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481  Making use of the Maxwell distribution function, calculate the number v of gas molecules reaching a unit area of a wall per unit time, if the concentration of molecules is equal to n, the temperature to T, and the mass of each molecule is m. 

482  Using the Maxwell distribution function, determine the pressure exerted by gas on a wall, if the gas temperature is T and the concentration of molecules is n. 

483  Making use of the Maxwell distribution function, find <1/v>, the mean value of the reciprocal of the velocity of molecules in an ideal gas at a temperature T, if the mass of each molecule is equal to m. Compare the value obtained with the reciprocal of the mean velocity. 

484  A gas consists of molecules of mass m and is at a temperature T. Making use of the Maxwell velocity distribution function, find the corresponding distribution of the molecules over the kinetic energies c. Determine the most probable value of the kinetic energy ep. Does ep correspond to the most probable velocity? 

485  What fraction of monatomic molecules of a gas in a thermal equilibrium possesses kinetic energies differing from the mean value by dn = 1.0 % and less? 

486  What fraction of molecules in a gas at a temperature T has the kinetic energy of translational motion exceeding e0 if e0 >> kT? 

487  The velocity distribution of molecules in a beam coming out of a hole in a vessel is described by the function F (v)= A V^3exp(mv^2/2kT), where T is the temperature of the gas in the vessel. Find the most probable values of (a) the velocity of the molecules in the beam; compare the result obtained with the most probable velocity of the molecules in the vessel; (b) the kinetic energy of the molecules in the beam. 

488  An ideal gas consisting of molecules of mass m with concentration n has a temperature T. Using the Maxwell distribution function, find the number of molecules reaching a unit area of a wall at the angles between θ and θ + dθ to its normal per unit time. 

489  From the conditions of the foregoing problem find the number of molecules reaching a unit area of a wall with the velocities in the interval from v to v+dv per unit time. 

490  Find the force exerted on a particle by a uniform field if the concentrations of these particles at two levels separated by the distance dh = 3.0 cm (along the field) differ by n = 2.0 times. The temperature of the system is equal to T = 280 K. 

491  When examining the suspended gamboge droplets under a microscope, their average numbers in the layers separated by the distance h = 40 μm were found to differ by η = 2.0 times. The environmental temperature is equal to T = 290 K. The diameter of the droplets is d = 0.40 μm, and their density exceeds that of the surrounding fluid by Δρ = 0.20 g/cm3. Find Avogadro's number from these data. 

492  Suppose that η0 is the ratio of the molecular concentration of hydrogen to that of nitrogen at the Earth's surface, while η is the corresponding ratio at the height h = 3000 m. Find the ratio η/η0 at the temperature T = 280 K, assuming that the temperature and the free fall acceleration are independent of the height. 

493  A tall vertical vessel contains a gas composed of two kinds of molecules of masses m1 and m2, with m2 > m1. The concentrations of these molecules at the bottom of the vessel are equal to n1 and n2 respectively, with n2 > n1. Assuming the temperature T and the freefall acceleration g to be independent of the height, find the height at which the concentrations of these kinds of molecules are equal. 

494  A very tall vertical cylinder contains carbon dioxide at a certain temperature T. Assuming the gravitational field to be uniform, find how the gas pressure on the bottom of the vessel will change when the gas temperature increases η times. 

495  A very tall vertical cylinder contains a gas at a temperature T. Assuming the gravitational field to be uniform, find the mean value of the potential energy of the gas molecules. Does this value depend on whether the gas consists of one kind of molecules or of several kinds? 

496  A horizontal tube of length L = 100 cm closed from both ends is displaced lengthwise with a constant acceleration w. The tube contains argon at a temperature T = 330 K. At what value of w will the argon concentrations at the tube's ends differ by n = 1.0%? 

497  Find the mass of a mole of colloid particles if during their centrifuging with an angular velocity ω about a vertical axis the concentration of the particles at the distance r2 from the rotation axis is η times greater than that at the distance r1 (in the same horizontal plane). The densities of the particles and the solvent are equal to ρ and to ρ0 respectively. 

498  A horizontal tube with closed ends is rotated with a constant angular velocity ω about a vertical axis passing through one of its ends. The tube contains carbon dioxide at a temperature T = 300 K. The length of the tube is l = 100 cm. Find the value ω at which the ratio of molecular concentrations at the opposite ends of the tube is equal to η = 2.0. 

499  The potential energy of gas molecules in a certain central field depends on the distance r from the field's centre as U(r) = ar^2 , where a is a positive constant. The gas temperature is T, the concentration of molecules at the centre of the field is n0 . Find: (a) the number of molecules located at the distances between r and r + dr from the centre of the field; (b) the most probable distance separating the molecules from the centre of the field; (c) the fraction of molecules located in the spherical layer between r and r + dr; (d) how many times the concentration of molecules in the centre of the field will change if the temperature decreases n times. 

500  From the conditions of the foregoing problem find: (a) the number of molecules whose potential energy lies within the interval from U to U+dU; (b) the most probable value of the potential energy of a molecule; compare this value with the potential energy of a molecule located at its most probable distance from the centre of the field. 
