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The diffusion coefficient for \(\mathrm{CO}_{2}\) at \(273 \mathrm{K}\) and 1 atm is \(1.00 \times 10^{-5} \mathrm{m}^{2} \mathrm{s}^{-1}\). Estimate the collisional cross section of \(\mathrm{CO}_{2}\) given this diffusion coefficient.

a. The diffusion coefficient of sucrose in water at \(298 \mathrm{K}\) is \(0.522 \times 10^{-9} \mathrm{m}^{2} \mathrm{s}^{-1} .\) Determine the time it will take a sucrose molecule on average to diffuse an rms distance of \(1 \mathrm{mm}\) b. If the molecular diameter of sucrose is taken to be \(0.8 \mathrm{nm}\) what is the time per random walk step?

a. The diffusion coefficient of the protein lysozyme \((\mathrm{MW}=14.1 \mathrm{kg} / \mathrm{mol})\) is \(0.104 \times 10^{-5} \mathrm{cm}^{2} \mathrm{s}^{-1} .\) How long will it take this protein to diffuse an rms distance of \(1 \mu \mathrm{m} ?\) Model the diffusion as a three-dimensional process. b. You are about to perform a microscopy experiment in which you will monitor the fluorescence from a single lysozyme molecule. The spatial resolution of the microscope is \(1 \mu \mathrm{m} .\) You intend to monitor the diffusion using a camera that is capable of one image every 60 s. Is the imaging rate of the camera sufficient to detect the diffusion of a single lysozyme protein over a length of \(1 \mu \mathrm{m} ?\) c. Assume that in the microscopy experiment of part (b) you use a thin layer of water such that diffusion is constrained to two dimensions. How long will it take a protein to diffuse an rms distance of \(1 \mu \mathrm{m}\) under these conditions?

A solution consisting of \(1 \mathrm{g}\) of sucrose in \(10 \mathrm{mL}\) of water is poured into a 1 L graduated cylinder with a radius of \(2.5 \mathrm{cm} .\) Then the cylinder is filled with pure water. a. The diffusion of sucrose can be considered diffusion in one dimension. Derive an expression for the average distance of diffusion \(x_{\text {ave}}\) b. Determine \(x_{\text {ave}}\) and \(x_{r m s}\) for sucrose for time periods of \(1 \mathrm{s}\) \(1 \mathrm{min},\) and \(1 \mathrm{h}\)

An advertisement for a thermopane window company touts Kr-filled windows and states that these windows provide ten times better insulation than conventional windows filled with Ar. Do you agree with this statement? What should the ratio of thermal conductivities be for \(\mathrm{Kr}\left(\sigma=0.52 \mathrm{nm}^{2}\right)\) versus \(\operatorname{Ar}\left(\sigma=0.36 \mathrm{nm}^{2}\right) ?\)

a. Determine the ratio of thermal conductivity for \(\mathrm{N}_{2}\) \(\left(\sigma=0.43 \mathrm{nm}^{2}\right)\) at sea level \((T=300 . \mathrm{K}, \mathrm{P}=1.00 \mathrm{atm})\) versus the lower stratosphere \((\mathrm{T}=230 . \mathrm{K}, \mathrm{P}=0.25 \mathrm{atm})\) b. Determine the ratio of thermal conductivity for \(\mathrm{N}_{2}\) at sea level if \(P=1\) atm, but the temperature is \(100 .\) K. Which energetic degrees of freedom will be operative at the lower temperature, and how will this affect \(C_{V, m} ?\)

a. The viscosity of \(\mathrm{Cl}_{2}\) at \(293 \mathrm{K}\) and 1 atm is \(132 \mu \mathrm{P}\) Determine the collisional cross section of this molecule based on the viscosity. b. Given your answer in part (a), estimate the thermal conductivity of \(\mathrm{Cl}_{2}\) under the same pressure and temperature conditions.

The Reynolds' number (Re) is defined as \(\mathrm{Re}=\rho\left\langle\mathrm{v}_{x}\right\rangle d / \eta,\) where \(\rho\) and \(\eta\) are the fluid density and viscosity, respectively; \(d\) is the diameter of the tube through which the fluid is flowing; and \(\left\langle\mathrm{v}_{x}\right\rangle\) is the average velocity. Laminar flow occurs when \(\operatorname{Re}<2000\), the limit in which the equations for gas viscosity were derived in this chapter. Turbulent flow occurs when \(\mathrm{Re}>2000 .\) For the following species, determine the maximum value of \(\left\langle\mathrm{v}_{x}\right\rangle\) for which laminar flow will occur: a. \(\mathrm{Ne}\) at \(293 \mathrm{K}(\eta=313 \mu \mathrm{P}, \rho=\) that of an ideal gas) through a 2.00 -mm-diameter pipe. b. Liquid water at \(293 \mathrm{K}\left(\eta=0.891 \mathrm{cP}, \rho=0.998 \mathrm{g} \mathrm{mL}^{-1}\right)\) through a 2.00 -mm-diameter pipe.

An Ostwald viscometer is calibrated using water at \(20^{\circ} \mathrm{C}\left(\eta=1.0015 \mathrm{cP}, \rho=0.998 \mathrm{g} \mathrm{mL}^{-1}\right) .\) It takes \(15.0 \mathrm{s}\) for the fluid to fall from the upper to the lower level of the viscometer. A second liquid is then placed in the viscometer, and it takes 37.0 s for the fluid to fall between the levels. Finally, \(100 .\) mL of the second liquid weighs 76.5 g. What is the viscosity of the liquid?

a. Derive the general relationship between the diffusion coefficient and viscosity for a gas. b. Given that the viscosity of Ar is \(223 \mu \mathrm{P}\) at \(293 \mathrm{K}\) and \(1 \mathrm{atm}\) what is the diffusion coefficient? c. What is the thermal conductivity of Ne under these same conditions (the collisional cross section of Ar is 1.5 times that of \(\mathrm{Ne}\) )?