An ideal gas flows within a small diameter tube. Derive an expression for the transition density of the gas p_c below which microscale effects must be accounted for. Express your result in terms of the gas molecule diameter, universal gas constant, Boltzmann’s constant, and the tube diameter. Evaluate the transition density for a D = 10-m-diameter tube for hydrogen, air, and carbon dioxide. Compare the calculated transition densities with the gas density at atmospheric pressure and T = 23°C.