Kinetic Theory

Consider a rectangular block of wood moving with a velocity v₀ in a gas at temperature T and mass density ρ. Assume the velocity is along x-axis and the area of cross-section of the block perpendicular to v₀ is A. Show that the drag force on the block is 4ρ 0 kT Av m , where m is the mass of the gas molecule

A box of 1.00m³ is filled with nitrogen at 1.50 atm at 300K. The box has a hole of an area 0.010 mm². How much time is required for the pressure to reduce by 0.10 atm, if the pressure outside is 1 atm.

Ten small planes are flying at a speed of 150 km/h in total darkness in an air space that is 20 × 20 × 1.5 km³ in volume. You are in one of the planes, flying at random within this space with no way of knowing where the other planes are. On the average about how long a time will elapse between near collision with your plane. Assume for this rough computation that a safety region around the plane can be approximated by a sphere of radius 10m.

Consider an ideal gas with the following distribution of speeds

(a) Calculate V_rms and hence T(m=3 $\times {10}^{-26}\mathit{k}\mathit{g}$ )

(b) If all the molecules with speed 1000m/s escape from the system , calculate new V_rms and hence T.

Explain why

(a) There is no atmosphere on moon.

(b) There is fall in temperature with altitude. 13.28 Consider an ideal gas with following distribution of speeds

(i) Calculate Vrms and hence T. 26 ( 3.0 10 kg) m − = ×

(ii) If all the molecules with speed 1000 m/s escape from the system, calculate new V_rms and hence T.

13.14 Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:

[Hint : Assume the atoms to be ‘tightly packed’ in a solid or liquid phase, and use the known value of Avogadro’s number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally. Because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few Å].

13.13 A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

n₂ = n₁ exp [ -mg (h₂ – h₁)/ k_BT]

where n₂, n₁ refer to number density at heights h₂ and h1 respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

n₂ = n₁ exp [ -mg N_A ( $\mathit{\rho }$ - ${\mathit{\rho }}^{\mathit{\text{'}}}$ ) (h₂ –h₁₎/ ( $\mathit{\rho }$ RT)]

where $\mathit{\rho }$ is the density of the suspended particle, and  that of surrounding medium. [N_A is Avogadro’s number, and R the universal gas constant.] [Hint : Use Archimedes principle to find the apparent weight of the suspended particle.]

13.12 From a certain apparatus, the diffusion rate of hydrogen has an average value of 28.7 cm³ s^–1. The diffusion of another gas under the same conditions is measured to have an average rate of 7.2 cm³ s^–1. Identify the gas.

[Hint : Use Graham’s law of diffusion: R1/R2 = ( M₂ /M₁ )^1/2, where R₁, R₂ are diffusion rates of gases 1 and 2, and M₁ and M₂ their respective molecular masses. The law is a simple consequence of kinetic theory.]

13.11 A metre long narrow bore held horizontally (and closed at one end) contains a 76 cm long mercury thread, which traps a 15 cm column of air. What happens if the tube is held vertically with the open end at the bottom?

13.10 Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 $\mathit{?}$ . Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of ${\mathit{N}}_2$ = 28.0 u).