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- A molecule in a gas undergoes about 1.0 × 109 collisions in each second. Suppose that (i) every collision is effective in deactivating the molecule rotationally and (ii) that one collision in 10 is effective. Calculate the width (in hertz) of rotational transitions in the molecule.Find the moment of inertia in kg-m^2 of a uniform gas sphere-mass density p of 0.687 kg/m^3 about an axis with a radius of 5m through its center.State, for each of the following physical quantities, if it is a scalar or a vector: Volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- The oxygen molecule has a mass of 5.30 × 10-26 kg and a moment of inertia of 1.94×10-46 kg m2 about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.A molecule in a gas undergoes about 1.0 × 109 collisions in each second. Suppose that (a) every collision is effective in deactivating the molecule rotationally and (b) that one collision in 10 is effective. Calculate the width (in cm-1) of rotational transitions in the molecule.Most of the mass of an atom is in its nucleus. Model the mass distribution in a diatomic molecule as two spheres of uniform density, each of radius 2.00 x 10-15 m and mass 1.00 x 10-26 kg, located at points along the y axis as in 42.5a, and separated by 2.00 x 10-10 m. Rotation about the axis joining the nuclei in the diatomic molecule is ordinarily ignored because the first excited state would have an energy that is too high to access. To see why, calculate the ratio of the energy of the first excited state for rotation about the y axis to the energy of the first excited state for rotation about the x axis.
- Although an ordinary H2 molecule consists of two identical atoms, this is not the case for the molecule HD, with one atom of deuterium (Le., heavy hydrogen, 2H). Because of its small moment of inertia, the HD molecule has a relatively large value of E: 0.0057 eV. At approximately what temperature would you expect the rotational heat capacity of a gas of HD molecules to "freeze out," that is, to fall significantly below the constant value predicted by the equipartition theorem?For a uniform cube, find I about one edge.Consider two immiscible liquids such as water and oil. If a spherical oil molecule of radius r is taken out of the oil phase and placed in the water phase, the unfavorable energy of this transfer is proportional to the area of the solute (oil) molecule newly exposed to the solvent (water) multiplied by the interfacial energy, i, of the oil-water interface. The interfacial energy of the bulk cyclohexane-water interface is i = 50 mJ m-2, and the radius of a cyclohexane molecule is 0.28 nm. Using Boltzmann distribution, estimate the solubility of cyclohexane in water at 25 C in units of mol L-1.The concentration of water in water phase is 55.5 mol L-1.
- A high-carbon steel with a fully pearlitic microstructure was used to form a high-strength bolt (H.-C. Lee et al., J. Mater. Proc. Tech. 211, 1044 (2011)). It was found that the bolt head had an average interlamellar spacing of 257 nm whereas the average spacing in the body of the bolt was 134 nm. Assuming that dislocation pileup is the primary mechanism responsible for the strength of this alloy, what ratio of strength (or hardness) might be expected in the head and body of the bolt?Consider two isolated systems of noninteracting spins with Na=4 and Nb= 16. Their initial energies are Ea= -2uB and Eb= -2ub. (a) What is the total number of microstates available to the composite system? (b) If the two systems are now allowed to exchange energy with one another, what is the probability that system A has energy Ea?A tube of length L is filled completely with an incompressible liquid of mass M and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity w. Find the force exerted by the liquid at the other end?