Irodov Solutions → Thermodynamics and Molecular Physics → Kinetic Theory of Gases. Boltzmann's Law and Maxwell's Distribution

 2.62. Modern vacuum pumps permit the pressures down to p = 4*10-15 atm to be reached at room temperatures. Assuming that the gas exhausted is nitrogen, find the number of its molecules per 1 cm3 and the mean distance between them at this pressure. Free solution >> 2.63. A vessel of volume V = 5.0 l contains m = 1.4 g of nitrogen at a temperature T = 1800 K. Find the gas pressure, taking into account that η = 30% of molecules are disassociated into atoms at this temperature. Free solution >> 2.64. Under standard conditions the density of the helium and nitrogen mixture equals ρ = 0.60 g/l. Find the concentration of helium atoms in the given mixture. Free solution >> 2.65. A parallel beam of nitrogen molecules moving with velocity v = 400 m/s impinges on a wall at an angle θ = 30° to its normal. The concentration of molecules in the beam n = 0.9*1019 cm-3. Find the pressure exerted by the beam on the wall assuming the molecules to scatter in accordance with the perfectly elastic collision law. Free solution >> 2.66. How many degrees of freedom have the gas molecules, if under standard conditions the gas density is ρ = 1.3 mg/cm3 and the velocity of sound propagation in it is v = 330 m/s. Free solution >> 2.67. Determine the ratio of the sonic velocity v in a gas to the root mean square velocity of molecules of this gas, if the molecules are (a) monatomic; (b) rigid diatomic. Free solution >> 2.68. A gas consisting of N-atomic molecules has the temperature T at which all degrees of freedom (translational, rotational, and vibrational) are excited. Find the mean energy of molecules in such a gas. What fraction of this energy corresponds to that of translational motion? Free solution >> 2.69. Suppose a gas is heated up to a temperature at which all degrees of freedom (translational, rotational, and vibrational) of its molecules are excited. Find the molar heat capacity of such a gas in the isochoric process, as well as the adiabatic exponent γ, if the gas consists of (a) diatomic; (b) linear N-atomic; (c) network N-atomic molecules. Free solution >> 2.70. An ideal gas consisting of N-atomic molecules is expanded isobarically. Assuming that all degrees of freedom (translational, rotational, and vibrational) of the molecules are excited, find what fraction of heat transferred to the gas in this process is spent to perform the work of expansion. How high is this fraction in the case of a monatomic gas? Free solution >> 2.71. Find the molar mass and the number of degrees of freedom of molecules in a gas if its heat capacities are known: cv = 0.65 J/(g*K) and cp = 0.91 J/(g*K). Free solution >> 2.73. Find the adiabatic exponent γ for a mixture consisting of ν1 moles of a monatomic gas and ν2 moles of gas of rigid diatomic molecules. Free solution >> 2.75. Calculate at the temperature t = 17 °C: (a) the root mean square velocity and the mean kinetic energy of an oxygen molecule in the process of translational motion; (b) the root mean square velocity of a water droplet of diameter d = 0.10 μm suspended in the air. Free solution >> 2.76. A gas consisting of rigid diatomic molecules is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of the molecules η = 1.50 times? Free solution >> 2.77. The mass m = 15 g of nitrogen is enclosed in a vessel at a temperature T = 300 K. What amount of heat has to be transferred to the gas to increase the root mean square velocity of its molecules η = 2.0 times? Free solution >> 2.79. A gas consisting of rigid diatomic molecules was initially under standard conditions. Then the gas was compressed adiabatically η = 5.0 times. Find the mean kinetic energy of a rotating molecule in the final state. Free solution >> 2.80. How will the rate of collisions of rigid diatomic molecules against the vessel's wall change, if the gas is expanded adiabatically η times? Free solution >> 2.81. The volume of gas consisting of rigid diatomic molecules was increased η = 2.0 times in a polytropic process with the molar heat capacity C = R. How many times will the rate of collisions of molecules against a vessel's wall be reduced as a result of this process? Free solution >> 2.82. A gas consisting of rigid diatomic molecules was expanded in a polytropic process so that the rate of collisions of the molecules against the vessel's wall did not change. Find the molar heat capacity of the gas in this process. Free solution >> 2.85. Determine the gas temperature at which (a) the root mean square velocity of hydrogen molecules exceeds their most probable velocity by Δv = 400 m/s; (b) the velocity distribution function F(v) for the oxygen molecules will have the maximum value at the velocity v = 420 m/s. Free solution >> 2.86. In the case of gaseous nitrogen find: (a) the temperature at which the velocities of the molecules v1 = 300 m/s and v2 = 600 m/s are associated with equal values of the Maxwell distribution function F(v); (b) the velocity of the molecules v at which the value of the Maxwell distribution function F(v) for the temperature T0 will be the same as that for the temperature η times higher. Free solution >> 2.89. At what temperature of a gas will the number of molecules, whose velocities fall within the given interval from v to v + dv, be the greatest? The mass of each molecule is equal to m. Free solution >> 2.92. From the Maxwell distribution function find , the mean value of the squared vx projection of the molecular velocity in a gas at a temperature T. The mass of each molecule is equal to m. Free solution >> 2.95. 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. Free solution >> 2.100. 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. Free solution >> 2.103. 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. Free solution >> 2.104. 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. Free solution >> 2.105. 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 free-fall acceleration g to be independent of the height, find the height at which the concentrations of these kinds of molecules are equal. Free solution >> 2.106. 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. Free solution >> 2.107. 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? Free solution >> 2.109. 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. Free solution >> 2.110. 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. Free solution >>