⁻⁵ m³/mol. Using these values, graph six curves of pressure vs. volume between V = 2 x 10⁻⁵ m³ and 2.0 x 10⁻⁴ m³ , for 1 mol of oxygen gas at temperatures of 80 K, 100 K, 120 K, 130 K, 150 K, and 170 K. From the graphs determine approximately the critical temperature for oxygen.